Derived Mackey functors

Deriv ed Mac k ey functors D. Kaledin ∗ T o Pierr e Deligne, o n the o c c as ion of his 65th birthday Con ten ts 1 Homological preliminaries. 8 1.1 Generalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Fibrations and base c hange. . . . . . . . . . . . . . . . . . . . 10 1.3 Bar-resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Semiorthogonal decomp ositions. . . . . . . . . . . . . . . . . . 13 1.5 A ∞ -structures. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 Algebras and m o dules. . . . . . . . . . . . . . . . . . . 15 1.5.2 Homotop y . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.3 Coalgebras. . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.4 Categories. . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 2-cate gories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Recollection on Mack ey functors. 27 2.1 Deﬁnitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 F unctorialit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 2.3 Pro du cts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 The derived versio n. 31 3.1 The quotien t construction. . . . . . . . . . . . . . . . . . . . . 32 3.2 Example: the trivial group. . . . . . . . . . . . . . . . . . . . 33 3.3 W reat h pro d u cts. . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Additivization of the quotien t construction. . . . . . . . . . . 39 3.5 The pro ofs of the comparison results. . . . . . . . . . . . . . . 41 3.6 Deriv ed Burn s ide rings. . . . . . . . . . . . . . . . . . . . . . 44 ∗ P artially supported b y gran t NS h-1987.2008.1 1 4 W aldhausen-t yp e description. 45 4.1 Heuristic explanation. . . . . . . . . . . . . . . . . . . . . . . 46 4.2 The S -construction. . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Digression: complemen tary pairs. . . . . . . . . . . . . . . . . 51 4.4 The comparison theorem. . . . . . . . . . . . . . . . . . . . . 55 4.5 Additivit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 F unctorialit y and pro ducts. 57 5.1 F unctorialit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Pro du cts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 Categories of Galois type. 63 6.1 Galois-t yp e catego ries and ﬁ xed p oints. . . . . . . . . . . . . 63 6.2 Filtration b y sup p ort. . . . . . . . . . . . . . . . . . . . . . . 68 6.3 DG mo dels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3.1 The complexes. . . . . . . . . . . . . . . . . . . . . . . 73 6.3.2 Coaction. . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3.3 Com ultiplication and higher op erations. . . . . . . . . 76 6.4 The comparison theorem. . . . . . . . . . . . . . . . . . . . . 77 6.5 Induction and pro du cts. . . . . . . . . . . . . . . . . . . . . . 79 7 T ate homology description. 80 7.1 Generalized T ate cohomology . . . . . . . . . . . . . . . . . . . 81 7.2 Adapted complexes. . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 T ate cohomology and ﬁ xed p oin ts fu nctors. . . . . . . . . . . 85 7.4 In vertible ob jects. . . . . . . . . . . . . . . . . . . . . . . . . 87 7.5 The case of Mac k ey fu nctors. . . . . . . . . . . . . . . . . . . 88 7.6 Cyclic group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8 Relation to stable homotop y . 94 8.1 Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.2 Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.3 Equiv arian t homology . . . . . . . . . . . . . . . . . . . . . . . 98 8.4 A d ictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 In tro duc tion. The notion of a “Mac k ey functor” asso ciated to a ﬁ n ite group G is a stan- dard to ol b oth in algebraic top ology and in grou p theory . It w as originally 2 in tro du ced by Dress [Dr] and later clariﬁed by several p eople, in particular b y Lindn er [Li]; the reader can ﬁ nd mo dern exp ositions in the top ological con text e.g. in [LMS], [M], [tD ], or a more algebraic treatmen t in [T2]. In this pap er, we will b e mostly concerned with applicatio ns to algebraic top ology . Of these, the m ain one is the follo wing: the category of Mac ke y functors is the natural target for equiv arian t homology and cohomology . Namely , assume giv en a CW complex X equ ip p ed with a cont inuous action of a ﬁnite group G . Then if the action is n ice enough, the cellular homology complex C q ( X, Z ) inherits a G -ac tion, so that w e can treat h omol- ogy as a functor f r om G -equiv ariant CW complexes to the deriv ed categ ory D ( G, Z ) of r epresen tations of the group G . Ho w ev er, this loses some essen- tial information. F or example, for an y subgroup H ⊂ G , the homotop y type of the subs p ace X H ⊂ X of H -ﬁxed p oin ts is a G -homotop y in v arian t of X in a suitable sense; bu t once we forget X and remember only the ob j ect C q ( X, Z ) ∈ D ( G, Z ), there is n o wa y to r eco v er th e homology H q ( X H , Z ). Th us , the Mac k ey f unctors: certain algebraic gadgets designed to re- mem b er not only the homology H q ( X, Z ) as a representat ion of G , b ut also all the groups H q ( X H , Z ), H ⊂ G , with whatev er natur al group action they p ossess, and some n atural m aps b et w een them. W e recall the pr ecise deﬁn i- tions in Section 2. F or n o w, it suﬃces to sa y that in the standard approac h, Mac k ey functors f orm a tensor ab elian cate gory M ( G ) suc h that, among other things, (i) for an y G -equiv ariant CW complex X , we hav e n atural homology ob- jects H G q ( X, Z ) ∈ M ( G ), (ii) there is a forgetful exact tensor functor fr om M ( G ) to the category Z [ G ]-mo d of represen tations of G wh ic h reco ve rs H q ( X, Z ) with th e natural G -acti on w h en app lied to H G q ( X, Z ), (iii) for any su bgroup H ⊂ G , the homology H q ( X H , Z ) can also b e reco v- ered from H G q ( X, Z ) ∈ M ( G ), (iv) H G q ( X, Z ) is compatible with stabilizatio n and the tensor pro duct, and extends to the “gen uine G -equiv arian t stable homotop y category” of [LMS], h ere denoted by StHom ( G ). More precisely , for every subgroup H ⊂ G , one has an exact f u nctor from M ( G ) to the category of ab elian groups which asso ciates an ab elian group M H to ev ery M ∈ M ( G ); then in (iii), th er e is a fun ctorial isomorp hism H q ( X H , Z ) ∼ = H G q ( X, Z ) H . 3 One can use the corresp ondence M 7→ M H to visualize the str u cture of the categ ory M ( G ) in the follo wing wa y . F or any s ubgroup H ⊂ G , let M H ( G ) ⊂ M ( G ) b e the full sub category spann ed by su ch M ∈ M ( G ) that • M H ′ = 0 un less H ′ con tains a conjugate of H . Then this is a Serre ab elian sub category , and the sub categories M H ( G ), H ⊂ G , form an increasing “ﬁltration by su pp ort” of the categ ory M ( G ). The top asso ciated graded quotien t of this ﬁltration is equiv alent to the catego ry Z [ G ]-mo d ules — that is, we ha ve M ( G ) / hM H ( G ) i { e }6 = H ⊂ G ∼ = Z [ G ]-mo d , where hM H ( G ) i { e }6 = H ⊂ G ⊂ M ( G ) is the Serre sub category generated by M H ( G ) for all subgroup s H ⊂ G except for the trivial sub group { e } ⊂ G . One can also co mpu te other quotie nts; f or example, the smallest sub category M G ( G ) ⊂ G corresp onding to G itself is equiv alen t to th e category Z -mo d of ab elian groups . More generally , for any n ormal subgroup N ⊂ G there exists a fu lly faithfu l exact inﬂation functor Inﬂ N G whic h giv es an equ iv alence Inﬂ N G : M ( G/ N ) ∼ = M N ( G ) ⊂ M ( G ) . Analogous s tr uctures also exist on the categ ory StHom ( G ). Namely , for an y G -sp ectrum X ∈ StHom ( G ), one h as the so-called L ewis-May ﬁxe d p oints sp e ctrum X H , so that one can deﬁne the sub catego ries StHom H ( G ) b y ( • ). T hen for a normal s ubgroup N ⊂ G , one has a f ully faithful embed- ding StHom ( G/ N ) ∼ = StHom N ( G ) ⊂ StHom ( G ). In p articular, the smallest sub categ ory S tHom G ( G ) ⊂ StHom ( G ) is equiv alen t to the non-equiv arian t stable homotop y category StHom . Another imp ortan t feature of the stable catego ry is the ge ometric ﬁxe d p oints functor Φ H : StHom ( G ) → StHom ; on the leve l of Mack ey functors, this corresp ond s to the pr o j ections ont o the asso ciated graded quotien ts of the ﬁltration by supp ort. T o a p erson trained in homologica l algebra, a natural n ext thing to do is to consider the derive d category D ( M ( G )) of the ab elian catego ry M ( G ), and try to extend all of the ab o v e to the “deriv ed lev el”: one w ould lik e to ha v e a natural equiv arian t homology fu nctor C G q ( − , Z ) : G - StHom → D ( M ( G )), and one wo uld exp ect the category D ( M ( G )) to imitate the natural structur e of the category StHom ( G ). Unfortunately , an d th is came as a nast y sur prise to the author, th is program do es not w ork: the d eriv ed category D ( M ( G )) is not th e righ t thing to consider. The sp eciﬁc pr oblem is that the in ﬂ ation fun ctor Inﬂ N G 4 is not fully faithful on the lev el of deriv ed categories. Already in the case G = Z /p Z , p prime, when the only subgroups in G are the trivial subgroup { e } ⊂ G and G itself, we can consid er th e full triangulated sub category D G ( M ( G )) ⊂ D ( M ( G )) spanned b y such M ∈ D ( M ( G )) th at M { e } = 0. Then while we d o ha v e th e equiv ale nce D ( M ( G )) / D G ( M ( G )) ∼ = D ( G, Z ), the functor D ( Z -mo d) ∼ = D ( M G ( G )) → D G ( M ( G )) is not an equiv alence. The category D G ( M ( G )) w hic h ough t to b e equiv a- len t to the d eriv ed category of ab elian group s is in fact rather complicated and b ehav es badly . So, wh ile one might b e able to constr u ct a homology functor S tHom ( G ) → D ( M ( G )), it do es not s eem to reﬂect the structure of StHom ( G ) to o closely , and in p articular, one cannot exp ect an y reasonable compatibilit y w ith the geometric ﬁxed p oin t f unctors Φ H . But fortunately , a moment’ s r eﬂection on the deﬁn ition of a Mac k ey functor (to wit, the argum en t in the b eginning p aragraphs of Section 3 ) sho ws why this is so, and in fact suggests what the correct “ca tegory of deriv ed Mac ke y fun ctors” should b e. Th is is the su b ject of the pr esen t pap er. F or any ﬁn ite group G , w e construct a tensor triangulated catego ry D M ( G ) of “deriv ed Mac key f u nctors” wh ic h enjoys the follo wing prop erties. (i) F or any su b group H ⊂ G and any M ∈ D M ( G ), there exists a fun c- torial “ﬁx ed p oin ts” ob j ect M H ∈ D ( Z -mo d). (ii) F or an y subgroup H ⊂ G , let D M H ( G ) ⊂ D M ( G ) b e th e full trian- gulated sub category span n ed b y M ∈ D M ( G ) satisfying ( • ). Then D M H ( G ) ⊂ D M ( G ) is admiss ible in the sense of [BK] – that is, the em b edd ing functor D M H ( G ) ֒ → D M ( G ) has a left and a righ t-adjoin t – and we h a v e an equ iv alence (0.1) D M H ( G ) / hDM H ′ ( G ) i H ⊂ H ′ ⊂ G,H 6 = H ′ ∼ = D ( W H , Z ) , where W H = N H /H is the qu otient of the normalizer N H ⊂ G of H ⊂ G by H itself. (iii) F or an y normal sub group N ⊂ G , we hav e an equiv alence c Inﬂ N G : DM ( G/ N ) ∼ = D M N ( G ) ⊂ D M ( G ) . Informally sp eaking, th e sub catego ries D M H ( G ) form a ﬁltration of the catego ry DM ( G ) indexed by the lattice of conjugacy classes of sub groups 5 in G ; this ﬁltration giv es rise to a “semiorthogonal d ecomp osition” in the sense of [BK], and the graded p ieces of the ﬁltration are naturally iden tiﬁed with D ( W H , Z ), H ⊂ G . W e explicitly construct f unctors e Φ [ G/H ] : DM ( G ) → D ( W H , Z ) that give the pro jections ont o these graded pieces. W e also compute the gluing functors b et w een D ( W H , Z ) ⊂ D M ( G ); these are n aturally expressed in terms of a certain generalizatio n of T ate cohomology of ﬁn ite group s. The reader will notice that the pr op erties of the category D M ( G ) are sligh tly stronger than what we ha v e men tioned for the ab elian category M ( G ), in that we identify the asso ciated graded pieces of the ﬁltration by supp ort, and d escrib e h ow these pieces are glued. If one lo calizes the cate- gory M ( G ) b y inv erting the order of the group G , then the corresp ond in g statemen t for the lo calized category f M ( G ) is a theorem of Thevenaz [T1 ] (in this case, th ere is no gluing: the categ ory M ( G ) is semisimple, and it splits into a d irect s u m of the catego ries of repr esen tations of th e groups W H , H ⊂ G ). I do n ot kno w wh ether an ything is kno wn for M ( G ) in the general non-semisimple case. I also do not know whether analogous state- men ts are kno ws for the category StHom ( G ) — namely , w h ether the graded pieces of the ﬁltration by the su b categories StHom H ( G ) ⊂ StHom ( G ) ha v e b een computed and/or whether the gluin g fun ctors h a v e b een iden tiﬁed. Moreo v er, in this pap er w e construct a n atur al equiv ariant homology functor h G from G -sp ectra to our derived Mac ke y functors such that (i) the fun ctor h G is tens or, (ii) for an y sub grou p H ⊂ G , th e fu nctor h G sends th e sub category StHom H ( G ) ⊂ StHom ( G ) in to D M H ( G ) ⊂ D M ( G ), and (iii) for an y sub grou p H ⊂ G and a G -sp ectrum X , the und erlying com- plex of e Φ [ G/H ] ( h G ( X )) ∈ D ( W H , Z ) computes the homology of the geometric ﬁxed p oint sp ectrum Φ H ( X ). Unfortunately , w e can only do all of the ab ov e for ﬁnite G -CW sp ectra X . I do not kn o w wh ether the corresp ondin g s tatemen ts are tru e for the whole catego ry StHom ( G ). I also exp ect that for any G -sp ectrum X , h G ( X ) H is the homology of the Lewis-Ma y ﬁxed p oin ts sp ectrum X H , b ut I do not present ly kn o w ho w to p ro v e it. The pap er is organized as follo ws. In Section 1, we give some necessary standard facts from homological algebra and catego ry theory (this includ es 6 the notions r elated to semiorthogonal decompositions of tr iangulated cat- egories). In Section 2, w e recall the usual d eﬁnition of Mac k ey fun ctors and s ome of their basic pr op erties. Th en we giv e our derived version. W e actually giv e not one bu t t w o deﬁnitions. First, w e give a rather explicit deﬁnition using A ∞ -categ ories and bar resolutions – this is Section 3 (in fact, we work in a s ligh tly larger generalit y of a sm all category C whic h has ﬁb ered pr o ducts – for Mac k ey fu nctors, this is th e category of ﬁn ite G -sets). Then in Section 4, w e giv e a more inv arian t deﬁnition somewhat in th e spir it of W aldhausen’s S -construction, and we sh o w that the tw o deﬁnitions are equiv alent. I n Section 5, w e sh o w ho w to extend the basic prop erties of Mac k ey functors given in Section 2 to the d eriv ed setting (in particular, we construct th e tensor pro d uct, the inﬂation f u nctors c Inﬂ N G and the geometric ﬁxed p oints fun ctors Φ [ G/H ] ). F ormally , this is wh ere the pap er migh t ha v e ended ; ho wev er, neither of our t w o equiv alen t deﬁnitions is s uitable for computations. F or this reason, w e giv e a third rather explicit description of the same category , and this is the sub ject of the length y Section 6. Essen tially , we d o a sort of Koszul d u alit y — we try to d escrib e th e same category D M ( G ) u sing th e geometric ﬁxed p oints functors Φ [ G/H ] as a “ﬁb er functor”. T his is sur prisingly delicate; in p articular, w e ha ve to work with A ∞ -comod ules ov er an A ∞ -coalg ebra instead of the more usu al A ∞ -mo dules o v er an A ∞ -algebra. One imm ed iate adv an tage is that the fu nctors Φ [ G/H ] are tensor, so that this new description is b etter compatible with the te nsor pro d uct in D M ( G ). Ho w ev er, the r eal rewa rd comes in Section 7: w e are able to prov e the equiv alences (0.1), thus describing the category of d eriv ed Mac k ey fu nctors as a s uccessiv e extension of represen tation cate gories of the sub quotien t groups W H of the group G , and w e express the gluing d ata b et w een these represent ation catego ries in terms of a generalization of T ate (co)homology . This “ge neralized T ate cohomology” has th e great adv antag e of b eing trivial in many cases; ev en when it is n ot trivial, it is u s ually p ossib le to compute it. In the ﬁn al Section 8, we describ e the relation b etw een the category of deriv ed Mac key functors and the G -equiv ariant stable homotop y category . Not b eing an exp ert on stable homotop y , I ha v e k ept the exp osition to an absolute minim um; ho we ver, I hop e that Section 8 do es sho w that th e cat- egory of derived Mac k ey f unctors imitates th e equiv ariant stable homotop y catego ry in a satisfactory w a y . It should b e stressed that a top ologist would learn almost nothing from this pap er: pr ett y m uc h ev erything that w e pro v e about derive d Mac k ey 7 functors is well-kno wn for equiv ariant sp ectra (p ossibly in a d iﬀeren t lan- guage). In a sense, the whole p oint of the pap er is that s o muc h sur viv es in the purely homological theory , w hic h is usually prett y trivial compared to the r ic hness of the sp here sp ectrum. On the other hand, th in gs standard in algebraic top ology are not alwa ys well-kno wn outside of it; a reader with a more geomet ric and/or homological bac kground can treat the pap er as an extended exercise in the theory of cohomologica l descen t. As suc h, it migh t ev en b e u seful, e.g. in th e th eory of Artin motiv es. In the in terest of full disclosure, I should men tion that my p ersonal m ain reason for doing this researc h was its application to the so-calle d “cyclot omic sp ectra”, an d a comparison th eorem b etw een T op ological Cyclic Homology , on one hand , and a syn tomic version of th e P erio dic C yclic Homology , on the other h and. Needless to sa y , in the end all of this h ad to b e take n out and relegated to a separate pap er, wh ic h is “in preparation”. Ac kno wledgemen ts. It is a p leasure and an honor to d edicate this pap er to Pierre Deligne, on the o ccasion of h is 65-th birthd a y . Th e pap er would b e imp ossible without his elegan t theory of cohomological descen t. I am v ery grateful to A. Be ilinson, R. Bez ru k avnik o v, M. Kon tsevic h, A. Kuznetso v, G. Merzon, D. O rlo v, L. Po sitselsky and V. V ologo dsky for us efu l d iscussions. It wa s A. Beilinson who originally asked whether one can do an y gluing b y means of T ate h omology , and attracted my atten tion to the p roblem of descen t for Artin motiv es. And last but f oremost, this pap er o w es its existence to con v ersations with L. Hesselholt, who in tro d uced me to Mac ke y functors and stable equiv arian t theory , and generally encouraged m e to think ab out the sub ject. A part of this work w as don e du ring m y s tay at th e Univ ersit y of T okyo; the hospitalit y of this wo nder f ul p lace and my host Prof. Y u. K a w amata is gratefully ac kno wledged. I am extremely grateful to the referee for many usefu l su ggestions, friend ly criticism, and a lot of explanations ab out the top ological p art of the story . 1 Homological preliminaries. 1.1 Generalities. Th roughout the p ap er w e w ill ﬁx once and for all an ab elian category Ab; this is th e category w here our Mac key fu nctors will tak e v alues. W e will assu me th at Ab is suﬃciently nice, in that it is a Grothendiec k ab elian category with enough pro jectiv es and injectiv es. W e will also assume that Ab is equipp ed with a symmetric tensor pro du ct. In addition, we will assu m e ﬁxed some fun ctorial DG enh an cement for Ab, so 8 that for an y M , M ′ ∈ Ab we h a v e a complex RHom q ( M , M ′ ) of ab elian group s whose h omology compu tes Ext q ( M , M ′ ), and moreov er, RHom q ( − , − ) is functorial in b oth arguments and equipp ed with fun ctorial and asso ciativ e comp osition maps RHom q ( M , M ′ ) ⊗ R Hom q ( M ′ , M ′′ ) → RHom q ( M , M ′′ ) for any M , M ′ , M ′′ ∈ Ab. F or example, Ab ma y b e th e category of vec tor spaces o ve r a ﬁeld k , or the category of m o dules o ve r a comm utativ e ring R , or ind eed simply the category of ab elian groups. W e will denote b y D (Ab) the (unboun ded) deriv ed category of Ab. W e will tacitly assume th at th e DG enhancemen t is extended to unb ounded complexes, so that f or an y t w o complexes M q , M ′ q w e hav e a complex of ab elian groups RHom q ( M q , M ′ q ) whose 0-th homolog y classes are in one-to-one corresp ondence with maps from M q to M ′ q in D (Ab) (this is slightl y d elicate – to do it prop erly , one has to use either h -pro j ectiv e or h -injectiv e replacemen ts in the sense of [Ke1], [Sp]). Throughout the pap er, we will w ork with many small cat egories. Usually w e will denote ob ject s in a s mall category C by small r oman letters; for any c, c ′ ∈ C , w e will denote b y C ( c, c ′ ) the set of morphisms from c to c ′ . F or any small category C , we will denote b y F un( C , Ab) th e category of all functors from C to Ab. This is also a Grothendiec k ab elian catego ry with enough pro jectiv es and injectiv es. W e will denote its deriv ed category b y D ( C , Ab). A fu nctor γ : C → C ′ induces a restriction fu nctor γ ∗ : F un( C ′ , Ab) → F un( C , Ab), and this h as a left-adjoin t γ ! : F un( C , Ab ) → F un( C ′ , Ab) and a right- adjoint γ ∗ : F un ( C , Ab) → F un ( C ′ , Ab), kno wn as the left and right Kan extensions . By adj unction, γ ∗ is exact, γ ! is r igh t-exact, and γ ∗ is left- exact, so that we ha ve derived fu nctors L q γ ! , R q γ ∗ : D ( C , Ab ) → D ( C ′ , Ab). In particular, if C ′ = pt is the p oin t category , w e ha v e F un( C ′ , Ab) = F un( pt , Ab) ∼ = Ab, an d if τ : C → pt is the pro jection to the p oin t, th e Kan extensions τ ! and τ ∗ are giv en by the direct and in verse limit o ve r the catego ry C . Their deriv ed functors are known as homolo gy H q ( C , − ) and c ohomolo gy H q ( C , − ) of the category C . On the other hand, if we c ho ose an ob ject c ∈ C and let i c : pt → C b e the functor whic h sends pt to c ∈ C , then the Kan extension functors 9 i c ! , i c ∗ : Ab ∼ = F un( pt , Ab) → F un ( C , Ab) are exact. T o sim p lify notation, w e will denote M c = i c ! ( M ), M c = i c ∗ ( M ) for an y ob ject M ∈ Ab. Explicitly , the functors M c , M c ∈ F un( C , Ab) are giv en b y (1.1) M c ( c ′ ) = M C ( c,c ′ ) M , M c ( c ′ ) = Y C ( c ′ ,c ) M , for any c ′ ∈ C (the sum and the pro d uct of copies of M in dexed by elemen ts in the corresp onding Hom-sets). If M is pro jectiv e, then M c is pr o jectiv e in F un( C , Ab); if M is inj ective , then M c is injectiv e in F u n ( C , Ab ). F or an y M ∈ C , a map f : c → c ′ induces natural maps M c ′ → M c , M c ′ → M c . 1.2 Fibrations a nd base change. In general, it is r ather cum b ers ome to compute the Kan extensions explicitly; how ev er, there is on e situati on in tro du ced in [SGA] wh ere the computations are simp liﬁed. Namely , assume giv en a f unctor γ : C ′ → C b et w een small categories C , C ′ . A morphism f ′ : c ′ 0 → c ′ 1 in C ′ is called Cartesian with resp ect to γ if it h as the follo wing unive rsal p rop erty: • any morphism f ′′ : c ′′ 0 → c ′ 1 suc h that γ ( f ′′ ) = γ ( f ′ ) factorizes uniqu ely as f ′′ = f ′ ◦ i through a morp hism i : c ′′ 0 → c ′ 0 suc h that γ ( i ) = id . The functor γ is called a ﬁbr ation if (i) for an y morphism f : c 0 → c 1 in the category C and an y ob ject c ′ 1 ∈ C ′ with γ ( c ′ 1 ) = c 1 there exists a Cartesian map f ′ : c ′ 0 → c ′ 1 suc h that γ ( f ′ ) = f , (ii) and moreo v er, the comp osition of t w o Cartesian maps is Cartesian. Example 1.1. L et C b e a category with ﬁb ered pro ducts, let C ′ b e th e catego ry of diagrams c 0 → c 1 in C , and let γ : C ′ → C b e the f u nctor which sends a diagram c 0 → c 1 to c 1 . Then γ is a ﬁbration. F or any ob ject c ∈ C , denote b y C ′ c the ﬁb er of the functor γ o v er the ob ject c ∈ C – that is, the category of ob jects c ′ ∈ C ′ suc h th at γ ( c ′ ) = c and those morphism s i b et we en them for which γ ( i ) = id . Note that by the univ ersal prop ert y of Cartesian maps, th e map f : c ′ 0 → c ′ 1 in (i) is unique u p to a unique isomorp h ism, so that, if γ is a ﬁ bration, setting f ∗ ( c ′ 1 ) = c ′ 0 deﬁnes a functor f ∗ : C ′ c 1 → C ′ c 0 whic h we will call the tr ansi- tion functor corresp ond ing to f . The s ame un iversal prop er ty provi des a canonical map ( f ◦ g ) ∗ ∼ = f ∗ ◦ g ∗ for any tw o comp osable morphisms f , g 10 in C , and the condition (ii) insures that this is an isomorphism . Th ese iso- morphisms in tur n satisfy a compatibilit y condition for comp osable triples, and th e whole thin g has b een axiomatized by Grothend iec k und er the name of a “pseud o-functor” from C to the 2-category C at of small categories; w e refer the reader to [SGA] for d etails. Grothendiec k also pro ved the inv erse statemen t: eve ry ﬁbration ov er C is uniquely deﬁ n ed b y the corresp ond in g con tra v arian t pseudo-fun ctor f r om C to Cat. No wada ys this is usu ally called “the Grothendiec k construction”. Assume given a ﬁb ration γ : C ′ → C , another small category C 1 , and a functor η : C 1 → C . Deﬁne a sm all catego ry C ′ 1 as a ﬁb ered pro d uct C ′ 1 η ′ − − − − → C ′ γ ′   y   y γ C 1 η − − − − → C . Then we ha ve a pair of adjoint b ase change isomorphisms η ∗ ◦ R q γ ∗ ∼ = R q γ ′ ∗ ◦ η ′ ∗ , L q η ′ ! ◦ γ ′ ∗ ∼ = γ ∗ ◦ L q η ! . F or the p ro of, see e.g. [Ka, Lemma 1.7]. Dually , γ : C ′ → C is a c oﬁbr ation if the corresp onding functor γ opp : C opp ′ → C opp b et wee n the opp osite categories is a ﬁbr ation. Un der the Grothendiec k construction, coﬁbrations corresp ond to co v arian t p seudofun c- tors. If γ is a coﬁbration, we ha ve a base change isomorph ism η ∗ ◦ L q γ ! ∼ = L q γ ′ ! ◦ η ′ ∗ . In particular, for any functor E ∈ F un ( C ′ , Ab), the v alue L q ( E )( c ) at an ob ject c ∈ C is canonically giv en b y (1.2) L q ( E )( c ) ∼ = H q ( C ′ c , E ) . 1.3 Bar-resolutions. Another computational to ol that we will need is the so-called b ar-r esolution . Assume giv en a s m all category C . Then ev ery functor E ∈ F un( C , Ab) h as a canonical resolution P q ( C , E ) with terms P i − 1 ( C , E ) = M c 1 →···→ c i E ( c 1 ) c i , where the sum is tak en ov er all the d iagrams c 1 → · · · → c i in the category C , and the usu al diﬀerentia l δ = d 1 − d 2 + · · · ± d i , where d l drops the ob ject c l 11 from the diagram, and acts as the identit y m ap if 1 < l < i , as th e natur al map E ( c 1 ) c i → E ( c 1 ) c i − 1 induced by c i − 1 → c i if l = i , and as the m ap E ( c 1 ) c i → E ( c 2 ) c i induced by the map E ( c 1 ) → E ( c 2 ) if l = 1. T o see that this is in deed a r esolution, one ev aluates P q ( C , E ) at some ob ject c ∈ C . By deﬁnition, the resulting complex is giv en by P i − 1 ( C , E )( c ) = M c 1 →···→ c i → c E ( c 1 ) , where the sum is no w o v er all th e diagrams en d ing at c ∈ C . If one adds the term E ( c ) in degree − 1 corresp ondin g to the diagram consisting of c itself, then the resulting complex is ob viously c hain-homotopic to 0 — the con tracting homotop y h sends the term corresp onding to a diagram c 1 → · · · → c i → c to the term corresp onding to c 1 → · · · → c i → c → c , where the last map c → c is th e ident it y m ap. Since all the ob jects M c , c ∈ C , M ∈ Ab are obvio usly acyclic f or the homology fu nctor H q ( C , − ), the bar r esolution can b e used to compu te the homology H q ( C , E ). T h is results in the b ar-c omplex C q ( C , E ) with terms (1.3) C i − 1 ( C , E ) = M c 1 →···→ c i E ( c 1 ) . This has the follo wing standard prop erties. (i) The bar-complex C q ( C , E ) is functorial with resp ect to E . (ii) F or an y fu nctor γ : C → C ′ and any E : C ′ → Ab, there is a natural map γ E ∗ : C q ( C , γ ∗ E ) → C q ( C ′ , E ) whic h ind uces the natural adjun ction map H q ( C , γ ∗ E ) → H q ( C ′ , E ) on h omology , and for any comp osable pair of functors γ : C → C ′ , γ ′ : C ′ → C ′′ , we ha v e γ γ ′ ∗ E ∗ ◦ γ ′ E ∗ = ( γ ′ ◦ γ ) E ∗ . (iii) F or an y t w o cate gories C , C ′ and fu n ctors E : C → Ab, E ′ : C ′ → Ab, w e ha v e a K ¨ unneth-type quasiisomorphism (1.4) C q ( C , E ) ⊗ C q ( C ′ , E ′ ) → C q ( C × C ′ , E ⊠ E ′ ) , and this is asso ciativ e with r esp ect to triple pro d ucts. Of th ese, only (iii) is s ligh tly non-obvio us; the requ ired qu asiisomorphism is giv en by the s huﬄe pro duct. In addition, the bar-resolution can b e used to giv e a canonical DG en- hancemen t to the category F un( C , Ab). I ndeed, for any E , E ′ ∈ F un ( C , Ab ) 12 w e can compute Ext q ( E , E ′ ) by the bar-resolution; this results in a bicom- plex RHom q , q ( E , E ′ ) = RHom q , q C ( E , E ′ ) with terms (1.5) RHom i − 1 , q ( E , E ′ ) = Y c 1 →···→ c i RHom q ( E ( c 1 ) , E ′ ( c i )) . W e denote b y RHom q ( E , E ′ ) the total complex of this bicomplex. Giv es three ob jects E , E ′ , E ′′ ∈ F un( C , Ab), we hav e a natur al comp osition map RHom q ( E , E ′ ) ⊗ RHom q ( E ′ , E ′′ ) → RHom q ( E , E ′′ ), and this is asso ciativ e in the ob vious s en se. W e w ill also need a sligh tly more reﬁn ed version of th e bar resolution. F or any i , the diagrams c 1 → · · · → c i and isomorp hisms b et we en them form a group oid C i . Denote by σ i , τ i : C i → C the fu nctors whic h s end a d iagram c 1 → · · · → c i to c 1 ∈ C resp . c i ∈ C . F or an y E ∈ F un( C , Ab), consider the complex e P i − 1 , q ( C , E ) = τ i ! P q ( C i , σ ∗ i E ) . F orgetting one vertex in a d iagram giv es a fun ctor C i → C i − 1 , and this construction is strictly asso ciativ e. Th erefore by the pr op erties (i), (ii) of the bar complex, we can turn the collection e P q , q ( C , E ) in to a bicomplex, with the second diﬀerentia l giv en by the same formula δ = d 1 − d 2 + · · · ± d i as in the case of P q ( C , E ). T h e total complex e P q ( C , E ) is th en also a resolution on the fun ctor E . T o see this, one again ev aluates at an ob ject c ∈ C , and uses the same con tracting h omotop y h as in the case of P q ( C , E ). In the case of tw o f u nctors E , E ′ ∈ F un( C , Ab), we can use the resolution e P q ( C , E ) to compute RHom q ( E , E ′ ); this r esults in the triple complex with terms (1.6) RHom i, q , q ( E , E ′ ) = C q ( C i , RHom q ( σ ∗ i E , τ ∗ i E ′ )) , a r eﬁnement of the double complex (1.5). Its tota l complex is functorially quasiisomorphic to RHom q ( E , E ′ ). 1.4 Semiorthogonal decomp ositions. W e will also need some tec hnol- ogy for working with tr iangulated categories; the stand ard reference here is [BK]. In ligh t of r ecen t adv an ces in axiomatic homotop y theory , it is p erhaps b etter to state explicitly that in this pap er, our n otion of a triangulate d c ate gory is the original notion of V erd ier. A full triangulated sub catego ry D ⊂ D ′ in a triangulated cate gory D ′ is called lo c alizing if the quotien t D ′ / D exists (in spite of the set-theoretic diﬃculties of the V erd ier construction). 13 A full triangulated s u b category D ⊂ D ′ is called left resp. right admissible if the em b edding fun ctor D ֒ → D ′ admits a left r esp. right adjoint ; it is admissible if it is adm iss ible b oth on the left and on the r igh t. Th e left orthogonal ⊥ D ⊂ D ′ consists of ob jects M ∈ D ′ suc h th at Hom( M , N ) = 0 for any N ∈ D . This is also a fu ll triangulated su b category in D ′ , and it kno wn that D ⊂ D ′ is left-admissible if and only if D and ⊥ D generate the whole D ′ . In this case, one sa ys that h ⊥ D , D i is a semi-ortho gonal de c omp osition of the triangulated catego ry D ′ . One s ho ws that D ′ is then generated b y D and ⊥ D in the follo wing strong sense: for an y M ′ ∈ D ′ , there exists a un ique and fu nctorial distinguished triangle (1.7) ⊥ M − − − − → M ′ − − − − → M − − − − → with M ∈ D and ⊥ M ∈ ⊥ D . Moreo ve r, th e category D ⊂ D ′ is lo calizing, and w e hav e a n atural ident iﬁcation ⊥ D ∼ = D ′ / D . Analogously , the righ t orth ogonal D ⊥ ⊂ D ′ consists of ob jects M ∈ D ′ suc h that Hom( N , M ) = 0 f or an y N ∈ D , and D ⊂ D ′ is righ t-admissible if and on ly D ′ is generated b y D and D ⊥ ; in this case, hD , D ⊥ i is a semiorthog- onal decomp osition of the category D ′ . W e h a v e the follo wing standard fact. Lemma 1.2. A ssume give n a left-admissible triangulate d sub c ate gory D ⊂ D ′ . Then the natur al pr oje ction D ⊥ → D ′ / D ∼ = ⊥ D is ful ly faithful, and it is an e quivalenc e if and only if D ⊂ D ′ is right-admissible.  Giv en an ad m issible sub category D ⊂ D ′ , one deﬁn es the gluing f u nctor R : D ′ / D → D as the comp osition D ′ / D ∼ − − − − → D ⊥ ⊂ D ′ − − − − → D , where the second fun ctor D ′ → D is left-adjoin t to the emb edding D ⊂ D ′ . This is a triangulated fu nctor deﬁned up to a canonical isomorphism. Ob jects M ′ ∈ D ′ are in natural one-to -one corresp ondence with triples h M ⊥ , M , r i of an ob ject M ⊥ ∈ D ⊥ , an ob ject M ∈ D , and a gluin g map r : R ( M ⊥ ) → M [1]. W e note that it is not p ossible to r eco v er the category D ′ from D , D ′ / D and the gluing fun ctor R : D ′ / D → D (w e can reco v er ob jects, bu t n ot morphisms). Ho w ev er, there is the follo wing useful fact. Lemma 1.3. A ssume g i ven triangulate d c ate gories D ′ 1 , D ′ 2 e q u ipp e d with left-admissible sub c ate gories D 1 ⊂ D ′ 1 , D 2 ⊂ D ′ 2 , and a triangulate d functor F : D ′ 1 → D ′ 2 . Mor e over, assume that 14 (i) F sends D 1 into D 2 , ⊥ D 1 into ⊥ D 2 , and the i nduc e d functors F : D 1 → D 2 , F : D ′ 1 / D 1 → D ′ 2 / D 2 ar e e qu ivalenc es, and (ii) D 1 ⊂ D ′ 1 is also right-admissible, and F sends D ⊥ 1 into D ⊥ 2 ⊂ D ′ 2 . Then F i s also an e quivalenc e. Pr o of. Since D ′ 2 is generated by D 2 = F ( D 1 ) and ⊥ D 2 ∼ = F ( ⊥ D 1 ), the functor F is essen tially surjectiv e, and it suﬃces to pr o v e that it is f ully faithful — that is, for an y M , N ∈ D ′ 1 , the map F : Hom( M , N ) → Hom( F ( M ) , F ( N )) is an isomorph ism. By (1.7), we ma y assume that M lies either in ⊥ D 1 or in D 1 . If M ∈ ⊥ D 1 and N ∈ D 1 , then b oth sides are 0. If M , N ∈ ⊥ D 1 , then the m ap is bijectiv e by (i) . Th us we m a y assume M ∈ D 1 . Decomposing N b y (1.7) with resp ect to the semiorthogonal d ecomp osition hD 1 , D ⊥ 1 i , we s ee that we may assu me that either N ∈ D 1 or N ∈ D ⊥ 1 . Then in the ﬁ rst case, the claim follo ws f rom (i), and in the second case, from (ii).  1.5 A ∞ -structures. T o construct triangulated categories, w e will u se the mac hinery of A ∞ -algebras and A ∞ -categ ories (this is ve ry w ell co v ered in the literature; a stand ard referen ce is, for example, B. Keller’s o v erview [Ke3]). W e br ieﬂy recall th e relev ant n otions. 1.5.1 Algebras and mo dules. An A ∞ -algebr a structur e on a graded free Z -mo dule A q is giv en by a co deriv ation δ of the f ree non-unital asso ciativ e coalge br a T q ( A q [1]) generated by A q shifted by 1 su c h that δ 2 = 0. Exp licitly , the structure is giv en b y a collection of op erations b n : A q ⊗ n → A q , n ≥ 1, and δ 2 = 0 is equiv alen t to (1.8) X i + j + l = n b i +1+ l ◦ ( id ⊗ i ⊗ b j ⊗ id ⊗ l ) = 0 for an y n ≥ 1. F or n = 1, this reads as b 2 1 = 0, so th at b 1 is a d iﬀerential whic h turns A q in to a complex of Z -mo dules. After add ing some signs de- p end in g on degrees of th e op erands, the h igher op erations b n , n ≥ 2, can b e arranged together into an op erad f Ass ∞ of complexes of Z -mo d u les. More- o v er, the action of symmetric grou p s on th e comp onent complexes f Ass ∞ is irrelev ant for the deﬁn ition of an A ∞ -algebra – th e op erad f Ass ∞ is in duced from an asymmetric op erad Ass ∞ in the sense of [Hi]. T h e asymm etric 15 op erad Ass ∞ is equipp ed with a canonical surjectiv e augmen tation quasiiso- morphism Ass ∞ → Ass onto the asso ciativ e asymmetric op erad A ss . If one forgets the diﬀerentia ls, As s ∞ is the f ree asymm etric op erad generated b y a single op eration b n for eac h n ≥ 2. Thus As s ∞ is coﬁbrant in the n atural closed mod el structure on the ca tegory of asym m etric op erads (see [Hi]) , and the augmen tation m ap Ass ∞ → Ass is a coﬁb ran t rep lacement f or A s s . F or an y A ∞ -algebra A q , th e op eration m 2 giv en by (1.9) m 2 ( x, y ) = ( − 1) deg ( x ) b 2 ( x, y ) induces an asso ciativ e multiplica tion on the homolog y groups H q ( A q ). A homolo gi c al unit in A q is an elemen t 1 ∈ A 0 suc h th at b 1 (1) = 0, and the cohomology class of 1 is the un it for the asso ciativ e algebra H q ( A q ). A homologica l unit 1 in d uces a con tracting homotop y h f or the diﬀeren tial δ on T q ( A q [1]) by setting h ( a 1 ⊗ · · · ⊗ a n ) = 1 ⊗ a 1 ⊗ · · · ⊗ a n . W e will assume that all A ∞ -algebras are equipp ed w ith a h omologica l un it. An A ∞ -morphism f b et w een A ∞ -algebras A q , A ′ q is a DG coalgebra morphism T q ( A q [1]) → T q ( A ′ q [1]). Explicitly , f is giv en b y a collection of maps f n : A ⊗ n q → A ′ q , n ≥ 1 such that (1.10) X i + j + l = n f i +1+ l ◦ ( id ⊗ i ⊗ b j ⊗ id ⊗ l ) = X i 1 + ··· + i s b s ◦ ( f i 1 ⊗ · · · ⊗ f i s ) for any n ≥ 1. In particular, f 1 is a map of complexes, and it ind uces an algebra map H q ( f 1 ) : H q ( A q ) → H q ( A ′ q ). The map f is unital if so is H q ( f 1 ). Ev ery DG algebra is automatically an A ∞ -algebra (with trivial b n , n ≥ 3). In p articular, for a complex M q of ob jects in an ab elian category Ab as in Su b section 1.1, End( M q ) is a DG algebra. The structure of an A ∞ mo dule over A q on the complex M q is giv en by a unital A ∞ -morphism A q → End( M q ). Explicitly , this giv en by a collectio n of maps b n : A ⊗ n − 1 q ⊗ M q → M q for all n ≥ 2 satisfying (1.8) (where b 1 is the diﬀerentia l on M q ). Equ iv- alen tly , an A ∞ -mo dule structure on M q is giv en by a diﬀeren tial δ on the cofree T q ( A q [1])-co mo d ule T q ( A q [1]) ⊗ M q whic h turns it into a DG como d ule. 1.5.2 Homotop y . T he homo topy c ate gory Ho ( A q , Ab) of A ∞ mo dules o v er A q is the full sub category in the c hain-homotop y catego ry of DG co- mo dules ov er T q ( A q [1]) spanned by A ∞ -mo dules (ob jects are complexes M q equipp ed w ith an A ∞ -mo dule structure, maps are c hain-homotopy classes 16 of maps b et w een the corresp onding DG como du les T q ( A q [1]). Explicitly , if we are giv en t w o complexes M q , M ′ q equipp ed with A ∞ -mo dule struc- tures o v er A q , th en th e graded group Hom q A q ( M q , M ′ q ) of maps b et w een the corresp ondin g T q ( A q [1])-co mo d ules can b e canonically wr itten down as (1.11) Hom q A q ( M q , M ′ q ) = Y n ≥ 0 Hom q − n ( A ⊗ n q ⊗ M q , M ′ q ) , and it has a natural diﬀeren tial giv en by d ( a ) = δ ◦ a − a ◦ δ . Maps in Ho ( A q , Ab) are the degree-0 h omology classes of this complex. T h e cate gory Ho ( A q , Ab) is obvio us ly triangulated. Inside it, w e ha v e the f u ll triangulated sub categ ory spanned by those M q whic h are acyclic as complexes of ob jects in Ab. Lemma 1.4. The sub c ate gory of acyclic A ∞ -mo dules in Ho ( A q , Ab) is lo- c alizing. Pr o of. As in the case of unboun ded complexes of ob jects studied in [Ke1], sa y that an A ∞ -mo dule M q is h - inje ctive if it is right-o rthogonal in Ho ( A q , Ab) to all acyc lic A ∞ -mo dules. Then it suﬃces to p ro v e that for ev ery M q ∈ Ho ( A ∞ , Ab ), there exists an h -injectiv e f M q ∈ Ho ( A q , Ab) equipp ed with a quasiisomorphism f M q → f M q . Cho ose a complex f M q of ob jects in Ab wh ic h is h -injectiv e an d equipp ed with an injectiv e quasiisomorphism M q → f M q . Then since w e assume th at A q is a complexes of fr e e Z -mo dules, th e map A ⊗ n q ⊗ M q → A ⊗ n q ⊗ f M q is an injectiv e quasiisomorphism for an y n . T hen w e can solve th e equations (1.8) by induction on n , to obtain an A ∞ -mo dule structure on f M q and an A ∞ -quasiisomorphism M q → f M q . Moreo v er, for an y acyclic A ∞ -mo dule N q , the terms in the complex Hom A q ( N q , f M q ) can b e rewritten as (1.12) Hom q ( A ⊗ n q ⊗ N q , M q ) ∼ = Hom q ( A ⊗ n q , Hom q ( N q , f M q )) , and since A q is a complex of free Z -mo dules and f M q is right -orthogonal to N q , these are acyclic complexes. Th us Hom A q ( N q , f M q ) is the limit of an in ve rse system of acyclic complexes of ab elian group s, and the transition maps in this system are su rjectiv e. Th er efore th e inv erse limit is also acyclic, and f M q is h -injective in Ho ( A q , Ab).  Deﬁnition 1.5. The derive d c ate gory D ( A q , Ab) is obtained b y lo calizing the category Ho ( A q , Ab) with resp ect to the su b category of acyclic A ∞ - mo dules. 17 W e hav e the obvious forgetful fu nctor D ( A q , Ab) → D (Ab). It has b oth a left and a r igh t-adjoin t, the fr e e and the c ofr e e mo du le fu nctors; they send a complex M q in to A ∞ -comod ules giv en by A q ⊗ M q , resp . Hom Z ( A q , M q ), with an A ∞ -mo dule structur e indu ced b y the stru ctur e maps b n of the A ∞ - algebra A . T o see the adjun ction, one n otes that the con tracting homotop y h giv en by the homological un it in A q induces a homotop y wh ic h con tracts Hom A q ( A q ⊗ M q , N q ) to Hom( M q , N q ) for an y A ∞ -mo dule N q , and similarly for the cofree mo du le Hom Z ( A q , N q ). In fact, we h a v e (1.13) Hom q ( A ⊗ n q ⊗ A q ⊗ M q , N q ) ∼ = Hom q ( A ⊗ n q , Hom q Z ( A q , N q )) for any n ≥ 0 and any tw o complexes M q , N q in Ab, so that the complexes Hom q A q ( − , − ) one has to con tract are exactly the same in the free and in the cofree case. The adju n ction also holds on the lev el of homotop y categories. In particular, for an y A ∞ -mo dule M q , the adjunction giv es an A ∞ -map A q ⊗ M q → M q . Iterating th is construction, we obtain a v ersion of the bar resolution for A ∞ -mo dules; in eﬀect, any A ∞ -mo dule M q is q u asiisomorphic to the direct limit lim n → M ( n ) q so that the tran s ition maps M ( n ) q → M ( n +1) q are injectiv e, and th eir cok ernels are f ree A ∞ -mo dules. Dually , we hav e the cobar r esolution, and ev ery M q can represen ted as an in v erse limit of a system with surj ectiv e transition maps with cofree ke rn els. More generally , for any A ∞ -map f : A q → A ′ q b et wee n A ∞ -algebras, we ha v e an obvious r estriction fu nctor f ∗ : D ( A ′ q , Ab) → D ( A q , Ab). Replacing an A ∞ -mo dule with its free r esp. cofree r esolution, one easily sh o ws that f ∗ has b oth a left-adjoin t f ! and a r igh t-adjoin t f ∗ . Lemma 1.6. If the A ∞ -map f : A ∞ → A ′ ∞ is a quasiisomo rphism, then f ∗ and f ! ar e mutual ly inverse e quivalenc es of c ate gories. Pr o of. Let M q , N q b e t wo A ∞ -mo dules ov er A ′ ∞ . Rewr ite the terms of the complex (1.11) as in (1.12). Then for an y n ≥ 0, the natural map f ∗ : Hom q ( A ⊗ n q , Hom q ( N q , f M q )) → Hom q ( A ′ ⊗ n q , Hom q ( N q , f M q )) 18 is a quasiisomorphism. This imp lies that f ∗ is fully faithful. T o see the it is essen tially surjectiv e, n ote that it comm utes w ith ﬁltered direct limits, and for an y complex M q , f ∗ ( A ′ q ⊗ M q ) is quasiisomorph ic to the free A ∞ -mo dule A q ⊗ M q .  1.5.3 Coalgebras. An A ∞ -c o algebr a is an A ∞ -algebra in the categ ory opp osite to that of ab elian groups. Exp licitly , an A ∞ -coalg ebra structure on a graded Z -mo d ule A q is giv en by a collection of op er ations b n : A q → A ⊗ n q , n ≥ 1 , sub ject to relations (1.8) (where the comp osition ◦ sh ould b e u ndersto o d in the r ev erse order). Equiv alen tly , this s tructure is enco ded b y a collectio n of maps A q ⊗ Ass ∞ → A ⊗ n q , where Ass ∞ is the asymmetric A ∞ -op erad. As in th e algebra case, we will only consider A ∞ -coalg ebras w hic h are fr ee as Z -mo dules. F or an y A ∞ - coalge br a A q , b 1 is a diﬀerentia l, that is, b 2 1 = 0, so that A q b ecomes a complex of fr ee ab elian group s . Th e dual complex ( A q ) ∗ = Hom Z ( A q , Z ) is naturally an A ∞ -algebra. A homolo gic al c ounit for an A ∞ -coalg ebra A q is a map 1 : A 0 → Z su c h that 1 ◦ b 1 = 0, and 1 ∈ ( A 0 ) ∗ is a homologica l un it for ( A q ) ∗ . W e will only consider counital coalgebras. An A ∞ -morphism f : A ′ q → A q of A ∞ -coalg ebras is give n by a collection of maps f n : A ′ q → A ⊗ n q , n ≥ 1 satisfying (1.10). Suc h a m ap is a quasiisomorphism if so is its comp onent f 1 . The dual m aps f ∗ n giv e an A ∞ -map f ∗ : ( A q ) ∗ → ( A ′ q ) ∗ ; f is c ounital if f ∗ is unital. As in the algebra case, we will only consider counital A ∞ -maps. An A ∞ -c omo dule ov er an A ∞ -coalg ebra A q in an ab elian category Ab is giv en by a graded ob j ect M q in Ab together with maps (1.14) b n : M q → M q ⊗ A ⊗ n − 1 q , n ≥ 1 , again sub ject to (1.8). Again, b 1 is a diﬀeren tial on M q . Moreo v er, M q is automatica lly an A ∞ -mo dule ov er ( A q ) ∗ , and H q ( M q ) is a mo dule ov er the cohomology algebra H q (( A q ) ∗ ). If this mo d ule is un ital, the A ∞ -comod ule M q is called counital. W e w ill only consider counital como dules. Equiv alen tly , an A ∞ -structure on A q can b e describ ed as a sq u are-zero deriv atio n of the completed tensor alg ebra b T q ( A q [ − 1]), and an A ∞ -comod ule 19 structure on a complex M q is the same as a diﬀerent ial on the completed tensor pro duct (1.15) e T q ( A q [ − 1]) b ⊗ M q = lim n ← T q ( A q [ − 1]) /T ≥ n ( A q [ − 1]) ⊗ M q whic h turns it in to a DG b T q ( A q [ − 1])-mod ule. The homotop y catego ry Ho ( A q , Ab) of A ∞ -comod ules o ve r A q in Ab is the full sub categ ory in the c hain-homotop y category of top ologic al DG b T q ( A q [ − 1])-mod ules spann ed b y DG mo du les of the form (1.15). Give n t wo A ∞ -comod ules M q , M ′ q , w e deﬁne a complex Hom q A q ( M q , M ′ q ) as Y n ≥ 0 Hom q ( M q , A ⊗ n ⊗ M ′ q ) , with the diﬀerenti al d ual to that of (1.11). Then the space of maps from M q to M ′ q in Ho ( A q , Ab) is the 0-th homology group of th is complex. An ob ject M q ∈ Ho ( A q , Ab) is called acyclic if it is acyclic as a complex in Ab. Lemma 1.7. A ssume that Ab is the c ate gory of mo dules over a ring R . Then the sub c ate g ory of acyclic c omplexes in Ho ( A q , Ab) is lo c alizing. Pr o of. W e adopt the metho d of [W , Prop osition 10.4.4]; I am grateful to the referee for suggesting this r eferen ce. As in [W, Prop osition 10.4.4], it suﬃces to p r o v e that for any A ∞ - comod ule M q ∈ Ho ( A q , Ab), there exists a set A of quasiisomorphisms r α : M α q → M q , α ∈ A , such that for any quasiisomorph ism r : M ′ q → M q , one of the maps r α factors through r . L et κ b e an inﬁ nite cardinal larger than the cardinalit y of L M r . There is at most a set of qu asiisomorphisms r α : M α q → M q suc h that th e cardinalit y of L M α i is at most κ ; tak e them all. Assume giv en a quasiisomorph ism r : M ′ q → M q . Ev ery elemen t m ∈ M ′ i , i ∈ Z , lies in an at m ost coun table A ∞ - sub comod ule M m q ⊂ M ′ i – indeed, w e can take the ab elian su bgroup gen- erated b y m , ad d to it all the left-hand sid es of the structur e maps b n of (1.14), and rep eat the pro cedure by induction. Therefor there exists an A ∞ -sub como d ule M (1) q ⊂ M ′ q of cardinalit y at m ost κ s u c h that the map r (1) i : H i ( M (1) q ) → H i ( M q ) induced b y r is surj ective for ev ery i ∈ Z . Rep eating the pr o cedure, we obtain a system of sub como d ules M ( n ) q ⊂ M ′ q , n ≥ 1, s uc h that M ( n +1) q con tains M ( n ) q , th e map r ( n ) i : H i ( M ( n ) q ) → H i ( M q ) 20 is still surj ectiv e for any n ≥ 2, i ∈ Z , and th e natural map H i ( M ( n ) q ) → H i ( M ( n +1) q ) induced b y the em b edd in g M ( n ) q → M ( n +1) q annihilates Ker r ( n ) i for any i ∈ Z . Let M ′′ q = S M ( n ) q ⊂ M ′ q . Then the natural map r ′′ : M ′′ q → M q is a quasiisomorphism b y constru ction, and it is of the form r α for s ome α in the indexing set A .  Remark 1.8. The pr o of can probably b e mo diﬁ ed so that it only requ ires our original assumptions on Ab , but I ha ve n’t pu r sued it for lac k of in ter- esting examples. Deﬁnition 1.9. The derive d c ate gory D ( A q , Ab) of A ∞ -comod ules o ver A q in Ab is the quotien t of the h omotop y catego ry Ho ( A q , Ab) by the sub cate- gory of acyclic ob jects. As in th e algebra case, an A ∞ -comod ules M q ∈ Ho ( A q , Ab) is called h -inje ctive if it r igh t-orthogonal to all acyclic ob jects. W e ha v e an obvious forgetful functor Ho ( A q , Ab) → Ho (Ab) onto the chain-homoto py category of complexes in Ab , and it has a righ t-adjoint wh ic h sends M q ∈ Ho (Ab) in to the cofree comodu le A q ⊗ M q , with the como dule structure maps b n giv en by th e structure map s of A ∞ . As in the algebra case, the counit on A q induces a homotop y wh ic h con tracts the complex Hom q A q ( N q , A q ⊗ M q ) on to Hom q ( N q , M q ) for an y N q , M q ∈ Ho ( A q , Ab), and this giv es the adjun c- tion. In particular, if M q ∈ D (Ab) is h -injectiv e, the cofree A ∞ -comod ule A q ⊗ M q is h -injectiv e in Ho ( A q , Ab). One is tempted no w to d ualize the b ar- construction and obtain an h -injectiv e r eplacemen t for any M q Ho ( A q , Ab ). Ho w ev er, this do e s not work . The reason is the follo wing: to b e h -injectiv e, the cobar resolution of an A ∞ -comod ule M q has to b e a pro jectiv e limit of h -injectiv e como d u les. Th us as a graded ob ject in Ab, it is of the f orm A q ⊗ Y n ≥ 0 ( A ⊗ n q ⊗ M q ) . 21 Ho w ev er, this is d iﬀeren t from Y n ≥ 0 ( A q ⊗ A ⊗ n q ⊗ M q ) , and it is th e latter, n ot the former which can b e con tracted on to M q b y the homologica l counit of A ∞ . This is w h y th e existence of D ( A q , Ab) has to b e pr o v ed b y an indirect metho d. And even when Ab is as in L emm a 1.7, it is not clear at p resent whether for an arbitrary A q , any M q ∈ Ho ( A q , Ab) is quasiisomorphic to an h -injectiv e M ′ q . Remark 1.10. I f A is simply a coalgebra, not an A ∞ or DG coalgebra, then more is kno wn, sin ce a very compr ehensiv e study of the h omologica l prop erties of unb ounded complexes of como du les has b een done recen tly by L. P ositelski [P]. In particular, it h as b een pr ov ed in [P] th at h -injectiv e replacemen ts do exist. Ho w ev er, the pr o of is very ind irect, and it is not clear at all whether it can b e generalized to the A ∞ -case. As in the algebra case, an A ∞ -map f : A q → A ′ q induces a n atural corestriction functor f ∗ : D ( A q , Ab) → D ( A ′ q , Ab), assumin g that Ab is as in Lemm a 1.7 so that b oth catego ries are w ell-deﬁned. If A ′ q = Z and f is the coun it, this is the forgetful functor D ( A q , Ab) → D (Ab), and it has a righ t-adjoint gi ven b y the cofree comod ule construction. In general, it is not clear whether f ∗ admits a right- adjoint (and the situation with the left-adjoin t is even worse). Let us list some other things that d o n ot w ork for coalgebras. (i) There are no free comod ules, only the cofree ones ((1.13) d o es not w ork, since it would inv olve double du alization). (ii) The pro of of Lemma 1.4 breaks do wn b ecause an analog of (1.12) is not an isomorph ism. (iii) F or those intereste d in su c h things, the category of A ∞ -comod ules for a general A q do es not admit a closed mo del str u cture (or at least, none suc h is kno wn ). (iv) Finally , Lemma 1.6 fails. No t only d o es its pro of break do wn, the statemen t itself is false. In fact, one case where this happ ens w ill b e the m ain s ub ject of Section 6. 22 1.5.4 Cate gories. Inf ormally , a (small) A ∞ -categ ory is an A ∞ -algebra “with man y ob jects”. T o k eep trac k of the com binatorics, it is con ve nient to use the app r oac h of [Le]. F or an y set S , d enote b y Z S -mo d the category of S -g raded Z -mo dules. Fix a s et S , an d consider the catego ry Z S × S -mo d. Equip it with a tensor pro duct by setting ( A ′ ⊗ A ′′ ) s ′ ,s ′′ = M s ∈ S A ′ s ′ ,s ⊗ A ′′ s,s ′′ for an y A ′ , A ′′ ∈ Z S × S -mo d. T h is is not symmetric, but neither are the op erads Ass and As s ∞ , so that sp eaking ab out asso ciativ e and A ∞ -algebras in Z S × S -mo d mak es p erfect sense. F or any small add itive category B with the set of ob jects S , th e sum B S = M s,s ′ B ( s, s ′ ) is an asso ciativ e algebra in Z S × S -mo d. Then an A ∞ -c ate gory B q consists of (i) a small graded additiv e catego ry B q , with a certain set of ob jects S , and (ii) an A ∞ -algebra B q in Z S × S -mo d equipp ed with an isomorph ism H q ( B q ) ∼ = B S q of graded asso ciativ e algebras in Z S × S -mo d. Our reason for making th is sligh tly con v oluted deﬁn ition is that it automat- ically tak es care of th e units. As in the algebra case, w e will assume that B q ( b, b ′ ) is a complex of fr e e Z -mo dules for any tw o ob jects b, b ′ ∈ B q . F or an y map of sets f : S → S ′ , we ha v e an obvious pseudotensor restriction functor f ∗ : Z S ′ × S ′ -mo d → Z S × S -mo d; an A ∞ -functor b et w een A ∞ -categ ories B q , B ′ q joists of a functor f : H q ( B q ) → H q ( B ′ q ) and an A ∞ - morphism B q → f ∗ B ′ q . F or any ab elian categ ory Ab, the categ ory Ab S of S -graded ob jects in Ab is naturally a mo du le category o v er the tensor category Z S × S -mo d. An A ∞ -functor f r om an A ∞ -categ ory B q to Ab is then an A ∞ -mo dule ov er B q in C S suc h that u n its act by identi ty maps on the corresp onding B q -mo dule H q ( M q ). As in the algebra case, we ha v e th e homotop y category Ho ( B q , Ab) and the deriv ed category D ( B q , Ab ) of A ∞ -functors from B q to Ab. The role 23 of free and cofree mo dules is pla y ed by representable and corepresentable functors: for an y ob ject b ∈ B q and an y M q ∈ Ho (Ab), these are giv en by M b q ( b ′ ) = M q ⊗ B q ( b, b ′ ) , M q b ( b ′ ) = Hom Z ( B q ( b ′ , b ) , M q ) for any b ′ ∈ B q . W e also h a v e th e bar and cobar r esolution, so that the de- riv ed category D ( B q , Ab) is generated by r epresen table resp. corepr esen table functors in the same sense as the cate gory of A ∞ -mo dules is generated by free resp . cofree mo du les. F or any A ∞ -functor f : B q → B ′ q b et wee n t w o A ∞ -categ ories, w e hav e the r estriction f unctor f ∗ : D ( B ′ q , Ab) → D ( B q , Ab) and its t wo adj oints f ! : D ( B q , Ab) → D ( B ′ q , Ab) , f ∗ : D ( B q , Ab) → D ( B ′ q , Ab) . An A ∞ -functor f : B q → B ′ q is a quasie quivalenc e if the corresp ond ing fun c- tor H q ( B q ) → H q ( B ′ q ) is an equiv alence, and the natural map f 1 : B q ( b, b ′ ) → B ′ q ( f ( b ) , f ( b ′ )) is a quasiisomorphism for an y b, b ′ ∈ B q . It is not d iﬃcult to generalize Lemma 1.6 and show that for a qu asiequiv alence f , the f u nctors f ∗ and f ! are m utually inv erse equiv ale nces of the derived categories. W e note that an y small category C d eﬁ nes an additiv e category Z [ C ] with the same ob jects, and morphisms giv en by Z [ C ]( c, c ′ ) = Z [ C ( c, c ′ )], where Z [ S ] for a set S means the free ab elian group generated b y S . Th en Z [ C ] can b e treated as an A ∞ -categ ory (placed in homologica l d egree 0), and w e of course hav e D ( C , Ab) ∼ = D ( Z [ C ] , Ab ). W e will also n eed a version of this for coalgebras. In fact, it will more con v enient to use a slightly m ore reﬁned n otion. Assume give n a small catego ry C , with the set of ob jects C ob and the set of morph isms C mor . In tro d uce a tensor p ro du ct on the category Z C mor -mo d by setting ( A ′ ⊗ A ′′ ) f = M f ′ ,f ′′ ∈C mor ,f ′ ◦ f ′′ = f A ′ f ′ ⊗ A ′′ f ′′ for an y A ′ , A ′′ ∈ Z C mor -mo d and f ∈ C mor . Then Z C mor -mo d is a m onoidal catego ry , and for an y ab elian catego ry Ab as in Sub section 1.1, Ab C ob is a mo dule category o v er Z C ob -mo d. Deﬁnition 1.11. A C -gr ade d A ∞ -c o algebr a A q is an A ∞ -coalg ebra in the monoidal category Z C mor -mo d. An A ∞ -c omo dule in Ab o ver A q is an A ∞ - mo dule o ve r A q in the m o dule category Ab C mor . 24 As in the A ∞ -categ ory case, w e will alw a ys assume that our graded coalge br as consist of free Z -mo d ules; we will also assume that coa lgebras and como dules are (homologica lly) counital in the same sense as in th e n on- graded case. Exp licitly , a C -graded A ∞ -coalg ebra A q is giv en by a coll ection A q ( f ) of complexes of free Z -mo dules n umb ered by morphism s f of the catego ry C , tog ether with a homological counit map A q ( id c ) → Z for any ob ject c ∈ C and a com ultiplication map b n : A q ( f 1 ◦ · · · ◦ f n ) → A q ( f 0 ) ⊗ · · · ⊗ A q ( f n ) for any n ≥ 2 and an y n -tuple of comp osable maps f 1 , . . . , f n in C , sub ject to (1.8). An Ab-v alued A q -comod ule E q in the category Ab is a collection of complexes E q ( c ) in Ab, on e for eac h ob j ect c ∈ C , together w ith map s b n : E q ( c ) → A q ( f 1 ) ⊗ · · · ⊗ A q ( f n ) ⊗ E q ( c ′ ) for an y n ≥ 1 and any n -tup le of comp osable maps f 0 , . . . , f n , f 0 ◦ · · · ◦ f n : c → c ′ in C , again sub ject to (1.8). As in the non-graded case, an y C -graded A ∞ -coalg ebra A q pro du ces th e homotop y and the d eriv ed categories of A q -comod ules, denoted Ho ( A q , Ab) resp. D ( A q , Ab). F or an y ob ject c ∈ C and any complex M q in Ab, we ha v e the corepresen table A q -comod ule M c q giv en by (1.16) M c q ( c ′ ) = Y f : c ′ → c A q ( f ) ⊗ M q . F or an y C -graded A ∞ -map f : A q → A ′ q , we ha v e the restriction functor f ∗ : D ( A q , Ab) → D ( A ′ q , Ab). As in the coalg ebra case, it d o es not ha v e to b e an equiv alence ev en if f is a quasiisomorphism in a su itable sens e. On the other hand, assume giv en another small categ ory C ′ and a functor ρ : C ′ → C . Deﬁne a C ′ -graded A ∞ -coalg ebra ρ ∗ A q b y setting ρ ∗ A q ( f ) = A q ( ρ ( f )) for any morphism f in C ′ , with the same structur e maps b n . W e then h a v e an ob vious pu llbac k functor ρ ∗ : D ( A q , Ab) → D ( ρ ∗ A q , Ab) . I do n ot kn o w under what assumptions, if any , either of the functors ρ ∗ , f ∗ has a righ t adjoint. 25 1.6 2 -categories. T o pro du ce A ∞ -categ ories, we will use 2-catego ries; we end the pr eliminaries with a br ief sket c h of the corresp ond ing construction. Assume giv en a small monoidal category C , and consider the bar complex C q ( C , Z ). If C is s tr ictly asso ciativ e, then the tensor pro d uct f unctor m : C × C → C ind uces an asso ciativ e DG algebra structure on C q ( C , Z ) (apply the pr op erties (ii) and (iii) of S ubsection 1.3). More generally , if we also ha v e an ob ject T ∈ F un ( C , Z ) and a map (1.17) T ⊠ T → m ∗ T whic h is associativ e on trip le pro du cts, then C q ( T ) b ecomes an asso ciativ e DG algebra (plug in the pr op ert y (i)). Ho w ev er, monoidal categ ories in nature are usually asso ciativ e only up to an isomorphism – there is an associativit y isomorphism m ◦ m 12 ∼ = m ◦ m 23 satisfying the p enta gon equation. W e observ e that in this case C q ( C , Z ) is no longer a DG algebra, but it has an A ∞ -algebra structure. Indeed, for any n ≥ 2, let I n b e th e group oid whose ob jects are all p ossible n -ary op erations obtained from a single b in ary operation, and whic h has exactly one m orp hism b et wee n ev ery t w o ob jects. Then I n , n ≥ 2 form an asymmetric op erad of categories, and an y we akly asso ciativ e monoidal catego ry C is an algebra o v er th is op erad: w e ha ve natural f unctors I n × C n → C for ev ery n . T h e bar complexes C q ( I n , Z ) form an asymmetric op erad of com- plexes of Z -mo dules, C q ( C , Z ) is an algebra o v er th is op erad, and th e op erad itself is a resolution of the trivial asymmetric op erad As s . Th e asymmetric op erad Ass ∞ is another s uc h resolution, and it is coﬁbran t. Ther efore the augmen tation map Ass ∞ → Ass factors through a map Ass ∞ → C q ( I q , Z ). Fixing this map once and for all, we turn the bar complex C q ( C , Z ) f or an y monoidal catego ry C in to an A ∞ -algebra. Analogously , a (w eakly) monoidal functor b et w een mon oidal catego ries C , C ′ induces an A ∞ -map b et wee n the A ∞ -algebras C q ( C , Z ), C q ( C ′ , Z ). The same construction ob viously wo rks for bar complex C q ( C , T ) with co eﬃcien ts, where T ∈ F un( T , Z ) is equipp ed with an associativ e m ap (1.17). Moreo v er, if w e h a v e a 2-ca tegory Q with a certain set of ob jects { c } and categories of 1-morph isms Q ( c, c ′ ), c, c ′ ∈ C , then th e same construction pro du ces an A ∞ -categ ory with the same ob jects, and with the bar com- plexes C q ( Q ( c, c ′ ) , Z ) as complexes of morphisms. This is also fun ctorial with resp ect to 2-functors, and has an obvious v ersion with co eﬃcien ts. 26 2 Recollection on Mac k ey functors. This ends the preliminaries. F or the con v enience of the reader, we start the pap er itself by b r ieﬂy recalling the deﬁnitions and known facts ab out Mac k ey functors (w e more-or-less follo w the exp ositions in [M] and [T2]). 2.1 Deﬁnitions. Assume giv en a group G , and let Γ G b e the category of ﬁnite sets equipp ed with a G -action. This category ob viously has p ullbac ks. Deﬁne a b igger category Q Γ G as follo ws: ob jects are the s ame as in Γ G , maps from S 1 to S 2 are isomorphism classes of d iagrams S 1 ← S ′ 1 → S 2 , comp osition of S 1 ← S ′ 1 → S 2 and S 2 ← S ′ 2 → S 3 is giv en b y the diagram S 1 ← S ′ 1 × S 2 S ′ 2 → S 3 . The category Q Γ G is self-du al, Q Γ G ∼ = Q Γ opp G . Moreo v er, disjoint un ions of sets giv e ﬁnite copro du cts b oth in th e category Γ G and in the category Q Γ G . Ev ery G -ﬁnite set S ∈ Γ S can b e canonically decomp osed int o suc h a disj oint union (2.1) S = a p ∈ S/G S i of subs ets S p on which G act s transitiv ely; w e call them G -orbits . T h is decomp osition is v alid b oth in Γ G and in Q Γ G . Deﬁnition 2.1. A G -Mackey fu nc tor M is a f unctor M : Q Γ G → Ab to the category Ab of ab elian groups whic h is add itiv e in th e follo wing sense: for an y S ∈ Q Γ S , th e natural map M p ∈ S/G M ( S p ) → M ( S ) induced by the decomp osition (2.1) is an isomorph ism. Mac k ey fu nctors obviously form an ab elian category which w e denote b y M ( G, Ab), or simply by M ( G ). By deﬁnition, M ( G, Ab ) is a full sub categ ory in F un ( Q Γ G , Ab), and one c hec ks easily that the em b edd in g M ( G, Ab) → F un ( Q Γ G , Ab) admits a left adjoin t, wh ic h w e call additiviza- tion and d enote by Add : F un( Q Γ G , Ab ) → M ( G, Ab) (in fact, Add is also righ t-adjoin t to the em b edding). F or any coﬁnite subgroup H ∈ G , the v alue M ([ G/H ]) of a G -Mac key functor M on the G -orbit [ G/H ] is usually denoted b y M H . By the ad- ditivit y prop erty , M H for all H ⊂ G completely deﬁne M . Explicitly , a G -Mac key functor M is giv en by 27 (i) an ab elian group M H for an y coﬁnite sub group H ∈ G , and (ii) t w o maps f ∗ : M H 1 → M H 2 , f ∗ : M H 2 → M H 1 for any t wo coﬁn ite subgroups H 1 , H 2 ⊂ G an d a G -equiv ariant map f : [ G/H 1 ] → [ G/H 2 ], suc h that f ∗ ◦ g ∗ = ( g ◦ f ) ∗ and g ∗ ◦ f ∗ = ( g ◦ f ) ∗ for an y tw o comp osable maps f : [ G/H 1 ] → [ G/H 2 ], g : [ G/H 2 ] → [ G/H 3 ], and for an y tw o maps f : [ G/H 1 ] → [ G/H ], g : [ G/H 2 ] → [ G/H ], we hav e (2.2) g ∗ ◦ f ∗ = X p ∈ S/G f p ∗ ◦ g ∗ p , where we let S = [ G/H 1 ] × [ G/H 2 ], ta ke its d ecomp osition (2.1), and let g p : S p → [ G/H 1 ], f p : S p → [ G/H 2 ] b e the n atural pro jections. W e note that s in ce S/G = G \ ( G × G ) / ( H 1 × H 2 ) ∼ = H 1 \ G/H 2 , the comp onents S p corresp ond to doub le cosets H 1 g H 2 ⊂ G ; for this reason, (2.2) is known as the double c oset formula . This is the original deﬁn ition of Mac k ey f unctors in tro du ced by Dress [Dr]; the v ersion with the category Q Γ G is due to Lin d - ner [Li]. T h e coll ection h f ∗ , f ∗ i without the condition (2 .2) is sometimes called a bifunctor (from the category of ﬁn ite G -orbits to Ab). Example 2.2. Rep resen tation r ing: setting [ G/H ] 7→ R H , the represen- tation r ing of the group H , deﬁnes a Mac k ey fu nctor, with f ∗ giv en by restriction and f ∗ giv en b y induction. T his is the origin of the notion and the name: the double coset form ula f or R H w as foun d by Mac k ey . Example 2.3. C ohomology: setting [ G/ H ] 7→ H q ( H , Z ), f ∗ giv en by r e- striction, f ∗ giv en by corestriction, d eﬁ nes a (graded) Mack ey f unctor. T o obtain a th ir d useful deﬁnition of Mac k ey f u nctors, one considers an additive category B G deﬁned as follo ws: ob j ects of B G are ﬁ nite G - orbits [ G/H ], and the set of maps from S 1 to S 2 is the f ree ab elian group generated by isomorp hism classes of d iagrams S 1 ← S → S 2 , where S is another ﬁnite G -orbit. Comp osition g ◦ f of t w o maps f : [ G/H 1 ] → [ G/H 2 ], g : [ G/H 2 ] → [ G /H 3 ] represen ted by diagrams [ G/H 1 ] ← S f → [ G/ H 2 ], [ G/H 2 ] ← S g → [ G/H 3 ] is giv en by g ◦ f = X p ( g ◦ f ) p , where we consider the decomp osition (2.1) of the ﬁb ered pro d uct S = S f × [ G/H 2 ] S g , and let ( g ◦ f ) p b e the map represen ted by the diagram 28 [ G/H 1 ] ← S p → [ G/H 3 ]. Th en a G -Mac k ey fun ctor M is ob viously the same thing as an additiv e functor B G → Ab. The catego ry B G can b e describ ed more explicitly in terms of th e so-called “Burnside rings”. Deﬁnition 2.4. The Burnside ring A G of a group G is the ab elian group generated b y isomorp hism classes [ S ] of ob jects S ∈ Γ S in the category Γ G , mo dulo the relations [ S 1 ] + [ S 2 ] = [ S 1 ` S 2 ], and with the pro duct giv en b y [ S 1 ] · [ S 2 ] = [ S 1 × S 2 ]. Then the endomorphism ring of the trivial G -orbit [ G/ G ] ∈ B G ob viously coincides with the Burn s ide r ing A G . And more generally , giv en tw o ﬁnite orbits S 1 , S 2 ∈ B G , w e h av e (2.3) B G ( S 1 , S 2 ) = M p ∈ ( S 1 × S 2 ) /G A H p , where S p = [ G/H p ] are the comp onents in the decomp osition (2.1) of the pro du ct S = S 1 × S 2 . Remark 2.5. Normally the deﬁn itions in this sub s ection are giv en for a ﬁnite group G ; how ev er, ev erything w orks in a slightl y wid er generalit y , and this is s ometimes useful. Of course, the category M ( G ) as deﬁn ed here only d ep ends on the proﬁnite completion of the group G . There is also a v ersion of Mac ke y f unctors f or top ologica l group s su c h as ﬁ nite-dimensional Lie groups, see e.g. [tD]; how ev er, this is b ey ond the scop e of the present pap er. 2.2 F unctorialit y . F or an y coﬁnite su bgroup H ⊂ G of a group G and a ﬁ nite H -set S , the pro duct S × H G = ( S × G ) /H is n atur ally a ﬁnite G -set, with the G -actio n through the second factor. This deﬁnes a fu nctor γ G H : Γ H → Γ Q . In f act, if we denote S = [ G/H ], then γ G H is an equiv alence b et wee n Γ H and the category Γ G /S of ﬁnite G -sets equip p ed with a m ap to S . The fun ctor γ G H ob viously comm utes w ith ﬁb ered pr o ducts, th us extends to a fun ctor γ G H : Q Γ H → Q Γ G . It also commutes with disjoin t unions, so that w e can d eﬁne an exact functor Restr G H : M ( G ) → M ( H ) whic h sends M : Q Γ H → Ab to γ H G ◦ M : Q Γ G → Ab. F or an y G -Mac k ey functor M ∈ M ( G ), the H -Mac key functor Restr G H ( M ) is called the r estric- tion of M to H ⊂ G . 29 Assume that a sub group H ⊂ G is n ormal, and let N = G/H b e the quotien t. Th en any N -set is also a G -set, so that w e hav e an ob vious full em b edd ing Γ N → Γ G whic h in duces a full em b edding Q Γ N → Q Γ G com- patible with disjoint un ions . T h is induces a functor Ψ H : M ( G ) → M ( N ) (usually Ψ H ( M ) is denoted simply by M H , bu t th is migh t cause confu s ion). Ho w ev er, we also ha ve a fu nctor Inﬂ N G : M ( N ) → M ( G ) called inﬂation and giv en by “extension by 0”: we set Inﬂ N G ( M ) K = ( M K/H , H ⊂ K, 0 , otherwise . This is an exact full embedd ing. It has a left-adjoin t which is d enoted b y Φ H : M ( G ) → M ( N ). If the sub group H ⊂ G is not normal, w e can co nsid er its normalizer N H ⊂ G . Assume that the normalizer N H ⊂ G is coﬁnite in G . Then w e can d eﬁne the fu nctor Φ H : M ( G ) → M ( N H /H ) b y ﬁr st r estricting to N G ( H ): Φ H ( M ) = Φ H ( Restr G N H ( M )) . Analogously , M H has a natural structur e of a ( N H /H )-Ma c ke y functor. 2.3 Pro ducts. Both in Examp le 2.2 an d Example 2.3, the Mac k ey fun c- tors ha ve an additional stru cture — an asso ciativ e pro du ct. This is axiom- atized as follo ws (this deﬁnition is tak en fr om [tD, S ubsection 6.2]). Deﬁnition 2.6. A Gr e en functor is a Mac key functor M ∈ M ( G ) equipp ed with an asso ciativ e pro d uct in eac h M H suc h that (i) for any f , the map f ∗ preserve s the pr o duct, (ii) for any f : [ G/H 1 ] → [ G/H 2 ], x ∈ M H 2 , y ∈ M H 1 , we h a v e x · f ∗ ( y ) = f ∗ ( f ∗ ( x ) · y ) , f ∗ ( y ) · x = f ∗ ( y · f ∗ ( x )) . W e note that the Cartesian p ro du ct of ﬁnite s ets deﬁnes a fun ctor m : Q Γ G × Q Γ G → Q Γ G ; this ind uces a symmetric tensor pro d uct on the cate- gory M ( G ) by M ⊗ N = Add ( m ! ( M ⊠ N )) . A Green fun ctor is then the same as a Mac key functor equipp ed with an algebra structure in the symmetric tensor category M ( G ). F or example, 30 to see the condition (i), one can argue as follo ws. C on s ider the natural em b edd ing i : Γ opp G → Q Γ G . Then for an y M , N , K ∈ M ( G ) we hav e Hom( M ⊗ N , K ) = Hom( Add ( m ! ( M ⊠ N )) , K ) = Hom( m ! ( M ⊠ N ) , K ) = Hom( M ⊠ N , m ∗ K ) , and since m ∗ comm utes with i ∗ , eve ry map M ⊗ N → K induces a map i ∗ M ⊗ i ∗ N → i ∗ K , where the tensor pr o duct on F un(Γ opp G , Ab) is again giv en by the d irect image m ! with r esp ect to the pro d uct functor m : Γ opp G × Γ opp G → Γ opp G . Ho w ev er, on Γ opp G , this pr o duct functor is left-adjoin t to th e diagonal em b edding δ : Γ opp G → Γ opp G × Γ opp G ; therefore m ! ∼ = δ ∗ , and the tensor structur e on F un(Γ opp G , Ab) is in f act giv en b y the p oin t wise pro d uct, so that for any Green functor M ∈ M ( G ), the restriction i ∗ M is simp ly a functor from Γ opp G to the category of rings. Since the p oin t orbit [ G/G ] ∈ Q Γ G is the u nit ob ject for the pr o duct functor m , the Mack ey fun ctor A , [ G/H ] 7→ B G ([ G/G ] , [ G/ H ]) it repr esen ts is th e unit ob ject for the tensor pro du ct of Mac key functors and in p articular, a Green fu nctor. This Green functor A is called the Burnside ring Gr e en functor — in d eed, b y (2.3), the comp onent A H is exactly the Burnside ring of the ﬁnite group H . Every Mac k ey fun ctor M ∈ M ( G ) is then a mo d ule o v er the Burnsid e ring Green fu nctor A . 3 The d eriv ed v ersion. Since the catego ry M ( G ) of G -Mac k ey fun ctors is ab elian, one can consider its deriv ed category D ( M ( G )). Ho wev er, as we ha v e explained in the In - tro duction, its formal p rop erties are somewhat deﬁ cient. Th e goal of this pap er is to suggest a cure for this b y deﬁning a certain tr iangulated category whic h cont ains M ( G ) bu t diﬀers from D ( M ( G )), and has all the p rop erties one w ould like to h av e. The idea b ehind the construction is very simple. In the deﬁnition of the Burnside ring A G , and more generally , in the d eﬁnition of the additiv e catego ry B G , w e take ab elian group s spanned by isomorphism classes of certain ob jects – in other words, w e tak e the 0-th homology group H 0 of a certain group oid. The correct thing to tak e at the lev el of triangulated catego ries is the f ull homology , not j ust its degree-0 part. There are sev eral w a ys to make this precise. In this section, w e giv e a construction whic h u s es A ∞ metho ds and bar-resolutions. 31 3.1 The quotien t construction. Assume giv en a small category C wh ich has ﬁb ered pro d ucts. Th en w e can ob viously deﬁne a catego ry Q C as follo ws: ob jects are ob j ects of C , maps from c 1 ∈ Q C to c 2 ∈ Q C are give n b y isomorphism classes of diagrams c 1 ← c → c 2 , and comp ositions are giv en b y the ﬁb ered pro du cts, as in Sub section 2.1. Ho w ev er, we can reﬁne the constru ction. Let QC b e the 2 -c ate gory w hose ob jects are aga in the ob jects of C , and such that for an y c 1 , c 2 ∈ C , the catego ry QC ( c 1 , c 2 ) of maps from c 1 to c 2 in QC is th e category of diagrams c 1 ← c → c 2 and their isomorphisms. If C also h as a terminal ob j ect, th us all pro du cts, we can equiv alen tly set Q ( c 1 , c 2 ) = Q ( c 1 × c 2 ), where Q ( c ) for an ob ject c ∈ C is the category of ob jects in C equipp ed with a map to c and their isomorphisms . The comp osition is again giv en by ﬁb ered pr o ducts. Since ﬁb ered p ro du cts are asso ciativ e up to a canonical isomorphism, QC is a w ell-deﬁned 2-ca tegory . W e denote by B C q the A ∞ -categ ory in the sense of Subs ection 1.5 associated to QC b y the pro cedure describ ed in Subsection 1.6 . Thus th e ob j ects in B C q are again th e same as in C , m aps from c 1 to c 2 are giv en by B C q ( c 1 , c 2 ) = C q ( QC ( c 1 , c 2 ) , Z ) , where Z in the righ t-hand side is the constant functor with v alue Z , and com- p ositions are ind uced by the 2-category stru cture on QC . W e note that by deﬁnition, this A ∞ -categ ory is concent rated in non -p ositiv e cohomological degrees. Deﬁnition 3.1. The derived catego ry D ( B C q , Ab) of A ∞ -functors f rom B C q to Ab is denoted b y D Q ( C , Ab). W e note that every map f : c 1 → c 2 in the ca tegory C canonicall y deﬁnes a 1-map fr om c 1 to c 2 in the 2-category QC , and w e h a v e a 2-functor C → QC , where C is u ndersto o d as a discrete 2-category . The bar complex of a discrete category with the s et of ob jects S is canonically quasiisomorphic to the f ree ab elian group Z [ S ] generated b y S ; therefore by restriction, w e obtain a canonical fun ctor D Q ( C , Ab) → D ( C , Ab). Analogously , w e hav e a canonical functor D Q ( C , Ab) → D ( C opp , Ab). Assume th at the categ ory C , in add ition to ﬁb ered p ro du cts, has ﬁnite copro ducts. Deﬁnition 3.2. An A ∞ -functor M q ∈ D Q ( C , Ab) is additive if its restric- tion M q ∈ D ( C opp , Ab) is additiv e in the sense of S ubsection 2.1 : for any 32 c, c ′ ∈ C , the natural map M q ( c ` c ′ ) → M q ( c ) ⊕ M q ( c ′ ) induced b y the em b eddin gs c → c ` c ′ , c ′ → c ` c ′ is a qu asiisomorphism. Th e full sub categ ory in D Q ( C , Ab ) spanned b y additive A ∞ -functors is denoted b y D Q add ( C , Ab) ⊂ D Q ( C , Ab). In p articular, let G b e a group, and let Γ G b e the category of ﬁnite G -sets. Deﬁnition 3.3. A derive d G -Mackey functor is an additive ob ject in the catego ry DQ (Γ G , Ab ). Th e categ ory D Q add (Γ G , Ab) of d eriv ed G -Mac k ey functors is d enoted by D M ( G, Ab). If Ab = Z -mo d is th e categ ory of ab elian groups D M ( G, Z -mo d) is denoted simply by D M ( G ). 3.2 Example: the trivial group. T o illustrate the general notion of a deriv ed Mac k ey f unctor, consider the case of the trivial group G = { e } , so that Γ G = Γ . Of course, M ( { e } , Ab) = Ab; w e w ould exp ect the same to hold of the deriv ed level. Let us chec k that this is indeed so. T o do this, consider the sub category Γ + ⊂ Q Γ with the same ob jects, and th ose 1-morph isms S 1 ← S → S 2 for wh ic h the map S → S 1 is injectiv e. W e note that suc h diagrams ha v e no non-trivial automorphisms; therefore the 2-categ ory structure on Γ + is trivial and we can treat it as a usual catego ry . Here are t w o equiv alent descriptions of Γ + . (i) The category w hose ob jects are ﬁ nite sets S , and wh ose m orp hisms from S 1 → S 2 are “partial maps ” f : S 1 99K S 2 – that is, maps to S 2 deﬁned on a subset S ⊂ S 1 . (ii) The category of ﬁnite sets S with a ﬁxed p oin t 1 ∈ S . Here (i) is just a restatemen t of the d eﬁnition, and th e passage f rom (i) to (ii) is by formally adding the ﬁxed p oint (on m orphisms, all elements in the set S 1 where a partial m ap f : S 1 99K S 2 is und eﬁ ned go into the added ﬁxed p oint). Denote by λ ∗ : DQ (Γ , Ab) → D (Γ + , Ab) the fu n ctor giv en by restriction with resp ect to the emb edding λ : Γ + → Q Γ. F or an y ﬁ nite set S , denote by T ( S ) = Z [ S ] the f ree ab elian group it generates. Then the corresp ondence S 7→ T ( S ) ob viously d eﬁnes an ob ject 33 T ∈ D Q (Γ , Z -mo d ): for an y map f : S 1 → S 2 , the m ap f ∗ : T ( S 1 ) → T ( S 2 ) is indu ced by f , an d the map f ∗ : T ( S 2 ) → T ( S 1 ) is the adjoin t map, f ∗ ([ s ]) = X s ′ ∈ f − 1 ( s ) [ s ′ ] for any elemen t s ∈ S 2 . The ob ject T ∈ D Q (Γ , Z -mo d) is additiv e in the sense of Deﬁnition 3.2. Restricting it to Γ + giv es an ob ject λ ∗ ( T ) ∈ F un(Γ + , Z -mo d ) wh ich we will denote b y the same letter T b y abus e of notation. Lemma 3.4. (i) F or any M q ∈ D (Ab) and any M ′ q ∈ D Q (Γ , Ab) , the natur al map (3.1) RHom q D Q (Γ , Ab) ( M ′ q , T ⊗ M q ) → RHom q D (Γ + , Ab) ( λ ∗ ( M ′ q ) , T ⊗ M q ) is an isomorphism. (ii) The functor D (Ab) → D Q (Γ , Ab) given by M q 7→ T ⊗ M q is the ful l emb e dding onto D Q add (Γ , Ab) . Pr o of. W e will need some semi-ob vious facts on the structur e of the category F un(Γ + , Z -mo d ) (see e.g. [Ka , S ection 3.2]). A standard set of p ro jectiv e generators of this cat egory is giv en by representable fu nctors T n , n ≥ 0, explicitly describ ed b y T n ( S ) = Z [( S ` { 1 } ) n ] . W e ha ve T n = T ⊗ n 1 . In particular, T 0 = Z , the constan t f unctor with v alue Z . Moreo v er, we ha ve a direct sum decomp osition T 1 = T ⊕ T 0 . Therefore the tensor p ow ers T ⊗ n are also pro jectiv e, and giv e another set of generators for the category F un(Γ + , Z -mo d). These generators are semi-orthogonal: w e ha v e Hom( T ⊗ n , T ⊗ m ) = 0 w hen n > m . In addition, Hom( T 0 , T ⊗ n ) = 0 f or an y n ≥ 1. Explicitly , (3.2) T ⊗ n ( S ) = Z [ S n ] for an y S ∈ Γ + . W e also note that w e ha v e Hom( T , T ) = Z , which immediately implies that M 7→ T ⊗ M giv es f u lly faithful em b edd ings Ab ⊂ F un(Γ + , Ab), D (Ab ) ⊂ D (Γ + , Ab). The category D Q (Γ , Ab ) is generated by represen table A ∞ -functors M S q of the form M S q ( S ′ ) = C q ( Q ( S, S ′ ) , Z ) ⊗ M 34 for all M ∈ Ab, S ∈ Γ. Therefore it is suﬃcien t to pro ve (i) for ob jects of this form. Fix a ﬁn ite set S ′ ∈ Γ and an ob ject M ′ ∈ Ab, and let M ′ q = M ′ S ′ q . Explicitly , we h a v e M ′ q ( S ) = M e S ∈ Γ C q (Σ e S , M ′ ⊗ Z [Γ( e S , S × S ′ )]) , where Σ e S is the group of automorphisms of the ﬁn ite set e S . T his direct sum decomp osition is not fu n ctorial with resp ect to S . How ev er, if we restrict to Γ + , then the increasing ﬁltration F q on λ ∗ ( M ′ q ) giv en by F n λ ∗ ( M ′ q )( S ) = M | e S |≤ n C q (Σ e S , M ′ ⊗ Z [Γ( e S , S × S ′ )]) is functorial ( | e S | d enotes the cardinalit y of the set e S ). The asso ciated graded quotien t is giv en by (3.3) gr F n λ ∗ ( M ′ q ) = C q (Σ e S , M ′ ⊗ Z [Γ( e S , S ′ )] ⊗ T ⊗ n ) , where w e hav e used (3.2), and e S is the set of cardinalit y n . By semi-ortho- gonalit y of the generators T ⊗ n , th is implies that RHom q ( gr F n λ ∗ ( M ′ q ) , T ⊗ M q ) = 0 for n 6 = 1, so th at RHom q ( λ ∗ ( M ′ q ) , T ⊗ M q ) = RHom q ( gr F 1 λ ∗ ( M ′ q ) , T ⊗ M q ) = RHom q D (Γ + , Ab) ( T ⊗ M ′ , T ⊗ M q ) ⊗ Z [ S ′ ] = RHom q Ab ( M ′ , M q ) ⊗ Z [ S ′ ] . Since M ′ q = M ′ S ′ q is r epresent able, this is exactly the left-hand side of (3.1), so that w e h a v e prov ed (i). As for (ii) , (i) immed iately implies that the functor D (Ab) → D Q (Γ , Ab) is a full emb edding, and since T ∈ D Q (Γ , Z -mo d ) is additiv e, we in fact ha v e a full em b eddin g D (Ab) ⊂ D Q add (Γ , Ab). T o prov e th at it is essentia lly sur- jectiv e, it suﬃces to pro ve th at its con tains all ob jects M q ∈ D Q add (Γ , Ab) whic h are concen trated in a single cohomologic al degree. But such ob j ects are Mac k ey fu nctors in the usual non-d eriv ed sense, so they are all of the form T ⊗ M , M ∈ Ab.  35 3.3 W reath pro ducts. Deﬁnition 3.3 is a DG v ersion of the ﬁ rst deﬁ- nition of a Mac ke y f unctor giv en in S ubsection 2.1. T o get a m ore explicit description of the category D M ( G ), w e need to s omeho w use the additiv- it y condition an d rep lace the A ∞ -categ ory B Γ G q with an A ∞ -categ ory whose ob jects are G -orbits, not all ﬁnite G -sets. W e do it b y using the stru cture of a “wreath pro du ct” of th e category Γ G of ﬁ nite G -sets. F or any small catego ry C , b y the wr e ath pr o duct C ≀ Γ of C with the catego ry Γ of ﬁn ite s ets we will un d erstand the category of pairs h S, { c s }i of a ﬁnite set S and a colle ction of ob jects c s ∈ C , one for eac h element s ∈ S , with m aps from h S, { c s }i to h S ′ , { c ′ s }i b eing a pair h f , f s i of a map f : S → S ′ and a collectio n of maps f s : c s → c ′ f ( s ) , one for eac h s ∈ S . By deﬁn ition, we ha ve a forgetful fun ctor ρ : C ≀ Γ → Γ, h S, { c s }i 7→ S . The fun ctor ρ is a ﬁbration; its ﬁ b er ρ S o v er a ﬁn ite set S ∈ Γ is canonically iden tiﬁed with C S , th e p ro du ct of copies of C indexed by element s of th e set S . In particular, C itself is naturally embedd ed into C ≀ Γ as the ﬁ b er o ve r the one-elemen t set pt ∈ Γ. W e will denote this em b eddin g by j C : C → C ≀ Γ. Irresp ectiv e of the prop erties of the category C , the category C ≀ Γ h as ﬁnite copro du cts. In a sense, it is obtained by formally adjoining ﬁnite copro ducts to C – th is can b e form ulated precisely as a certain universal prop erty of wreath pro ducts, but we will n ot need this. Another w a y to c haracterize C ≀ Γ b y a u niv ersal pr op erty is th e follo wing: for an y category C ′ ﬁb ered ov er Γ, an y functor f : C ′ pt → C from the ﬁ b er C ′ pt o v er the one- elemen t set pt ∈ Γ extends uniquely to a Cartesian f unctor C ′ → C ≀ Γ. W e will need the follo wing easy corollary of this fact. Lemma 3.5. A ssume given a smal l c ate gory C and an obje ct S ∈ C ≀ Γ . Then the c ate gory ( C ≀ Γ ) /S of obje cts S ′ ∈ C ≀ Γ e quipp e d with a map S ′ → S is natur al ly e quivalent to the wr e ath pr o duct ( C /S ) ≀ Γ , wher e C /S is the c ate gory of obje cts c ∈ C e qu ipp e d with a map j C ( c ) → S . Pr o of. Th e pro jection ( C ≀ Γ) /S → Γ wh ic h sends S ′ → S to ρ ( S ′ ) is obvio usly a ﬁ bration; the univ ersal prop erty of wr eath pro ducts then giv es a Cartesian comparison fun ctor ( C ≀ Γ) /S → ( C /S ) ≀ Γ , whic h is obviously an equiv alence on all the ﬁb ers, thus an equiv alence.  In the assumptions of Lemma 3.5, denote b y Q ≀ ( S ) ⊂ ( C ≀ Γ) /S the sub categ ory with the same ob ject s as ( C ≀ Γ) /S and those maps whic h are Cartesian with r esp ect to the ﬁb ration ( C ≀ Γ) /S → Γ. Equiv alen tly , w e h a v e 36 Q ≀ ( S ) ∼ = C /S ≀ Γ, where C /S ⊂ C /S is the sub category whose ob j ects are all ob jects in C /S , and whose maps are isomorphisms in C /S . More generally , give n t wo ob jects S 1 , S 2 ∈ C ≀ Γ , denote b y ( C ≀ Γ) / ( S 1 , S 2 ) the category of ob jects S ∈ C ≀ Γ equipp ed w ith maps S → S 1 , S → S 2 , let C / ( S 1 , S 2 ) ⊂ ( C ≀ Γ ) / ( S 1 , S 2 ) b e the ﬁb er of the pro jection ( C ≀ Γ) / ( S 1 , S 2 ) → Γ giv en by h S 1 ← S → S 2 i 7→ ρ ( S ), so that ( C ≀ Γ) / ( S 1 , S 2 ) ∼ = ( C / ( S 1 , S 2 )) ≀ Γ, and let C / ( S 1 , S 2 ) ⊂ C / ( S 1 , S 2 ) b e the group oid of isomorphisms of the catego ry C / ( S 1 , S 2 ). Denote Q ≀ ( S 1 , S 2 ) = C / ( S 1 , S 2 ) ≀ Γ ⊂ ( C ≀ Γ) / ( S 1 , S 2 ) , and assu me that the category C ≀ Γ has ﬁb ered pro d ucts. Th en these ﬁb ered pro du cts deﬁn e asso ciativ e comp osition functors (3.4) m : ( C ≀ Γ) / ( S 1 , S 2 ) × ( C ≀ Γ) / ( S 2 , S 3 ) → ( C ≀ Γ) / ( S 1 , S 3 ) whic h indu ce comp osition fu nctors on the catego ries Q ≀ ( − , − ). This allo ws to deﬁne a 2-category whic h we den ote b y Q ≀ C : it ob jects are the ob jects of C ≀ Γ, and its categories of morphisms are Q ≀ ( − , − ). Note that for an y S 1 , S 2 ∈ C ≀ Γ, we h a v e a n atural em b edding F C ( S 1 , S 2 ) : Q ( S 1 , S 2 ) → Q ≀ ( S 1 , S 2 ), and these embedd ings are compatible with ﬁb ered pro du cts, thus glue together to a 2-fun ctor F C : Q ( C ≀ Γ) → Q ≀ ( C ) . Both 2-catego ries here hav e the same ob jects and the same 1-morphism s; the diﬀerence is that the right- hand side has more 2-morphisms – the em- b edd in gs F C ( − , − ) are identi cal on ob jects and faithfu l, bu t n ot fu ll. W e pro v e r igh t a wa y one tec hnical r esults on the categories Q ≀ ( − , − ) whic h w e will need later on. Any ob ject c ∈ C ⊂ C ≀ Γ deﬁn es a functor j c : Γ → C ≀ Γ which s en ds a ﬁnite set S to the union of S copies of c . T his functor pr eserves ﬁb ered pro ducts, thus giv es a 2-functor j c : Q Γ → Q ( C ≀ Γ), a restriction f unctor j c ∗ : D Q ( C ≀ Γ , Ab) → D Q (Γ , Ab ), and a right-a djoint functor j c ∗ : DQ (Γ , Ab) → D Q ( C ≀ Γ , Ab). Lemma 3.6. F or any E q ∈ D Q (Γ , Ab) and any S ∈ C ≀ Γ , we have a natur al quasiisomorph ism j c ∗ E q ( S ) ∼ = C q ( Q ≀ ( c, S ) opp , ρ opp ∗ E q ) , wher e E q in the right-hand side is r estricte d to Γ opp ⊂ Q Γ and then pul le d b ack to Q ≀ ( c, S ) opp by the opp osite ρ opp to the pr oje ction ρ : Q ≀ ( c, S ) = C / ( c, S ) ≀ Γ → Γ . 37 Pr o of. Denote the embedd ing Γ opp → Q Γ by λ , and let λ ! : D (Γ opp , Ab) → D Q (Γ , Ab) b e the left-adjoin t functor to the restriction fun ctor λ ∗ . Then for an y M q ∈ D (Ab) w e hav e RHom q ( M q , C q ( Q ≀ ( c, S ) opp , ρ opp ∗ E q )) ∼ = RHom q ( λ ! ρ opp ! τ ∗ M q , E q ) , where τ : Q ≀ ( c, S ) opp → pt is the tautological pro j ection. Thus by ad j unc- tion, it suﬃces to pr o v e that λ ! ρ opp ! τ ∗ M q ∼ = j c ∗ M S q . T o constru ct a map f : λ ! ρ opp ! τ ∗ M q → j c ∗ M S q , it su ﬃces by adjunction to construct a m ap τ ∗ M q → ρ opp ∗ λ ∗ j c ∗ M S q , that is, a co mpatible s y s tem of maps M q → ρ opp ∗ λ ∗ j c ∗ M S q ( S ′ ) ∼ = H q ( Q ( S, j c ( ρ ( S ′ ))) , M q ) for an y S ′ ∈ Q ≀ ( c, S ); th ese maps are indu ced by ob vious tautological maps Z → H q ( Q ( S, j c ( ρ ( S ′ ))) , Z ). T o pr o v e that the map f is an isomorphism, w e ﬁrst need a wa y to con trol the functor λ ! . F or an y ﬁ nite set S ∈ Γ, let Γ /S b e the category of all ﬁnite sets S ′ equipp ed w ith a map S ′ → S and isomorphisms b et w een them, and let κ S : ( Γ /S ) opp → Γ opp b e the natural p r o jection w hic h sends [ S ′ → S ] to S ′ Then we ob viously h a v e λ ∗ Z q S ∼ = κ S ∗ Z ∈ D (Γ opp , Z -mo d) , and b y adjun ction, this yields a canonical isomorphism λ ! N q ( S ) ∼ = H q (( Γ /S ) opp , κ S ∗ N q ) , for an y N q ∈ D (Γ opp , Ab). No w apply this to N q = ρ opp ! τ ∗ M q . S ince ρ opp is a coﬁbration, we may compute κ S ∗ ρ opp ! b y base change ; this giv es a canonical isomorph ism H q (( Γ /S ) opp , κ S ∗ ρ opp ! τ ∗ M q ) ∼ = H q ( Q ′ ( c, S, S ) opp , M q ) , where Q ′ ( c, S, S ) is the catego ry obtained as th e C artesian pro d u ct Q ′ ( c, S, S ) − − − − → Γ /S   y   y Q ≀ ( c, S ) − − − − → Γ . 38 It remains to notice that the categ ory Q ′ ( c, S, S ) is canonical ly identiﬁed with Q ( S, j c ( S )), so that H q ( Q ′ ( c, S, S ) opp , M q ) ∼ = H q ( Q ( S, j c ( S )) opp , M q ) ∼ = j c ∗ M S q ( S ) . Th us the map f b ecomes an isomorp hism after ev aluating at ev ery ob ject S ∈ Q Γ, as required.  3.4 Additivization of the quotien t construction. No w let us consider again the functor T ∈ D Q (Γ , Z -mo d ) of Subsection 3.2, and let us restrict it to an ob ject T ∈ F u n(Γ , Z -mo d) b y the em b edd ing Γ → Q Γ. This T ∈ F un(Γ , Z -mo d) is isomorp h ic to the functor Z [1] represent ed b y the set [1] ∈ Γ with a sin gle element. F or an y S ∈ Γ, let τ S : Z → Z [ S ] = T ( S ) b e the diagonal em b edding. Th e maps τ S are not f u nctorial with resp ect to arbitrary maps of ﬁn ite sets S ; how ev er, th ey are fun ctorial with resp ect to isomorphisms. Thus if w e denote by Γ ⊂ Γ the category of ﬁnite sets and their isomorphisms , then we hav e a map of functors (3.5) τ : Z → t ∗ T , where Z ∈ F un( Γ , Z -mo d) is the constan t functor with v alue Z , and t : Γ → Γ is the em b ed ding. F or any small category C , we will d enote th e pullb ack ρ ∗ T ∈ F un( C ≀ Γ , Z ) with resp ect to the forgetful fu n ctor ρ : C ≀ Γ → Γ by the same letter T . By base c hange, we hav e T ∼ = ρ ∗ Z [1] ∼ = j C ! Z C , where Z C ∈ F un( C , Z ) is the constan t functor with v alue Z , and L i j C ! Z C = 0 for i ≥ 1. Therefore the n atural map (3.6) H q ( C , Z C ) → H q ( C ≀ Γ , T ) is an isomorphism. Assume that the wreath pro duct category C ≀ Γ has ﬁb ered pro du cts, and consid er the 2-cate gory Q ≀ ( C ). Then for an y S 1 , S 2 ∈ C ≀ Γ, the category Q ≀ ( S 1 , S 2 ) ∼ = C / ( S 1 , S 2 ) ≀ Γ carries a n atur al Z -mo d-v alued f u nctor T ∼ = ρ ∗ T ∼ = j S 1 ,S 2 ! Z ∈ F un( Q ≀ ( S 1 , S 2 ) , Z -mo d) , where j S 1 ,S 2 : C / ( S 1 , S 2 ) → Q ≀ ( S 1 , S 2 ) denotes the natur al em b eddin g. Moreo v er, for any S 1 , S 2 , S 3 ∈ C ≀ Γ, the comp osition f u nctors (3.4 ) induce functors m = ( j S 1 ,S 2 × j S 2 ,S 3 ) ◦ m : C / ( S 1 , S 2 ) × C / ( S 2 , S 3 ) → Q ≀ S 1 , S 2 , 39 and by deﬁnition, all the maps in the category C / ( S 1 , S 2 ) are inv ertible, so that the comp osition ρ ◦ m actually go es in to Γ ⊂ Γ. Th erefore the canonical m ap τ of (3.5) in duces a map τ S 1 ,S 2 ,S 3 : Z → m ∗ T . These maps are asso ciativ e on triple pr o ducts in the ob vious sense. By adjun ction, they induce maps (3.7) µ S 1 ,S 2 ,S 3 : T ⊠ T ∼ = ( j S 1 ,S 2 × j S 2 ,S 3 ) 1 Z → m ∗ T , and these maps are also asso ciativ e on triple pro ducts. As in the Sub section 3.1, the p ro cedure of Su bsection 1.6 giv es an A ∞ - catego ry B ≀C q with the same ob jects as C ≀ Γ by setting B ≀C q ( S 1 , S 2 ) = C q ( Q ≀ C ( S 1 , S 2 ) , T ) , with comp ositions induced by the 2-cate gory stru cture on Q ≀ C and the canonical maps µ of (3.7). Deﬁnition 3.7. The d eriv ed category D ( B ≀C q , Ab) of A ∞ -functors from B ≀C q to Ab is denoted b y D Q ≀ ( C , Ab ). W e note that f or an y S 1 , S 2 ∈ C ≀ Γ , the natur al functor F : Q ( S 1 , S 2 ) → Q ≀ ( S 1 , S 2 ) again go es in to Γ ⊂ Γ when comp osed with the pro j ection ρ : Q ≀ ( S 1 , S 2 ) → Γ; therefore the map τ of (3.5) indu ces maps Z → F ∗ T , and the 2-functor F C : Q ( C ≀ Γ) → Q ≀ C extends to an A ∞ -functor F C : B C ≀ Γ q → B ≀C q . The main comparison result that we wa nt to pr o v e is th e follo wing. Prop osition 3.8. The functor F ∗ C : D Q ≀ ( C , Ab ) → D Q ( C ≀ Γ , Ab ) induc e d by F C gives an e qu ivalenc e D Q ≀ ( C , Ab ) → D Q add ( C ≀ Γ , Ab) . Remark 3.9. The app earance of the functor T in the deﬁnition of the A ∞ -categ ory B ≀C q ( − , − ) lo oks lik e a tric k. On e moti v ation for this comes from top ology . The categories QC ( − , − ) are sym metric m on oidal categorie s with r esp ect to the disj oin t union. Applying group completio n to their classifying spaces |QC ( − , − ) | , one obtains inﬁnite lo op spaces. Then the complex B ≀C q ( − , − ) simp ly computes the homology of the corresp onding Ω- sp ectrum (as opp osed to B C ( − , − ), which computes the homology of the classifying sp ace |QC ( − , − ) | ). W e do not pr o v e th is since w e w ill not need it, but the pro of is n ot d iﬃcult (for example, it can b e d one along the lines of [Ka, Section 3.2]). 40 3.5 The pro ofs of the comparison re sult s. Before we pro ve P rop osi- tion 3.8, let us explain why it is useful. Assume giv en a small category C suc h that C ≀ Γ has ﬁb ered p ro du cts, and denote by (3.8) e B C q ⊂ B ≀C q the full sub category spann ed by C ⊂ C ≀ Γ. Let g D Q ( C , Ab ) b e the derive d catego ry of A ∞ -functors from e B C q to Ab. Lemma 3.10. R estriction with r esp e ct to the natur al emb e dding e B C q → B ≀C q induc es an e quivalenc e R : D Q ≀ ( C , Ab ) ∼ − → g D Q ( C , Ab ) . Pr o of. Let E : g D Q ( C , Ab) → D Q ≀ ( C , Ab ) b e the left-adjoin t fu nctor to R . I t suﬃces to prov e that E is essentially surjectiv e, and that the adjun ction map Id → R ◦ E is an isomorp h ism. The second f act is ob vious: b y adjunction, E sends repr esen table fun ctors in to rep r esen table fu n ctors, and since e B C q ⊂ B ≀C q is a full su b category , E d o es not c hange the spaces of map s b etw een them. It remains to pr ov e that E is essen tially s u rjectiv e. Sin ce D Q ≀ ( C , Ab) is generated b y represen table functors M S q , S ∈ C ≀ Γ, M ∈ Ab, it suﬃces to pro ve that all these functors lie in the essen tial image of E . By in duction on the cardinalit y | ρ ( S ) | , it suﬃces to pro ve that M S 1 ` S 2 q ∼ = M S 1 q ⊕ M S 1 q for an y M ∈ Ab, S 1 , S 2 ∈ C ≀ Γ. Indeed, assume giv en suc h an M and S 1 , S 2 . Explicitly , we h a v e M S 1 ` S 2 q ( S ′ ) = C q ( Q ≀ ( S, S ′ ) , T ) ⊗ M for an y S ′ ∈ C ≀ Γ. W e ha ve (3.9) Q ≀ ( S 1 ` S 2 , S ′ ) ∼ = Q ≀ ( S 1 , S ′ ) × Q ≀ ( S 2 , S ′ ) , and (3.10) T ∼ = ( π ∗ 1 T ⊠ π ∗ 2 Z ) ⊕ ( π ∗ 1 Z ⊠ π ∗ 2 T ) , where π 1 and π 2 are the pro jections on to the factors of the d ecomp osition (3.9), and Z mea ns the constant f u nctor with v alue Z . The direct sum 41 decomp osition (3.10) is fun ctorial with resp ect to S ′ , thus in duces a certain direct sum decomp osition M S 1 ` S 2 q ∼ = M 1 q ⊕ M 2 q . Here M 1 q is given by M 1 q ( S ′ ) = C q ( Q ≀ ( S 1 , S ′ ) × Q ≀ ( S 2 , S ′ ) , π ∗ 1 T ⊠ π ∗ 2 Z ) , and b y the K ¨ unneth formula, this is canonically quasiisomorph ic to M S 1 q ( S ′ ) ⊗ C q ( Q ≀ ( S 2 , S ′ ) , Z ) . Since the category Q ≀ ( S 2 , S ′ ) is a w reath pro duct, it has an initial ob ject, th us no h omology with constan t co eﬃcients, H q ( Q ≀ ( S 2 , S ′ ) , Z ) = Z , and we conclude that M 1 q ∼ = M S 1 q . Analo gously M 2 q ∼ = M S 2 q .  Lemma 3.10 sho ws th at Prop osition 3.8 allo ws one to get rid of the additivit y assump tion in the deﬁnition of the category D Q add ( C ≀ Γ , Ab) and reduces ev erything to an A ∞ -categ ory whose ob jects are those of C , not of C ≀ Γ. This has the follo wing immediate corollary . Corollary 3.11. Assume that the smal l c ate gory C itself has ﬁb er e d pr o d- ucts. Then ther e exists a natur al e quivalenc e of triangulate d c ate gories D Q ( C , Ab) ∼ = D Q ≀ ( C , Ab ) . Pr o of. By Lemma 3.10, it suﬃces to construct an equiv alence D Q ( C , Ab) ∼ = g D Q ( C , Ab ), or equiv alen tly , a qu asiisomorphism of A ∞ -categ ories B C q ∼ = e B C q . Both A ∞ -categ ories hav e the same ob j ects, the ob jects of the cate gory C . F or an y c 1 , c 2 ∈ C , the natural em b edd ing Q ( c 1 , c 2 ) → Q ≀ ( c 1 , c 2 ) = Q ( c 1 , c 2 ) ≀ Γ induces a map B C q ( c 1 , c 2 ) = C q ( Q ( c 1 , c 2 ) , Z ) → e B C ( c 1 , c 2 ) = C q ( Q ≀ ( c 1 , c 2 ) , T ) , these maps are ob viously compatible with comp ositions, and they are all quasiisomorphisms by (3.6).  W e will now pro ve Prop osition 3.8. T o do this, r ecall that for any ob ject c ∈ C , we ha v e the em b edding j c : Q Γ → Q ( C ≀ Γ) and the corresp ond ing restriction functor j c ∗ : DQ ( C ≀ Γ , Ab ) → D Q (Γ , Ab ). Moreo ver, j c ob viously extends to an embed d ing j c ≀ : Q ≀ ( pt ) → Q ≀ ( C ), and w e hav e a restrictio n functor j c ∗ ≀ : DQ ≀ ( C , Ab) → D Q ≀ ( pt , Ab). These are compatible with the functors F ∗ C of Prop osition 3.8: w e h av e a commutativ e d iagram (3.11) D Q (Γ , Ab) j c ∗ ← − − − − DQ ( C ≀ Γ , Ab) F ∗ pt x   x   F ∗ C D Q ≀ ( pt , Ab) j c ∗ ≀ ← − − − − D Q ≀ ( C , Ab ) . 42 Of course, b y Corollary 3.11 we ha ve D Q ≀ ( pt , Ab) ∼ = D Q ( pt , Ab) ∼ = D (Ab), and j c ∗ ≀ : DQ ≀ ( C , Ab ) → D (Ab) is simp ly the ev aluation at c ∈ Q ≀ ( C ). Lemma 3.12. Denote b y F C ! , F pt ! the functors left-adjoint to F ∗ C and F ∗ pt . Then the b ase c hange map F pt ! ◦ j c ∗ → j c ∗ ≀ ◦ F C ! obtaine d b y adjunction fr om (3.11) is an isomorphism. Pr o of. S ince D Q ( C ≀ Γ , Ab) is generated b y representa ble ob jects M S q , it suﬃces to prov e that RHom q ( F pt ! j c ∗ M S q , M ′ q ) ∼ = RHom q ( j c ∗ ≀ F C ! M S q , M ′ q ) for an y M q , M ′ q ∈ D (Ab), S ∈ C . By adjun ction, F C ! M q S = M F C ( S ) q , so that the righ t-hand sid e is isomorphic to RHom q Q ≀ ( S,c ) ( T , Z )) ⊗ RHom q ( M q , M ′ q ) . The left-hand side by adjun ction is isomorph ic to RHom q ( j c ∗ M S q , F ∗ pt M ′ q ) ∼ = RHom q ( j c ∗ M S q , M ′ q ⊗ T ) , whic h is isomorph ic to RHom q Q ≀ ( S,c ) opp ( Z , T )) ⊗ RHom q ( M q , M ′ q ) b y Lemm a 3.6. It remains to notice that RHom q Q ≀ ( S,c ) opp ( Z , T )) = R Hom q Q ≀ ( S,c ) ( T , Z )) iden tically , if b oth sides are u ndersto o d in the sense of (1.5).  Pr o of of Pr op osition 3.8. It su ﬃ ces to chec k that (i) the adju nction map F C ! ◦ F ∗ C → Id is an isomorp hism, so that F ∗ C is fully faithful, and (ii) an additive ob ject M q ∈ D Q add ( C ≀ Γ , Ab) w ith trivial F C ! ( M q ) is itself trivial, so that F ∗ C is essentially s urjectiv e. By Lemm a 3.10, it suﬃ ces to c hec k (i) after ev aluating on all c ∈ C ⊂ C ≀ Γ, and by Lemma 3.12, this amount to c hec king (i) with C replaced by pt . Analogously , the restriction functor j c ∗ ob viously send s D Q add ( C ≀ Γ , Ab) in to D Q add (Γ , Ab), and an ob ject M q ∈ D Q ( C ≀ Γ , Ab) with trivial j c ∗ ( M q ) for all c ∈ C is itself trivial; thus b y Lemma 3.12, (ii) can also b e only c hec k ed for C = pt . C onclusion: it suﬃces to prov e the Prop osition for C = pt . This has b een done already – com bine Lemma 3.10 and Corollary 3.11, on one hand, and Lemma 3.4, on the other hand.  43 3.6 Deriv ed Burnside rings. No w again ﬁx a ﬁnite group G and tak e C = Γ G , the catego ry of ﬁ nite G -sets. Deﬁne a functor ρ : Γ S → Γ by setting S 7→ S/G , the set of G -orbits on S . Th is fu nctor is a ﬁbration, and moreo v er, w e actually ha v e an identi ﬁcation Γ G ∼ = O G ≀ Γ, where O G is the catego ry of ﬁn ite G -orbits, and ρ is the tautological pr o jection O G ≀ Γ → Γ. Therefore we can also co ns id er the 2- category Q ≀ ( O G ) and the associated A ∞ -categ ory e B O G q of (3.8) whose ob j ects are ﬁnite G -orbits. T o simplify notation, denote B G q = e B O G q . then the follo wing is a reformulat ion of Prop osition 3.8 and Lemma 3.1 0 (with C = O G ). Prop osition 3.13. The triangulate d c ate gory D M ( G ) of derive d G -M ackey functors is e quivalent to the derive d c ate gory of A ∞ -functors B G q → Ab .  This Prop osition allo ws us to do some computations in the ca tegory D M ( G ); in particular, sp elling ou t the d eﬁnitions, w e can no w deﬁne a deriv ed v ersion of the Burns id e rin g A G . Let T = ρ ∗ T ∈ F un(Γ G , Z -mo d ), so that T ( S ) = Z [ S/G ] = Z [ S ] G . F or ev ery S 1 , S 2 ∈ Γ G , let µ S 1 ,S 2 : Z [ S 1 ] G ⊗ Z [ S 2 ] G = Z [ S 1 × S 2 ] G × G → Z [ S 1 × S 2 ] G b e the natural embedd ing. T ak en together, these maps give a map (3.12) µ : T ⊠ T → m ∗ T , where, as in Section 2, m : Γ G × Γ G → Γ G is the pro d uct functor (this map µ is of course the sp ecial case of (3.7) f or S 1 = S 2 = S 3 = { pt } ). Deﬁnition 3.14. The derive d Bu rnside ring A G q of the group G is the com- plex C q ( Γ G , T ), with the A ∞ -structure indu ced by th e Cartesian pr o duct of G -sets and the canonical map m T ∗ ◦ µ : C q (Γ G × Γ G , T ⊠ T ) → C q (Γ G × Γ G , m ∗ T ) → C q (Γ G , T ) , where µ is as in (3.12). By d eﬁ nition, A G q is a A ∞ -algebra ov er Z (a nd its homology algebra H q ( A G q ) is comm utativ e). It is isomorp hic to B G ([ G/G ] , [ G/ G ]), where [ G/G ] is the trivial G -orbit (the p oin t set pt with the trivial G -action). Lemma 3.15. Assume given a gr oup G . 44 (i) The 0 -th homolo gy H 0 ( A G q ) of the derive d B u rnside ring A G q is isomor- phic to the usual Burnside ring A G , and the 0 -th homolo gy H 0 ( B G q ) of the A ∞ -c ate gory B G q is isomo rphic to th e additive c ate gory B G of Subse ction 2.1. (ii) F or any two G -orbits S 1 , S 2 , we have a natur al qu asiisomorph ism (3.13) B G q ( S 1 , S 2 ) = M p ∈ ( S 1 × S 2 ) /G A H p q , wher e S p = [ G/H p ] ar e the c omp onents i n the de c omp osition (2.1) of the pr o duct S = S 1 × S 2 , and this quasiisomorphism induc es the isomorph ism (2.3) on 0 -th homolo gy. Pr o of. The quasiisomorphism (3.6) in our n ew notation r eads as A G q ∼ = C q ( O G , Z ) , so that the 0-th homology of A G q is the 0-th homology of the group oid O G of G -orbits; this is precisely the Bur nside rin g A G . The decomp osition (3.13) follo ws fr om the d ecomp osition Q ≀ ( S 1 , S 2 ) ∼ = Y p ∈ ( S 1 × S 2 ) /G Q ≀ ( S p ) b y the same argumen t as in the pro of of Lemma 3.10. Combining together (3.13), (2. 3 ) and the isomorphism H 0 ( A G q ) ∼ = A G giv es the isomorphism H 0 ( B G q ) ∼ = B G . It remains to pro ve that this isomorphism is compatible with the comp ositions; this is a straight forward c hec k whic h we lea v e to the reader.  4 W a ldhausen-t yp e d escription. The construction of the triangulat ed category DM ( G ) of derive d Ma c key functors giv en in the last Sectio n is very explicit but somewhat deﬁcien t, since it relies on explicit resolutions and A ∞ metho ds. W e will no w give a more in v ariant constru ction. T o do th is, w e mo dify the quotien t construction Γ G 7→ Q Γ G of Sub s ection 3.1 in a wa y which is similar to the p assage from Quillen’s Q -construction to W aldh ausen’s S -construction in algebraic K - theory . 45 4.1 Heuristic explanation. Let us ﬁrst explain informally what w e are going to d o (this Subs ection is purely h euristic and m a y b e skipp ed, formally , nothing in the r est of the p ap er dep end s on it). Recall that a simplicial set X is b y deﬁ nition a con trav ariant functor fr om the category ∆ of ﬁnite non- empt y totally ordered sets to the category of all sets. F or any non-negativ e in teger n ≥ 1, we will d enote b y [ n ] ∈ ∆ th e set with n elemen ts, or, to b e sp eciﬁc, the set of all int egers i , 1 ≤ i ≤ n ; w e will also denote X i = X ([ i ]) for an y i ≥ 1. An Ab -value d she af M on X is a collecti on of (i) a fun ctor M n : X n → Ab for every n ≥ 1, and (ii) a map M ( f ) : M n ′ → X ( f ) ∗ M n for ev ery map f : [ n ] → [ n ′ ], sub ject to standard compatibilit y conditions. Here in (i), w e treat the set X n as a discrete categ ory , so that M n is eﬀectiv ely j ust an X n -graded ob ject in Ab; in (ii), X ( f ) : X n ′ → X n is v alue of the fun ctor X : ∆ opp → Sets on the map f . There is the f ollo wing con venien t w a y to pack together these data (i), (ii), and also the compatibilit y conditions. Let us not only treat the s ets X n as discrete categories, but also treat X as a f unctor ∆ opp → Cat. Th en w e can apply the Grothendiec k construction of Subsection 1.2. The r esult is a category S ( X ) ﬁb er ed ov er ∆; explicitly , ob jects in S ( X ) are pairs h n, x ∈ X n i , and a map from h n, x ∈ X n i to h n ′ , x ′ ∈ X n i is giv en b y a map f : [ n ] → [ n ′ ] suc h that X ( f )( x ′ ) = x . One immediately chec ks that in this n otation, a sheaf M on X is exa ctly the same thing as a fu nctor M : S ( X ) → Ab. No w assume giv en a small category C . Recall th at to C , one canonically asso ciates a simp licial set N ( C ) called the nerve of C , and to any functor E ∈ F un( C , Ab), one asso ciates an Ab-v alued sheaf e E on the nerv e – in other w ords, we hav e a natural embedd ing (4.1) F un( C , Ab ) → F un( S ( N ( C )) , Ab ) . Explicitly , N ( C ) n is the set of diagrams c 1 → · · · → c n in the category C ; the fu nctor e E : S ( N ( C )) → Ab s en ds a diagram c 1 → · · · → c n to E ( c n ). F or any map f : [ n ′ ] → [ n ], the m ap N ( C )( f ) sends a diagram c 1 → · · · → c n to the diagram c f (1) → · · · → c f ( n ′ ) , and the corresp onding map e E ( f ) : E ( c f ( n ′ ) ) → E ( c n ) is induces by the natural map c f ( n ′ ) → c n . The embed ding (4.1) is fully faithfu l bu t not essential ly surj ective – not all sh ea v es on N ( C ) come fr om fu nctors E ∈ F un( C , Ab). Indeed, for any sheaf of th e form e E , the m ap e E ( f ) is an isomorph ism whenev er the map 46 f : [ n ′ ] → [ n ] sends n ′ to n – that is, pr eserv es the last elements of the totally ordered sets. Ho wev er, this is the only condition: the essen tial image of embed d ing (4.1) consists of suc h M ∈ F un( S ( N ( C )) , Ab ) that M ( f ) is a quasiisomorphism wheneve r f p reserv es the last elemen ts. Indeed, it is easy to see that such a sheaf M is completely deﬁn ed b y th e follo wing part of (i) , (ii): (i) the fun ctor M 1 : N ( C ) 1 → Ab, and (ii) the map (4.2) M ( s ) : s ∗ M 1 → M 2 ∼ = t ∗ M 1 , where the maps s, t : [1] → [2] send 1 ∈ [1] to 1 ∈ [2], resp. to 2 ∈ [2], sub ject to some conditions. Th is is exactly the same as a functor from C to Ab: M 0 giv es its v alues on ob j ects of the catego ry C , and the map s ∗ M 1 → t ∗ M 1 enco des the action of its morph isms. The same co mparison result is true on th e lev el of derived categories: the functor (4.3) D ( C , Ab ) → D ( S ( N ( C )) , Ab) is a fully faithful em b edding, and its essential image consists of su c h M q ∈ B ( S ( N ( C )) , Ab ) that M q ( f ) is a quasiisomorph ism whenev er f preserves the last elemen ts. No w, the observ ation that w e would lik e mak e is that small 2-categ ories also ha v e n erv es. Of course, the nerve N ( C ) of a 2-categ ory C is a simp licial catego ry rather th an a simplicial set; ho wev er , the Grothend iec k construc- tion still app lies, and the ﬁ b ered category S ( N ( C )) / ∆ is p erfectly w ell de- ﬁned. T he only new thing is that the ﬁb ers of this ﬁ bration are no longer discrete. As it happ ens, this c hanges n othing: if we deﬁne the triangulated catego ry D ( C , Ab ) by the b ar-construction, as in Subsection 1.6, then w e still ha ve a fully faithfu l emb edding (4.3), w ith the same c haracterization of its essen tial image. Ho wev er , since S ( N ( C )) is a us u al category , not a 2-cate gory , the righ t-hand side D ( S ( N ( C )) , Ab ) can b e deﬁ n ed in the usual w a y , with no r ecourse to 2-categorical mac hinery and A ∞ metho ds. Return no w to the s itu ation of S ubsection 3.1 – assume giv en a sm all catego ry C which h as ﬁb ered p r o ducts. In prin ciple, w e could simple app ly the ab o ve d iscussion to the 2-categ ory Q ( C ). Ho w eve r, w e will actually 47 do something sligh tly diﬀeren t. Namely , w e can deﬁne n erv es in an ev en greater generalit y : ins tead of a 2-catego ry , w e can consider a “categ ory in catego ries”, wh er e not only morphisms form a catego ry , not ju st a set, but also obje cts do th e same th ing. It is p robably n ot wo rth the eﬀort to axiomatize the situation; instead, let us giv e the sp eciﬁc example we will use. Let C [2] b e the category of arrows c 1 → c 2 in C , with morphisms giv en b y Cartesian squ ares c 1 − − − − → c 2   y   y c ′ 1 − − − − → c ′ 2 . Consider the opp osite category C [2] opp . W e ha ve t w o pro jections s , resp. t from C [2] opp to C opp whic h sen d an arro w c 1 → c 2 to its source c 1 , resp. its target c 2 . Moreo v er, comp osition of the arro ws deﬁn es a fu nctor m : C [2] opp × C opp C [2] opp → C [2] opp , where in the left-hand side, the pro jection to C opp is by t in the left factor, and by s in the right factor. This is our “category in catego ries”: C opp is its catego ry of ob jects, C [2] opp is its category of m orphisms, and the fun ctor m deﬁnes the comp osition. A f unctor from this “category in catego ries” to Ab is, then, giv en by the follo wing data: (i) a fun ctor M 1 : C opp → Ab, and (ii) a map s ∗ M 1 → t ∗ M 1 (the analog of the action map (4.2)). W e now notice that these data deﬁne exactly a f unctor M from the quotien t catego ry Q C of S ection 3 to Ab . The fu nctor M 1 deﬁnes the v alues of M at ob jects of the category C and the actio n of the maps f ∗ , f a morp h ism of C , while the action m ap s ∗ M 1 → t ∗ M 1 adds the maps f ∗ to the picture. This description breaks the symmetry b et w een f ∗ and f ∗ manifestly present in the deﬁnition of the category Q C , but this is n ot necessarily a bad thing: constructions suc h as the tensor p ro du ct of S ubsection 2.3 also break th is symmetry , and one m ay hop e th at some constructions actually lo ok b etter in the new descrip tion. As we sh all see, this is indeed the case. W e no w r esume rigorous exp osition. W e start by constructing th e nerve of our “catego ry in categories”; we will d enote this nerv e simp ly b y S C . 48 4.2 The S -construction. Assume giv en a small ca tegory C which has ﬁb ered pro ducts. F or ev ery intege r n ≥ 2, let C [ n ] b e the category of dia- grams of the form c 1 → · · · → c n in the cate gory C — in other words, C [ n ] is the catego ry of C -v alued functors from the tot ally ordered set [ n ] with n elements consider ed as a small categ ory in the usual w a y . Sending a di- agram c 1 → · · · → c n to c n ∈ C deﬁnes a pro j ection C [ n ] → C , and s ince C has ﬁb ered pro ducts, th is pro jection is a ﬁbr ation in the sense of [SGA]. Denote b y C [ n ] ⊂ C [ n ] the sub category with the same ob jects as C [ n ] and the maps whic h are Cartesian with r esp ect to the ﬁbration C [ n ] → C . Th us for example, C [2] is the category whose ob jects are arro ws in C , and whose morphisms are Cartesian squares. As usual, denote b y ∆ the category of non-empty ﬁn ite totally ordered sets. Th en for any order-p r eserving map f : [ n ] → [ n ′ ], w e ha ve a natural functor f ∗ : C [ n ′ ] → C [ n ] . Moreo ver, we can also treat it as a functor f ∗ : C [ n ′ ] opp → C [ n ] opp b et wee n the op p osite cate gories. Then the col lection C [ n ] opp with the trans ition functors f ∗ forms a simplicial cate gory , th at is, a categ ory ﬁb ered o ve r ∆. W e d enote this catego ry by S C . Explicitly , ob jects of S C are pairs of a ﬁ nite non -emp t y totally order ed set [ n ] and a diagram c 1 → · · · → c n in C ; a map from h [ n ] , c 1 → · · · → c n i to h [ n ′ ] , c ′ 1 → · · · → c ′ n ′ i is giv en by an order-pr eserving map f : [ n ] → [ n ′ ] and a collection of maps f i : c ′ f ( i ) → c i in C , one for eac h i ∈ [ n ], such th at the square c ′ f ( i ) f i − − − − → c i   y   y c ′ f ( j ) f j − − − − → c j is comm utativ e and Cartesian for an y i, j ∈ [ n ], i ≤ j . In p articular, w e ha v e a natural em b ed ding C opp → S C which sends c ∈ C opp to h [1] , c i . W e can deﬁn e a natural pro jection functor ϕ : S C → Q C as follo ws: an ob ject h [ n ] , c 1 → · · · → c n i ∈ S C go es to c n ∈ Q C , and a map h f , { f i }i : h [ n ] , c 1 → · · · → c n i → h [ n ′ ] , c ′ 1 → · · · → c ′ n ′ i go es to a m ap represented by the diagram c n f n ← − − − − c ′ f ( n ) − − − − → c ′ n ′ . W e w ill sa y that a map f : [ n ′ ] → [ n ] in the category ∆ is sp e cial if it is an isomorphism b et w een [ n ′ ] and a ﬁnal segment of the ordinal [ n ] (in other w ords, w e hav e f ( l ) = n + l − n ′ for an y l ∈ [ n ′ ]). W e will sa y that a map h f , { f i }i in the category S C is sp e ci al if it is Cartesian with resp ect to the 49 ﬁbration S C → ∆ opp — that is, all the maps f i are in ve rtible — and the comp onent f : [ n ′ ] → [ n ] is s p ecial. One c hecks easily that if a map f is sp ecial, then ϕ ( f ) is inv ertible in Q C . Moreo v er, w e will sa y that a fun ctor F : S C → Ab is sp ecial if F ( f ) is in ve rtible for ev ery sp ecial f . Then setting F 7→ F ◦ ϕ giv es an equiv alence F un( Q C , Ab) ∼ = F un sp ( S C , Ab ) , where F u n sp ( S C , Ab ) ⊂ F un( S C , Ab) is th e full s ub categ ory spanned b y sp ecial functors. In this sense, the catego ry Q C is obtained from the category S C by inv erting all sp ecial maps. Explicitly , a functor M ∈ F un( S ( C ) , Ab ) is giv en by the follo wing data: (i) A fun ctor M n ∈ F un( C [ n ] opp , Ab) for eve ry [ n ] ∈ ∆. (ii) A transition map (4.4) ( f ∗ ) ∗ M n → M n ′ for an y map f : [ n ] → [ n ′ ], w here f ∗ : C [ n ′ ] opp → C [ n ] opp is the transition functor corresp ond ing to the map f . The fu nctor M is sp ecial if the transition map M ( f ) is an isomorp hism for ev ery sp ecial map f (it is clearly su ﬃcien t to c hec k this f or the maps f : [1] → [ n ], f (1) = n ). Consider no w the d eriv ed category D ( S C , Ab). Deﬁnition 4.1. An ob ject M ∈ D ( S C , Ab) is called sp e cial if f or an y sp ecial map f in the category S C , the corresp onding map M ( f ) is a quasiisomor- phism. The full su b category in D ( S C , Ab) spann ed b y s p ecial complexes is denoted by D S ( C , Ab) ⊂ D ( S C , Ab). The tr iangulated categ ory D S ( C , Ab) obvio usly con tains the deriv ed cat- egory D (F un sp ( S C , Ab )) ∼ = D ( Q C , Ab ); how ev er, there is no reason wh y they should b e same, or indeed, ev en why the fun ctor D ( Q C , Ab) ⊂ D S ( C , Ab) should b e f u ll and faithful. It turns out that in general, it is not. T h is is exactly the diﬀerence b et w een the naive deriv ed category D ( Q ( C ) , Ab ) and the triangulated category D Q ( C , Ab) of Sub section 3.1 . Theorem 4.2. F or any smal l c ate gory C which has ﬁb er e d pr o ducts, we have a natur al e qu ivalenc e D S ( C , Ab) ∼ = D Q ( C , Ab) . 50 4.3 Digression: complemen tary pairs. T o prov e Theorem 4.2, we need to dev elop some combinatoria l mac hinery for inv erting sp ecial maps in the category S C . Unfortunately , the class of sp ecial maps do es n ot adm it a calculus of fr actions in the u sual sense. Ho w eve r, there is the follo wing substitute. Deﬁnition 4.3. Assume giv en a catego ry Φ and t wo classes of maps P , I in Φ. Th en h P , I i is a c omplementar y p air if the f ollo wing conditions are satisﬁed. (i) The classes P an d I are closed under the comp osition and cont ain all isomorphisms. (ii) F or ev ery ob ject b ∈ Φ, the category Φ I b of diagrams i : b ′ → b , i ∈ I has an in itial ob ject i b : ι ( b ) → b . (iii) Ev ery map f in Φ factorizes as f = p ( f ) ◦ i ( f ), p ( f ) ∈ P , i ( f ) ∈ I , and su c h a factorization is u nique u p to a u nique isomorphism. (iv) Ev ery d iagram b 1 p ← b i → b 2 in Φ with p ∈ P , i ∈ I ﬁts in to a co cartesian square b p − − − − → b 1 i   y   y i ′′ b 2 p ′ − − − − → b 12 in Φ with sp ecial p ′ ∈ P , i ′ ∈ I . Remark 4.4. If Φ has an initial ob ject 0, then (ii) follo ws from (iii) (the map i c : ι ( c ) → c is a part of the decomp osition 0 → ι ( c ) → c of the map 0 → c ). Remark 4.5. A complementary pair in the sense of Deﬁn ition 4.3 is an example of a factorization system in the sense of [Bou]. Assume gi ven a small category Φ and a complementary pair h P , I i of classes of m aps in Φ. Let D I (Φ , Ab) b e the full s ub category in the d eriv ed catego ry D (Φ , Ab) spann ed by those M q ∈ D (Φ , Ab) for whic h M q ( i ) is a quasiisomorphism for an y i ∈ I . Let R Φ b e the cat egory of diagrams c 1 ← c → c 2 in Φ w ith maps i 1 : c → c 1 , i 2 : c → c 2 in the class I . Let π i , π 2 : R Φ → Φ , b e the pro jections w hic h send c 1 ← c → c 2 to c 1 , resp. to c 2 , and den ote by Sp : D (Φ , Ab) → D (Φ , Ab) th e fu nctor give n b y Sp = L q π 1! π ∗ 2 . 51 Lemma 4.6. Ther e exists a map of functors ψ : Id → Sp such that ψ ◦ Sp = id , and the map ψ : M q → Sp ( M q ) for an obje c t M q ∈ D (Φ , Ab) is a quasi- isomorph ism if and only if M q lies in D I (Φ , Ab) ⊂ D (Φ , Ab) . In other words, Sp : D (Φ , Ab) → D (Φ , Ab ) is the canonical pro jection on to the left-admissible full sub catego ry D I (Φ , Ab) — that is, the comp o- sition of the embedd ing D I (Φ , Ab) ⊂ D (Φ , Ab ) and th e left-adjoin t fu nctor D (Φ , Ab) → D I (Φ , Ab). Pr o of. Both pro jections π 1 , π 2 ha v e a common section δ : Φ → R Φ wh ic h sends c ∈ Φ to the diagram c ← c → c with identit y maps. The isomorp hism Id ∼ = δ ∗ π ∗ 2 induces by adjun ction a map L q δ ! → π ∗ 2 ; the functorial map ψ is obtained by applying L q π 1! to this map. T o pr o v e the required p rop erties of the map ψ , we start by noting th at due to the conditions (iii) and (iv ) of Deﬁnition 4.3, the p ro jection π 1 : R Φ → Φ is a coﬁb ration. Therefore f or an y M q ∈ D (Φ , Ab) and any ob j ect c ∈ Φ, we h a v e a canonical base change isomorph ism Sp ( M q )( c ) ∼ = H q ( R Φ c , π ∗ 2 M q ) , where R Φ c ⊂ R Φ is the ﬁb er of the coﬁbration π 1 — that is, the cate gory of diagrams c ← c ′ → c ′′ in Φ with maps i ′ : c ′ → c , i ′′ : c ′ → c ′′ lying in I . This ﬁb er R Φ c pro jects to the category Φ I c of Deﬁnition 4.3 (iii) by f orgetting c ′′ and i ′′ ; denote b y F Φ c the ﬁb er of this pro jection ov er the initial ob j ect ι ( c ) ∈ Φ I c . Explicitly , F Φ c is the category of ob jects c ′′ ∈ Φ equipp ed with a map i : ι ( c ) → c ′′ , i ∈ I . By deﬁnition, for an y sp ecial map i ′ : c ′ → c , the map i c : ι ( c ) → c canonically factorizes as i c = i ′ ◦ i o : ι ( c ) → c ′ → c . Moreo v er, it is easy to see that if we hav e tw o maps i 1 , i 2 ∈ Φ suc h that i 1 ∈ I and i 1 ◦ i 2 ∈ I , then i 2 ∈ I (tak e its factorizatio n i 2 = i ′ 2 ◦ p of (iii), comp ose with i 1 to obtain a factorizat ion of i 1 ◦ i 2 , use the uniqueness to deduce that p is inv ertible). This implies that ι ( c ) ∼ = ι ( c ′ ) canonically , and i o = i c ′ . T h en sending a diagram c ← c ′ → c ′′ to i ′′ ◦ i c ′ : ι ( c ) → c ′′ deﬁnes a pro jection π : R Φ c → F Φ c , and this pro j ection is right-a djoint to the em b edding ι : F Φ c → R Φ c . The pro jection π 2 factors as π 2 = π ◦ π : R Φ c → F Φ c → Φ, wher e π : F Φ c → Φ send s a d iagram ι ( c ) → c ′′ to c ′′ . Therefore we hav e H q ( R Φ c , π ∗ 2 M q ) ∼ = H q ( R Φ c , π ∗ π ∗ M q ) ∼ = H q ( R Φ c , L q ι ! π ∗ M q ) ∼ = H q ( F Φ c , π ∗ M q ) , so that Sp ( M q )( c ) ∼ = H q ( F Φ c , π ∗ M q ). The canonical map ψ : M q ( c ) → Sp ( M q )( c ) is then induced by the inclusion pt → F Φ c whic h sends the p oin t to the diagram h ι ( c ) → c i ∈ F Φ c . 52 No w, for ev ery map i : c → c ′ , i ∈ I , the categories F Φ c and F Φ c ′ are canonically equiv alen t; therefore the map Sp ( M q )( c ) → Sp ( M q )( c ′ ) is a quasiisomorphism, and we ha v e Sp ( M q ) ∈ D I (Φ , Ab). And if w e know in adv ance that M q ∈ D I (Φ , Ab), then the p ullbac k π ∗ M q ∈ D ( F Φ C , Ab) is constan t, s o that Sp ( M q )( c ) ∼ = M q ( c ) ⊗ H q ( F Φ c , Z ). Since F Φ c has an initial ob ject, we hav e H q ( F Φ c , Z ) ∼ = Z .  F or any tw o ob jects c 1 , c 2 ∈ Φ, let Q I ( c 1 , c 2 ) b e the cate gory of diagrams c 1 → c ← c 2 in Φ suc h that the map c 1 → c lies in the class I , and the map c 2 → c lies in the class P . Then for an y thr ee ob jects c 1 , c 2 , c 2 , w e ha v e natural comp osition fu nctors (4.5) Q I ( c 1 , c 2 ) × Q I ( c 2 , c 3 ) → Q I ( c 1 , c 3 ) giv en by the co cartesian squares whic h exist by Deﬁnition 4.3 (iv). These functors are asso ciativ e. S ay that an ob ject c ∈ Φ is simple if the canonical map i c : ι ( c ) → c is an isomorphism, and let Q I (Φ) b e the 2-catego ry whose ob jects are simple ob jects in Φ, and w h ose morphism categorie s are Q I ( c 1 , c 2 ), with comp osition fun ctors (4.5 ). App lying th e p ro cedur e of Subs ection 1.6, construct an A ∞ -categ ory B I q (Φ) with the same ob jects as Q I (Φ), and with morphism s giv en b y B I q ( c 1 , c 2 ) = C q ( Q I ( c 1 , c 2 ) , Z ) . Let D Q I (Φ , Ab) b e the triangulated category of A ∞ -functors from the A ∞ - catego ry B I q (Φ) to Ab. Prop osition 4.7. Ther e exists a natur al e qu ivalenc e of tria ngulate d c ate- gories D Q I (Φ , Ab ) ∼ = D I (Φ , Ab) . Pr o of. T o deﬁn e a comparison functor ϕ : D I (Φ , Ab) → D Q I (Φ , Ab), let e Φ b e Φ with a f orm ally added initial ob ject ∅ , and declare that the map ∅ → c is in the class P for ev ery c ∈ Φ. Then all the conditions of Deﬁnition 4.3 are satisﬁed for e Φ, so that we can form the 2-catego ry Q I ( e Φ). F or any s im p le c ∈ Φ, th e category Q I ( ∅ , c ) is the category F Φ c of the pro of of Lemma 4.6, and for an y simp le c 1 , c 2 ∈ Φ, w e ha ve th e comp osition fun ctors m c 1 ,c 2 : F Φ c 1 × Q I ( c 1 , c 2 ) = Q I ( ∅ , c 1 ) × Q I ( c 1 , c 2 ) − → F Φ c 2 = Q I ( ∅ , c 2 ) . 53 A functor M ∈ F un(Φ , Ab) giv es b y restriction a fu n ctor M c = π ∗ 2 M ∈ F un( F Q c , Ab ) for every simple c ∈ Φ. F or eve ry simp le c 1 , c 2 ∈ Φ, we ha v e a natural map µ c 1 ,c 2 : M c 2 ⊠ Z → m ∗ c 1 ,c 2 M c 1 , and these maps are asso ciativ e in the obvious sense. W e deﬁ ne ϕ ( M ) q ∈ D Q I (Φ , Ab) by ϕ ( M ) q ( c ) = C q ( F Φ c , M c ) , with the A ∞ -functor structure indu ced b y the fu nctors m and the maps µ . This extends to a f u nctor ϕ : D (Φ , Ab) → D Q I (Φ , Ab). Comparin g the d eﬁnitions of th e functors Sp and ϕ , w e see that the canonical map ψ : M q → Sp ( M q ) induces a quasiisomorphism ϕ ( M q ) ∼ = ϕ ( Sp ( M q )) for an y M q ∈ D (Φ , Ab), so that th e functor ϕ factors through the pro jection Sp : D (Φ , Ab) → D I (Φ , Ab). T o sho w that the fun ctor ϕ : D I (Φ , Ab) → D Q I (Φ , Ab) is an equiv a- lence, we ha ve to pro v e that it is full, faithful, and essentially surjectiv e. The triangulated catego ry D Q I (Φ , Ab) is generated b y the represent able A ∞ -functors M Q c , for all simple c ∈ Φ and complexes M ∈ Ho (Ab), giv en b y M Q c ( c ′ ) = B I ( c, c ′ ) ⊗ M . The tr iangulated category D Q I (Φ , Ab) is gen- erated by the ob jects S p ( M c ), for all c ∈ Φ and M ∈ Ho (Ab), wh ere M c ∈ F un(Φ , Ab) is the r ep resen table fun ctor giv en b y M c ( c ′ ) = Z [Φ( c, c ′ )] ⊗ M . Therefore it suﬃces to pr o v e that (i) for any c ∈ Φ and M ∈ Ho (Ab), w e ha v e Sp ( M c ) ∼ = Sp ( M ι ( c ) ), (ii) for any simple c ∈ Φ and M ∈ Ho (Ab), w e hav e ϕ ( M c ) ∼ = M Q c , and (iii) for any simple c, c ′ ∈ Φ an d M ∈ Ho (Ab), th e natural map RHom q ( Sp ( M c ) , Sp ( M c ′ )) → RHom q ( M Q c , M Q c ′ ) induced b y the fun ctor ϕ is a qu asiisomorphism. T o p r o v e (i), n ote that b y adjun ction, we hav e RHom q ( Sp ( M c ) , M ′ ) ∼ = RHom q ( M c , M ′ ) ∼ = RHom q ( M , M ′ ( c )) for an y c ∈ Φ , M ∈ Ho (Ab), and an y M ′ ∈ D I (Φ , Ab) ⊂ D (Φ , Ab). In particular, RHom q ( Sp ( M c ) , M ′ ) ∼ = RHom q ( Sp ( M ι ( c ) , M ′ ). Since this is true for any M ′ ∈ D I (Φ , Ab), this im p lies (i). T o p ro ve (ii), note that for an y 54 simple c ′ ∈ Φ, we ha ve an embed d ing j : Q I ( c, c ′ ) → F Φ c ′ , and by Deﬁni- tion 4.3 (iii) there is a natural isomorph ism M c | F Φ c ′ ∼ = j ! M const , where M const ∈ F u n( Q I ( c, c ′ ) , Ab) is the constan t fu nctor with v alue M . Therefore Sp ( M c )( c ′ ) ∼ = C q ( F Φ c ′ , j ! M const ) ∼ = C q ( Q I ( c, c ′ ) , M const ) = B I ( c, c ′ ) ⊗ M , as required. Finally , f or (iii), note that by adjunction RHom q ( Sp ( M c ) , Sp ( M c ′ )) ∼ = RHom q ( M c , Sp ( M c ′ )) ∼ = Sp ( M c ′ )( c ) , so that (iii) follo ws fr om (ii).  4.4 The comparison theorem. No w again, assume giv en a small cat- egory C wh ic h has ﬁb ered pro du cts, and tak e Φ = S C . Sa y that a m ap h f , { f i }i in S C is c o-sp e cial if th e underlyin g map f : [ n ] → [ n ′ ] sends the ﬁrst element in [ n ] to the ﬁr st elemen t in [ n ′ ], f (1) = 1. Let I , P b e the classes of sp ecial, resp. co-sp ecial m ap s in th e category S C . Lemma 4.8. The p air h P , I i is a c omplem entary p air i n the sense of Deﬁ- nition 4.3. Pr o of. Deﬁn ition 4.3 (i) is ob vious. F or (ii), let c = h [ n ] , c 1 → · · · → c n i ; then ι ( c ) = h [1] , c n i , with th e obvious map i c : ι ( c ) → c . F or (iii), tak e a map f = h f , { f l }i : h [ n ] , { c l }i → h [ n ′ ] , { c ′ l }i , and let n 0 = f (1) ∈ [ n ′ ]; then the decomp osition is giv en by h [ n ] , { c l }i − − − − → h [ n ′ − n 0 ] , { c ′′ l }i − − − − → h [ n ′ ] , { c ′ l }i , where c ′′ l = c ′ l + n 0 , 1 ≤ l ≤ n ′ − n 0 , with the obvi ous maps. Finally , for (iv), let b = h [ n ] , { b l }i , b 1 = h [ n 1 ] , { b 1 l }i , b 2 = h [ n 2 ] , { b 2 l }i . Then b 12 = h [ n 12 ] , { b 12 l }i is giv en by n 12 = n 1 + n 2 − n , b 12 l = b 1 l + n − n 2 for l = n − n 2 + 1 , . . . , n 12 , and for l = 1 , . . . , n − n 2 , b 12 l is obtained as the ﬁb ered pro duct b 12 l − − − − → b 2 l   y   y b 1 1 − − − − → b 1 ∼ = b 2 n 2 +1 − n 55 in the category C .  Pr o of of The or em 4.2. By Lemma 4.8, Prop osition 4.7 can b e applied to the catego ry S C . An ob ject c = h [ n ] , c i i is s imple if and only if n = 1, th us simple ob jects in S C are the same as ob jects in C ⊂ S C . Then comparing the deﬁ- nitions of A ∞ -categ ories B C q of Su b section 3.1 and B I q ( S C ) of Sub section 4.3, w e see that it remains to pro v e the follo wing: for an y c 1 , c 2 ∈ C ⊂ S C , there exists a natural quasiisomorph ism η c 1 ,c 2 : C q ( Q I ( c 1 , c 2 ) , Z ) ∼ = C q ( Q ( c 1 , c 2 ) , Z ) , and these quasiisomorphisms extend to an A ∞ -functor. The functor ϕ : S C → Q C of S u bsection 3.1 obviously extends to a 2-functor Q I ( S C ) → QC ; this give s maps η c 1 ,c 2 whic h form an A ∞ -functor. T o pro v e that η c 1 ,c 2 is a quasiisomorphism, it remains to n otice that the fun ctor ϕ : Q I ( c 1 , c 2 ) → QC ( c 1 , c 2 ) h as a left-adjoint wh ic h s en ds a diagram c 1 ← c → c 2 in C to the diagram c 1 → h [2] , c 2 → c 1 i ← c 2 in S C .  4.5 Additivit y . Assume no w give n a small category C such th at the wreath p ro du ct C ≀ Γ h as ﬁb ered pro d ucts. T h en we can form the category S ( C ≀ Γ) and th e deriv ed category D S ( C ≀ Γ , Ab). By d eﬁnition, ( C ≀ Γ) opp is embed d ed into S ( C ≀ Γ) as the ﬁb er ov er [1] ∈ ∆, so that w e hav e the restriction fun ctor D S ( C ≀ Γ , Ab) → D (( C ≀ Γ) opp , Ab). Deﬁnition 4.9. An ob ject M q ∈ D S ( C ≀ Γ) is called additive if its restriction to D (( C ≀ Γ) opp , Ab) is additive in th e sense of Deﬁn ition 3.2. Prop osition 4.10. Assume that the smal l c ate gory itself C has ﬁb er e d pr o d- ucts. Then the f ul l sub c ate gory D S add ( C ≀ Γ , Ab) ⊂ D S ( C ≀ Γ , Ab) sp anne d by additive obje cts is c anonic al ly e q u ivalent to the c ate gory D S ( C , Ab ) . Pr o of. T he statemen t immediately follo ws from Theorem 4.2, Prop osi- tion 3.8 and Corollary 3.11.  Deﬁnition 4.11. Assume giv en a small category C suc h that C ≀ Γ has ﬁb ered pro du cts. Th en the full sub categ ory D S add ( C ≀ Γ , Ab) ⊂ D S ( C ≀ Γ , Ab) spanned b y add itiv e ob jects is d enoted by D S ( C , Ab). W e n ote that by Prop osition 4.10, this is consistent w ith our earlier Deﬁnition 4.1. 56 Remark 4.12. F or the sak e of metho dological purit y , it w ould b e n ice to ha v e a direct pro of of Prop osition 4.10 whic h do es not use the material of S ection 3 . Unfortun ately , I was not able to ﬁ nd such a pro of. All I could come up with essentially rep eats the pro of of Prop osition 3.8, with additional complications (wh ic h arise b ecause w e do not hav e a W aldh au s en- t yp e interpretation of the category D Q ≀ ( C , Ab) of Deﬁnition 3.7). In the particular case C = O G , the category of ﬁn ite G -orbits for a group G , w e h a v e C ≀ Γ ∼ = Γ G . Combining Theorem 4.2 and Pr op osition 4.10 , we get the follo wing. Corollary 4.13. F or any ﬁnite gr oup G , let O G b e the c ate gory of G - orbits. Then the c ate gory D M ( G ) of derive d G -M acke y functors is natur al ly e q u iv- alent to the c ate gory D S ( O G , Ab) of Deﬁnition 4.11.  5 F unctoriali t y and pro ducts. W e will n o w describ e some basic p rop erties of d eriv ed Mac k ey functors, mostly analogous to the material in S ubsection 2.2 and Subsection 2.3. 5.1 F unctorialit y . Assume give n t w o small categories C , C ′ suc h that C ≀ Γ and C ′ ≀ Γ h a v e ﬁ b ered p ro du cts. T hen an y fu nctor γ : C ′ → C ≀ Γ un iquely extends to a copro duct-preserving fun ctor γ : C ′ ≀ Γ → C ≀ Γ. I n either of the t w o constructions of the catego ry D S ( − , Ab) ∼ = D Q add ( − ≀ Γ , Ab) w e ha v e the follo wing obvious functorialit y prop ert y . • I f the extended functor γ : C ′ ≀ Γ → C ≀ Γ preserve s ﬁb ered pr o ducts, then w e hav e a n atural r estriction fun ctor γ ∗ : D S ( C , Ab) ∼ = D Q add ( C ≀ Γ , Ab) → D S ( C ′ , Ab) ∼ = D Q add ( C ′ ≀ Γ , Ab) , and the left-adjoin t ind uction f unctor γ ! : D S ( C ′ , Ab) → D S ( C , Ab). It turns out that this yields the deriv ed versions of b oth the fun ctor Ψ and the functor Φ of Subs ection 2.2. In fact, the functor Ψ has already app eared in Section 3 under a diﬀeren t name. F or an y ob ject c ∈ C , w e hav e its embedd ing functor j c : pt → C ≀ Γ, and the extended functor j c : Γ → C ≀ Γ pr eserves ﬁ b ered pr o ducts f or semi-trivial reasons. 57 Deﬁnition 5.1. The naive ﬁxe d p oint functor Ψ c is the functor Ψ c = j c ∗ : DS ( C , Ab ) → D S ( pt , Ab) ∼ = D (Ab) . In the Mac k ey functor case C = O G , c = [ G/H ], the fun ctor Ψ c is the deriv ed version of the functor Ψ H of S ubsection 2.2. W e can also sligh tly r eﬁ ne the construction. Let h c i ⊂ C b e the group oid of ob jects in C isomorp hic to c , and inv ertible maps b etw een them. Th en j c extends to an embedd ing e j c : h c i → C ≀ Γ wh ose natural extension e j c : h c i ≀ Γ → C ≀ Γ still preserve s ﬁb ered pro d u cts. W e denote the corresp onding restriction fun ctor by (5.1) e Ψ c = e j c ∗ : DS ( C , Ab ) → D S ( h c i , Ab) ∼ = D (Aut( c ) , Ab) . It tak es v alues in the category DS ( h c i , Ab), whic h is obviously equiv alen t to the deriv ed catego ry D (Aut( c ) , Ab) of Ab-v alued r ep resen tations of the group Aut( c ) of automorphisms of the ob j ect c . T o p r o ceed further, we need to imp ose a restriction on the category C . Deﬁnition 5.2. A cate gory C is called Hom - ﬁnite if the set of maps C ( c, c ′ ) is ﬁnite for an y tw o ob jects c, c ′ ∈ C . Assume that th e catego ry C is Hom -ﬁ nite. Then so is the wreath pro du ct C ≀ Γ, and an ob ject c also repr esents a fun ctor τ c : C → Γ, τ c ( c ′ ) = C ( c, c ′ ) . Its natural extension τ c : C ≀ Γ → Γ is r ep resen ted b y the same ob ject c , thus preserve s ﬁ b ered pro du cts. Deﬁnition 5.3. The ge ometric ﬁxe d p oints functor Φ c is given b y Φ c = τ c ! : D S ( C , Ab) → D S ( pt , Ab) ∼ = D (Ab) . In th e case C = O G , c = [ G/H ], this is the d eriv ed v ersion of the fu nctor Φ H of Sub s ection 2.2. The adjoin t functor τ ∗ c : D (Ab) → D S ( C , Ab ) can b e explicitly describ ed as as f ollo w. Denote T c = τ ∗ c ( T ) ∈ F u n sp ( S ( C ≀ Γ) , Z -mo d) ⊂ D S ( C , Z -mo d) , where T ∈ F un sp ( S (Γ) , Z -mo d ) is the natural ge nerator of the category D Q add (Γ , Z -mo d) introdu ced in Lemma 3.4. Then f or any M q ∈ D (Ab), w e ha v e τ ∗ c ( M q ) ∼ = M q ⊗ T c . 58 This can also b e reﬁned to incorp orate Au t( c ). Indeed, the group Aut( c ) ob viously acts on the ob ject T c . W e deﬁne the inﬂation fu nctor Inﬂ c : F un(Aut( c ) , Ab ) → F un( S ( C ≀ Γ) , Ab) b y Inﬂ c ( M ) = ( M ⊗ T c ) Aut( c ) , and we note that the f unctor R q Inﬂ c sends D (Aut( c ) , Ab ) in to th e category D S ( C , Ab) ⊂ D ( S ( C ≀ Γ) , Ab). By abuse of notation, w e will d rop R q and d enote the deriv ed fun ctor sim p ly b y Inﬂ c : D (Aut( c ) , A ) ∼ = D ( h c i , Ab) → D S ( C , Ab) , and w e den ote by e Φ c : DS ( C , Ab ) → D (Au t( c ) , Ab ) ∼ = D ( h c i , Ab) its left-adjoin t. Remark 5.4. The names naive and ge ometric attac hed to ﬁxed p oin ts func- tors come from equiv arian t stable homotop y theory , whose part in th e s tory w e will explain in Section 8. It seems that the geomet ric ﬁ xed p oin t fu nc- tor is the more imp ortan t of the t w o; thus from n ow, “ﬁxed p oin t functor” without an adjectiv e will mean th e functor Φ c . W e can further reﬁn e this construction by the follo wing observ ation: the group Aut( c ) acts not only on the functor T c , but on the f unctor τ c , too – for any c ′ ∈ C ≀ Γ, Aut( c ) naturally act on the ﬁn ite set τ c ( c ) = C ( c, c ′ ). Th us w e actually ha ve a functor b τ c : C → Γ Aut( c ) = O Aut( c ) ≀ Γ which induces functors b Φ c = b τ c ! : D S ( C , Ab) → D S ( O Aut ( c ) , Ab) = D M (Aut( c ) , Ab ) , c Inﬂ c = b τ ∗ c : DM (Aut( c ) , Ab ) → D S ( C , Ab) . In the case C = O G , c = [ G/H ], w e ha v e Aut( c ) = N H /H , wh er e N H ⊂ G is the normalizer of the subgroup G . T hen the extended inﬂation fu nctor tak es the f orm c Inﬂ N G : DM ( N , Ab) → D M ( G, Ab ) , where N = N H /H . If H ⊂ G is n orm al, so that N H = G and N = G/H , this is the derived v ersion of the fully faithful functor Inﬂ N G of Su bsection 2.2. As we will pro ve in S ection 7, Lemma 7.13, in this case the fun ctor c Inﬂ N G is fully faithful. 59 5.2 Pro ducts. T o introd uce a tensor pr o duct on the category D S ( C , Ab), w e imp ose the same additional assu mption as in Su bsection 2.3 – we requir e that the small categ ory C has a terminal ob j ect. Then so do es C ≀ Γ; of course, we still assum e that C ≀ Γ has ﬁb ered pro d ucts, so that altogether, it has all ﬁnite limits (in particular, pr o ducts). It is con v enient to iden tify D S ( C , Ab) ∼ = D Q add ( C ≀ Γ , Ab) and u se the A ∞ metho ds of Section 3. By the deﬁnition of the 2-cate gory Q ( C ≀ Γ), we ha v e an isomorphism Q (( C ≀ Γ) × ( C ≀ Γ)) ∼ = Q ( C ≀ Γ) × Q ( C ≀ Γ) . Since the bar construction is compatible with p ro du cts, th is means that for an y t w o ob jects M q , M ′ q ∈ D Q ( C ≀ Γ , Ab), we ha v e a well-deﬁned b o x p ro du ct M q ⊠ M ′ q ∈ D Q (( C ≀ Γ) × ( C ≀ Γ) , Ab ) W e also h a v e th e p ro du ct fun ctor m : ( C ≀ Γ) × ( C ≀ Γ) → C ≀ Γ , and since it preserves ﬁb ered p r o ducts, it induces the restriction fu n ctor m ∗ : D S ( C , Ab) ∼ = D Q add ( C ≀ Γ , Ab ) ⊂ D Q ( C ≀ Γ , Ab) − → D Q (( C ≀ Γ) × ( C ≀ Γ) , Ab ) . Consider the left-adjoin t functor m add ! : DQ (( C ≀ Γ) × ( C ≀ Γ) , Ab) → D S ( C , Ab) . Deﬁnition 5.5. The tensor pr o duct M q ⊗ M ′ q of t w o ob jects M q , M ′ q ∈ D S ( C , Ab) is given by M q ⊗ M ′ q = m add ! ( M q ⊠ M ′ q ) . Under our assum p tions on Ab, th is ob viously give s a wel l-deﬁn ed sym- metric tensor pro duct stru cture on th e triangulated category D S ( C , Ab ). W e hav e the follo wing result on compatibilit y b et w een tensor p ro du cts and ﬁ xed p oin ts. Assu me that the small categ ory C is Hom-ﬁnite, so that for an y ob ject c ∈ C , w e ha ve a well -deﬁn ed geometric ﬁ x ed p oin ts fu n ctor Φ c : DS ( C , Ab ) → D (Ab). 60 Prop osition 5.6. F or any c ∈ C , the ge ometric ﬁxe d p oints f u nctor Φ c is a tensor functor – that is, for any two obje cts M q , M ′ q ∈ D S ( C , Ab) we have an isomorph ism Φ c ( M q ⊗ M ′ q ) ∼ = Φ c ( M q ) ⊗ Φ c ( M ′ q ) , and this isomorphism is functorial in M q and M ′ q . Pr o of. The functor τ c : C ≀ Γ → Γ, b eing representa ble, comm utes with pro du cts, so that we hav e a commutativ e d iagram Q ( C ≀ Γ) × Q ( C ≀ Γ) m − − − − → Q ( C ≀ Γ) τ c × τ c   y   y τ c Q (Γ) × Q (Γ) m − − − − → Q (Γ) , and m ∗ ◦ τ ∗ c ∼ = ( τ c × τ c ) ∗ ◦ m ∗ . By adjunction, this giv es a functorial isomor- phism Φ c ( M q ⊗ M ′ q ) ∼ = Φ c ( M q ) ⊗ Φ c ( M ′ q ) , where th e pr o duct in the right -hand side is tak en in the cate gory D S ( pt , Ab) in the sense of Deﬁnition 5.5. I t r emains to notice that the canonical equiv- alence of Lemma 3.4 iden tiﬁes this pr o duct with the tensor pro duct in the deriv ed category D (Ab).  Remark 5.7. The situation with the tensor pro duct in DQ ( C , Ab) is some- what remin iscent of the tensor pro duct of D -mo dules on a smo oth algebraic v ariety X : while a D -mo dules is simply a sheaf of m o dules o v er the rin g D X of diﬀerential op erators on X , this ring is not comm utativ e, and one has to tak e sp ecial care to deﬁne a symmetric tensor pr o duct on the category D X -mo d. The p ro du ct b ecomes m uc h more natur al if one passes to a Koszul- dual in terpretation and replaces D -mo d ules with shea v es of DG m o dules o v er the de Rham complex Ω q X . W e w ill obtain an analogous Koszul-dual in terpretation of Mac k ey fu nctors in Section 6. 5.3 Induction. W e ﬁn ish this section with one more fun ctorialit y resu lt. It seems th at it do es not app ear in the standard theory of Mac k ey functors; ho we ve r, it will b e v ery useful in Section 8. W e again ﬁx a small category C suc h that C ≀ Γ has ﬁb ered p ro du cts. By the deﬁnition of the 2-category Q ( C ≀ Γ), w e ha v e a natural functor C ≀ Γ → Q ( C ≀ Γ) w hic h is identica l on ob jects, and send s a map f : S ′ → S 61 to th e diagram S ′ id ← S ′ f → S . Comp osing this with th e natur al embedd ing j C : C → C ≀ Γ, we ob tain a functor q : C → Q ( C ≀ Γ) . Since the 2-c ategory Q ( C ≀ Γ) is manifestly self-dual, w e also ha v e the opp osite functor q opp : C opp → Q ≀ Γ, and the corresp ondin g restriction fun ctor q opp ∗ : D S ( C , Ab) ∼ = D Q add ( C ≀ Γ , Ab) ⊂ D Q ( C ≀ Γ , Ab ) → D ( C opp , Ab) . W e w ill call the adjoin t fu nctor q opp ! : D ( C opp , Ab) → D S ( C , Ab) the induction functor , and we will sa y that an ob ject M q ∈ D S ( C , Ab) is induc e d if M q ∼ = q opp ! ( E q ) for some E q ∈ D ( C opp , Ab). If the target category Ab is a s ymmetric tens or category , then the func- tor category F un( C opp , Ab) has a n atural “p oin t wise” tensor pro duct th at induces a tensor pro d uct on th e d eriv ed category D ( C opp , Ab). Prop osition 5.8. (i) Assume that the c ate gory C is Hom -ﬁnite. Then for any obje ct c ∈ C and any E q ∈ D ( C opp , Ab) , we have an isomor- phism Φ c ( q opp ! ( E q )) ∼ = E q ( c ) , and this isomorphism is functorial i n E q . (ii) Assume that the c ate gory C has a terminal obje ct and that the tar- get c ate gory Ab is a symmetric tensor c ate gory. Then the induction functor q opp ! is a tensor functor – we have an isomorphism q opp ! ( E q ⊗ E ′ q ) ∼ = q opp ! E q ⊗ q opp ! E ′ q for any E q , E ′ q ∈ D ( C opp , Ab) , and this isomorphism is functorial in E q and E ′ q . Pr o of. F or (i), note that b y adjun ction, it suﬃ ces to p ro v e that q opp ∗ T c ∈ F un( C opp , Z -mo d) is the ob ject co-represent ed by c ∈ C opp , that is, T c ( c ′ ) ∼ = Z [ C opp ( c ′ , c )] = Z [ C ( c, c ′ )] . This is the deﬁnition of the ob ject T c . F or (ii), we rep eat the argumen t of Sub section 2.3. Firstly , as noted sev eral times already , we hav e a natural equiv alence b et w een D ( C opp , Ab) 62 and th e fu ll sub category D add (( C ≀ Γ) opp , Ab) ⊂ D (( C ≀ Γ) opp , Ab) spann ed by additiv e ob jects. Secondly , we hav e a comm utativ e diagram ( C ≀ Γ) opp × ( C ≀ Γ) opp m − − − − → ( C ≀ Γ) opp q opp   y   y q opp Q ( C ≀ Γ) × Q ( C ≀ Γ) m − − − − → Q ( C ≀ Γ) . Therefore it suﬃces to pro ve that for any E q , E ′ q ∈ D ( C opp , Ab) ∼ = D add (( C ≀ Γ) opp , Ab) we ha ve a f unctorial isomorphism m ! ( E q ⊠ E ′ q ) ∼ = E q ⊗ E ′ q . This is ob vious: the pro d uct functor m : ( C ≀ Γ) opp × ( C ≀ Γ) opp → ( C ≀ Γ) opp is left-adjoin t to the diagonal em b edd ing δ : ( C ≀ Γ) opp → ( C ≀ Γ) opp × ( C ≀ Γ) opp , so that m ! ∼ = δ ∗ , and b y deﬁ nition, E q ⊗ E ′ q ∼ = δ ∗ ( E q ⊠ E ′ q ).  6 Categories of Galois t yp e. W e will now giv e y et another descrip tion of the category D S ( C , Ab) of Deﬁ- nition 4.11, to complement those giv en in Section 3 and Section 4. It is based on an explicit DG mo del, as in Section 3; how ev er, this n ew DG mo del is more economical and more con ve nient for applications. It will require some additional assu m ptions on C (whic h are satisﬁed in the Mac k ey fu nctor case C = O G ). I n addition, f r om no w w e w ill assume that the target category Ab is the catego ry of mo d ules o v er a r ing, so that Lemma 1.7 applies. 6.1 Galois-t yp e categories and ﬁxed p oin ts. W e b egin by imp osing our conditions on the small category C . Deﬁnition 6.1. A category C is lattic e- like if it is Hom-ﬁ nite, and all its morphisms are surj ectiv e. Lemma 6.2. In a lattic e- like c ate gory C , every right-inverse f ′ : c → c ′ to a morphism f : c ′ → c is also a left-i nverse. Mor e over, every endomorp hism f : c → c of an obje ct c ∈ C i s invertible. Pr o of. F or the ﬁrst claim, note that ( f ′ ◦ f ) ◦ f ′ = f ′ ◦ ( f ◦ f ′ ) = f ′ ◦ id = id ◦ f ′ ; since f ′ is surjectiv e, this imp lies f ′ ◦ f = id . F or the second claim, note that since f is s urjectiv e, th e natur al m ap C ( c, c ) −◦ f − − − − → C ( c, c ) 63 is injectiv e. Since C ( c, c ) is a ﬁn ite set, this map must also b e su r jectiv e, so that there exists f ′ ∈ C ( c, c ) su c h that f ′ ◦ f = id .  Deﬁnition 6.3. A category C is of Galois typ e if it is lattice-li ke, and the wreath pro duct C ≀ Γ has ﬁb ered p ro du cts. Example 6.4. Here are some examples of catego ries of Galois type. (i) C = O G , the category of ﬁn ite G -orbits for a group G . (ii) C is the category of ﬁnite separable extensions of a ﬁeld k . (iii) C is the category Γ − of ﬁ nite s ets and surjectiv e maps b et we en th em. Of course, Galois theory shows th at (ii) is a p articular case of (i), thus the name “Galois-t yp e”; h o w eve r, the v ery in teresting example (iii) is not of this form. Assume giv en a small category C of Galois t yp e, and consider the cat- egory D S ( C , Ab) of Deﬁnition 4.1 1. Since C is Hom-ﬁnite, we can apply the constructions of Subsection 5.1; thus for any c ∈ C , w e ha ve the ob ject T c ∈ D S ( C , Z -mo d), the in ﬂ ation functor Inﬂ c : D ( h c i , Ab) → D S ( C , Ab) and the left-adjoin t ﬁxed p oint functor (6.1) e Φ c : DS ( C , Ab ) → D ( h c i , Ab) . In general, computing the functors e Φ c explicitly seems to b e r ather diﬃcult; ho we ve r, in the case of Galois-t yp e categories, there is a drastic simpliﬁ- cation. Namely , let q : C → C ≀ Γ → Q ( C ) b e the natural embedd ing of Subsection 5. 3, and let q ∗ : D S ( C , Ab ) → D ( C , Ab) b e the corresp onding restriction fun ctor. Moreov er, deﬁne an emb edding ν c : D ( h c i , Ab) → D ( C , Ab) b y ν c ( M q )([ c ′ ]) = ( M q ( c ′ ) , c ′ ∈ h c i , 0 , otherwise , and let e ϕ c : D ( C , Ab) → D ( h c i , Ab) b e the left-adjoin t functor. Th en we ha v e the follo wing. 64 Prop osition 6.5. We have a fu nctorial isomorphism e Φ c ∼ = e ϕ c ◦ q ∗ . In ord er to p ro ve th is pr op osition, we need some preliminary r esults and constructions. Consider the simplicial catego ry S ( C ≀ Γ ) / ∆. Note that by the deﬁn ition of the w r eath pro d uct, w e hav e a natural ﬁbr ation S ( C ≀ Γ) → Γ, so that altoge ther, S ( C ≀ ∆) is ﬁb ered o v er ∆ × Γ. Let S ( C ≀ Γ) ⊂ S ( C ≀ Γ) b e the sub category w ith the same ob jects, and those morph isms whic h are Cartesian with resp ect to the ﬁb ration S ( C ≀ Γ) → ∆ × Γ. By d eﬁnition, S ( C ≀ Γ) is also ﬁ b ered ov er ∆, and we h a v e a natur al Cartesian emb edding (6.2) η : S ( C ≀ Γ) → S ( C ≀ Γ) . While the category S ( C ≀ Γ) is n ot of the form S ( C ′ ) for some s mall category C ′ , most of the material of S ection 4 still applies to it, w ith appropriate c hanges. In p articular, sa y that a map f in S ( C ≀ Γ) is sp e cial , resp. c o- sp e cial if so is η ( f ). Then it is easy to c heck that the classes of sp ecial and co-sp ecial maps form a complemen tary pair in the s ense of Deﬁnition 4.3. W e can th us consider the full su b category D S ≀ ( C , Ab) ⊂ D ( S ( C ≀ Γ) , Ab) spanned by sp ecial fu nctors. Since η is a Cartesian fun ctor whic h r esp ects sp ecial maps, we ha v e a restriction functor η ∗ : D S ( C ≀ Γ , Ab) → D S ≀ ( C , Ab ), and the adjoint fun ctor R q η ∗ : D ( S ( C ≀ Γ) , Ab) → D ( S ( C ≀ Γ) , Ab) sends sp ecial fu nctors into sp ecial functors. Moreo v er, w e also hav e a 2-category description of D S ≀ ( C , Ab). Namely , the ﬁb er S ( C ≀ Γ) 1 o v er [1] ∈ ∆ is ob viously equiv alent to ( C ≀ Γ) opp (where C ⊂ C is the sub category with the same ob jects and inv ertible maps b et w een them). Consider the 2-category Q ( C ≀ Γ) of S ection 3, and let Q ( C ≀ Γ) ⊂ Q ( C ≀ Γ) b e the 2- sub category with th e same ob jects, those 1-morphisms c 1 ← c → c 2 for whic h the map c → c 1 actually lies in C ≀ Γ ⊂ C ≀ Γ, and all 2-morphisms b et we en th em. O n e c hec ks easily that this condition is compatible with the p ullbac ks, so that we indeed ha v e a w ell-deﬁned 2-cate gory . Applying the mac hin ery of Sub section 1.6, we pro du ce th e A ∞ - catego ry B q and the d eriv ed catego ry D Q ( C ≀ Γ , Ab) of A ∞ -functors f r om B q 65 to Ab. Note that we h a v e a natural fun ctor ( C ≀ Γ) opp → Q ( C ≀ Γ ), thus a restriction functor D Q ( C ≀ Γ , Ab) → D (( C ≀ Γ) opp , Ab). By deﬁ n ition, we also ha v e a natural em b ed ding η : Q ( C ≀ Γ) → Q ( C ≀ Γ). Lemma 6.6. Ther e exists a natur al e quivalenc e D S ≀ ( C , Ab) ∼ = D Q ( C ≀ Γ , Ab) which is c omp atible with η ∗ and with the natur al r estrictions to ( C ≀ Γ) opp . Pr o of. Same as Theorem 4.2.  No w ﬁx an ob ject c ∈ C , and deﬁne a 2-fun ctor ε c : Q ( C ≀ Γ) → Q ( h c i ≀ Γ) as follo ws. On ob j ects, ε c sends h S, { c s }i ∈ C ≀ Γ to the formal un ion of those comp onents c s whic h are isomorp hic to c . On morphism s, ε c sends a diagram c 1 ← c ′ → c 2 to ε c ( c 1 ) ← ε c ( c ′ ) → ε c ( c 2 ) (if c 1 , c 2 ∈ h c i ≀ Γ, this is w ell-deﬁned). Moreo ve r, let T c = ε ∗ c ( T c h c i ) ∈ D Q ( C ≀ Γ , Z -mo d) b e the p ullbac k of th e stand ard ob ject T c h c i ∈ D Q ( h c i ≀ Γ , Z -mo d), T c h c i ( c ′ ) = Z [( h c i ≀ Γ)( c, c ′ )], an d deﬁn e an embedd ing ν ≀ c : D ( h c i , Ab) → D Q ( C ≀ Γ , Ab) b y ν ≀ c ( M q ) = ( T c ⊗ M q ) Aut( c ) (since T c ( c ′ ) is a f ree Aut( c )-mo dule for any c ′ ∈ C , the group Aut( c ) has no higher cohomolog y w ith co eﬃcients in T c ⊗ M q , so that taking Aut( c )- in v arian ts is wel l-deﬁn ed on the lev el of derived categ ories). Lemma 6.7. We have a natur al isomorphism T c ∼ = R q η ∗ T c ∈ D S ( C ≀ Γ , Ab) ∼ = D Q ( C ≀ Γ , Ab) . Pr o of. T o construct a map T c → R q η ∗ T c , it suﬃces b y adjunction to construct a m ap η ∗ T c → T c . Suc h a map is ob vious – it is iden tical on c ′ ∈ h c i ≀ Γ ⊂ C ≀ Γ, and 0 otherwise. T o pro v e that the in duced map T c → R q η ∗ T c is an isomorphism, use Theorem 4.2 and Lemma 6.6 to pass to the W aldhausen-type int erpr etation. T hen since b oth T c and R q η ∗ T c are sp ecial, it suﬃces to c hec k that T c → R q η ∗ T c is an isomorphism on the ﬁb er ( C ≀ Γ) opp o v er [1] ∈ ∆. This is again obvi ous: on this ﬁb er, T c is the fun ctor represent ed b y c ∈ ( C ≀ Γ) opp , and T c is the f unctor rep resen ted by the same c considered as an ob ject in S ( C ≀ Γ) 1 = ( C ≀ Γ) opp ⊂ ( C ≀ Γ) opp .  66 It r emains to analyse the 2-categ ory Q ( C ≀ Γ). As in Sub section 3.2, consider the fu ll sub catego ry in Q ( C ≀ Γ) wh ose 1-morphisms are diagrams S ← S 1 → S ′ with injective map S 1 → S . Th is is actually a 1-ca tegory whic h w e denote b y C ≀ Γ + b y abu se of notation. The embedd ing q : C → Q ( C ≀ Γ) factors through the embedd ing η : Q ( C ≀ Γ) b y m eans of the em b edd ings C j − − − − → C ≀ Γ + λ − − − − → Q ( C ≀ Γ) , and w e ha v e and the co rr esp onding restriction functors λ ∗ , j ∗ . Sa y that an ob j ect M q ∈ D S ≀ ( C , Ab ) ∼ = D Q ( C ≀ Γ , Ab) is additive if its restriction to C ≀ Γ opp is ad d itiv e in the sense of Deﬁnition 3.2. Lemma 6.8. A ssume given two obje cts M q , M ′ q ∈ D Q ( C ≀ Γ , Ab ) . If M ′ q is additive, then the natur al map (6.3) Hom( M q , M ′ q ) → Hom( λ ∗ M q , λ ∗ M ′ q ) is an isomorphism. If M q is also additive, then the natur al map Hom( λ ∗ M q , λ ∗ M ′ q ) → Hom( j ∗ λ ∗ M q , j ∗ λ ∗ M ′ q ) is also an isomorphism. Pr o of. As in the pro of of Lemma 3.4, w e ﬁ rst note that the catego ry F un( C ≀ Γ + , Z -mo d ) is generated by represen table fun ctors Z S , S ∈ C ≀ Γ + . F or any suc h S = ` c s , c s ∈ C , we h a v e Z S ∼ = O Z c s , and for an y c ∈ C , w e ha v e a canonical direct su m decomp osition Z c = T c ⊕ Z . This induces direct sum decomp ositions Z S = M S ′ ⊂ S T S ′ , where the ob jects T S = O T c s , S = a c s ∈ C ≀ Γ + giv e a smaller set of pro j ectiv e generators of the categ ory F un ( C ≀ Γ + , Z -mo d ). F or any S 1 , S 2 ∈ C ≀ Γ + , we hav e Z S 1 ` S 2 ∼ = Z S 1 ⊗ Z S 2 ; therefore the additivit y 67 condition on an ob ject E q ∈ D ( C ≀ Γ + , Ab) means that the natur al map Z S 1 ⊕ Z S 2 → Z S 1 ` S 2 = Z S 1 ⊗ Z S 2 induces an isomorphism RHom q ( Z S 1 ⊗ Z S 2 ⊗ M q , E q ) → RHom q ( Z S 1 ⊗ M q , E q ) ⊕ RHom q ( Z S 2 ⊗ M q , E q ) for any M q ∈ D (Ab). In terms of the generators T S , this is equ iv alent to sa ying th at RHom q ( T S ⊗ M q , E q ) = 0 unless the decomp osition S = ` c s has exactly one term, S = c for some c ∈ C ⊂ C ≀ Γ + : (6.4) RHom q ( T S ⊗ M q , E q ) = ( RHom q ( M , E q ( c )) , S = c ∈ C ⊂ C ≀ Γ + , 0 , otherwise . (If C = { pt } , as in Lemma 3.4, these are exactly the orthogonalit y conditions on the generators T ⊗ n .) T hen as in the p ro of of Lemma 3.4, it suﬃces to c heck the ﬁrst claim for a representa ble M q = M S q ∈ D S ( C , Ab ); w e ha v e a ﬁltration on Hom( M S q , M ′ q ) with asso ciated graded quotien t of the form (3.3), and c hec king that th e map (6.3) is an isomorph ism amount s to applying (6.4) to E q = λ ∗ M ′ q . T o pr o v e the second claim, it remains to notice that for any additive ob ject E q ∈ D ( C ≀ Γ + , Ab), the adj unction map L q j ! j ∗ E q → E q is an isomorp h ism.  Pr o of of Pr op osition 6.5 . By d eﬁnition, f or an y M q ∈ D S ( C , Ab), M ′ q ∈ D ( h c i , Ab) w e hav e Hom( e Φ c M q , M ′ q ) ∼ = Hom( M q , Inﬂ c ( M ′ q )) , and b y Lemma 6.7, we hav e Inﬂ c ( M ′ q ) ∼ = R q η ∗ ( T c ⊗ M ′ q ) Aut( c ) = R q η ∗ ν ≀ c ( M ′ q ) , so that Hom( e Φ c M q , M ′ q ) ∼ = Hom( η ∗ M q , ν ≀ c ( M ′ q )) . It r emains to apply Lemma 6.6 and Lemma 6.8, and notice that b y d eﬁnition, q ∗ ν ≀ c ( M ′ q ) ∼ = ν c ( M ′ q ).  6.2 Filtration by supp ort. W e will no w giv e some corollaries of Prop o- sition 6.5. First, th e follo wing sim p le observ atio n. Supp ose w e are not inte rested in the functor e Φ c of (6.1 ), bu t only in the ﬁxed p oin t functor Φ c of Deﬁn ition 5.3 — that is, we are prepared to forget the Aut( c )-act ion on e Φ c . Th en there 68 is a more con v enient wa y to compute Φ c : D S ( C , Ab) → D (Ab). Namely , denote by C c the category of ob jects c ′ ∈ C equip p ed with a map c ′ → c . W e ha v e a pr o jection s c : C c → C wh ic h forgets the map. Comp osing s c with the em b edding q giv es a fun ctor q c : C c → Q ( C ≀ Γ) and a corresp onding restriction functor q ∗ c : D S ( C , Ab) → D ( C c , Ab). Let T c ∈ F un( C c , Z -mo d) b e giv en by (6.5) T c ( c ′ ) = ( Z , c ′ ∼ = c, 0 , otherwise , and let ϕ c : D ( C c , Ab) → D (Ab) b e the fun ctor left-adjoin t to the emb edding D (Ab) → D ( C c , Ab), M q 7→ M q ⊗ T . Corollary 6.9. We have Φ c ∼ = ϕ c ◦ q ∗ c . In p articular, Φ c M q = 0 if q ∗ c M q = 0 . Pr o of. The right-a djoint to the forgetful functor D ( h c i , Ab ) → D (Ab) send s M q ∈ D (Ab) to M q ⊗ Z [Aut( c )], w h ere Z [Aut( c )] is the regular rep resen tation of the group Aut( c ). Th en b y Prop osition 6.5, it suﬃces to pro v e that R q s c ( T ) ∼ = ν c ( Z [Aut( c )]). This is clear: s c is ob viously a discrete ﬁbration, so that R q s c ∗ can b e compu ted ﬁ b erwise, T is non-trivial on the ﬁb er ov er c ′ ∈ C if and only if c ′ lies in h c i , and for eve ry such c ′ , the ﬁ b er is a torsor o v er Aut( c ).  No w, by Lemma 6.2, the set [ C ] of isomorp hism classes of ob j ects of the lattice -lik e category C has a natural partial order, give n by [ c ] ≥ [ c ′ ] if and only if there exists a map c → c ′ . Deﬁnition 6.10. A subset U ⊂ [ C ] is close d if it is closed w ith r esp ect to the standard T 0 -top ology associated to the p artial order – that is, for any [ c ] , [ c ′ ] ∈ [ C ] with [ c ] ≥ [ c ′ ], [ c ] ∈ U imp lies [ c ′ ] ∈ U . Deﬁnition 6.11. F or an y sub s et U ⊂ [ C ], an ob ject E q ∈ D S ( C , Ab ) is supp orte d in U if E q ( c ) = 0 f or an y c ∈ C whose isomorphism class is outside U . An ob ject E q ∈ D S ( C Ab) has ﬁnite su pp ort if it is supp orted in a ﬁ n ite closed subset U ⊂ [ C ] T he fu ll sub category in D S ( C , Ab) spanned b y ob jects supp orted in a su b set U ∈ [ C ] is d enoted by D S U ( C , Ab) ⊂ D S ( C , Ab) , 69 and the full sub category in D S ( C , Ab) spanned b y ob ject s with ﬁnite s u pp ort is denoted by D S f s ( C , Ab ) ⊂ D S ( C , Ab). F or an y tw o sub sets U ′ ⊂ U ⊂ [ C ], w e obviously ha ve a full em b edding D S U ′ ( C , Ab ) ⊂ D S U ( C , Ab ). Corollary 6.12. F or any close d subset U ⊂ [ C ] , the ﬁxe d p oints functor e Φ c : DS U ( C , Ab) → D ( h c i , Ab ) is trivial unless [ c ] ∈ U . If [ c ] ∈ U is maximal, so that U ′ = U \ { [ c ] } is also close d, then e Φ c ∼ = e Ψ c on D S U ( C , Ab) , wher e e Ψ c is as in (5.1) , the sub c ate g ory D S U ′ ( C , Ab) ⊂ D S U ( C , Ab ) is right-admissible, and e Φ c ∼ = e Ψ c factors thr ough an e quivalenc e (6.6) D S U ( C , Ab ) / D S U ′ ( C , Ab) ∼ = D ( h c i , Ab) . Pr o of. Consider an ob ject E q ∈ D S U ( C , Ab ). By deﬁnition, for an y c ∈ C whose isomorph ism class is n ot in U , w e ha v e E q ( c ) = 0. Since U is closed, we also hav e E q ( c ′ ) = 0 for any c ′ with [ c ′ ] ≥ [ c ], so that q ∗ c E = 0. By Corollary 6.9, this prov es that Φ c ( E q ) = 0; since th e forgetful fu nctor D ( h c i , Ab) → D (Ab) is obviously conserv ativ e, we also ha v e e Φ c ( E q ). This pro ve s th e ﬁrst claim. F or an ob ject c ∈ C w h ose isomorph ism class is maximal in U , we ha ve q ∗ c ( E q )( c ′ ) = 0 unless c ′ ∈ C c is isomorphic to c , so that q ∗ c ( E q ) ∼ = Ψ c ( E q ) ⊗ T c . Since the corresp ond ence E q 7→ E q ⊗ T c is obviously a fu ll emb edding from D (Ab) to D ( C c , Ab), we conclude th at ϕ c ( q ∗ c E q ) ∼ = Ψ c ( E q ) , so that b y Corollary 6.9, the natural map Ψ c ( E q ) → Φ c ( E q ) is an isomor- phism. Since the forgetful f unctor D ( h c i , Ab ) → D (Ab) is conserv ativ e, this pro ve s th e second claim. T o prov e th e rest of the claims, note that the inﬂ ation fu nctor Inﬂ c : D ( h c i , Ab) → D S ( C , Ab) obviously send s D ( h c i , Ab ) in to the full sub cate- gory D S U ( C , Ab ). By deﬁnition, Inﬂ c is right-a djoint to Φ c ∼ = Ψ c . Mo reov er, w e obvio usly ha v e Ψ c ◦ Inﬂ c ∼ = Id . Therefore Inﬂ c : D ( h c i , Ab) → D S U ( C , Ab ) is a fu lly faithful em b edd ing with left-admissible image. But by deﬁnition, for any E ∈ D S U ( C , Ab ), Ψ c ( E ) = 0 if and only if E lies in D S U ′ ( C , Ab ). Thus the orthogo nal ⊥ Inﬂ c ( D ( h c i , Ab)) is exactly D S U ′ ( C , Ab ), and the category D S U ( C , Ab ) h as a semiorthogonal decomp osition h Inﬂ c ( D ( h c i , Ab )) , D S U ′ ( C , Ab ) i , whic h ﬁn ishes the pro of.  70 Remark 6.13. I n the particular case C = O G , c = [ G/H ] for a coﬁnite subgroup H ⊂ G , and U = { [ c ′ ] ∈ [ C ] | [ c ′ ] ≤ [ c ] } , (6.6) is exactly (0.1) of the In tro d uction. Corollary 6.14. F or any two ﬁnite close d su bsets U ′ ⊂ U ⊂ [ C ] , the c at- e g ory DS U ′ ( C , Ab) is an admissible ful l sub c ate gory in D S U ( C , Ab ) , and the left ortho g onal ⊥ D S U ′ ( C , Ab ) c onsists of those obje cts E ∈ D S U ( C , Ab ) which ar e supp orte d in U \ U ′ . Pr o of. The fact that D S U ′ ( C , Ab) ⊂ D S U ( C , Ab ) is r igh t-admissible imm e- diately follo ws by induction from Corollary 6.12 . Moreo v er, the semiorthog- onalit y statemen t of Corollary 6.12 implies that D S U ′ ( C , Ab ) is generated b y ob jects of the f orm Inﬂ c ′ ( M ), c ′ ∈ U ′ , M ∈ D ( h c ′ i , Ab). Th us E ∈ D S U ( C , Ab ) lies in the orth ogonal ⊥ D S U ′ ( C , Ab ) if and only if it is orthog- onal to all su c h Inﬂ c ′ ( M ). By adjunction this is equiv alen t to e Φ c ′ ( E ) = 0, c ′ ∈ U ′ , whic h by deﬁn ition means exactly that E is sup p orted in U \ U ′ . Finally , to prov e that D S U ′ ( C , Ab ) ⊂ D S U ( C , Ab) is left-admissib le, it suﬃces b y induction to consider the case U ′ = U \ { [ c ] } for a maximal c ∈ U . By Lemma 1.2, it suﬃces to prov e that the pro jection ⊥ D S U ′ ( C , Ab ) → D S U ( C , Ab ) / D S U ′ ( C , Ab) ∼ = D ( h c i , Ab) is essen tially surjectiv e. Deﬁne a functor I : F un(Aut( c ) , Ab ) → F un( C ≀ Γ , Ab) by I ( M ) = ( T c ⊗ M ) Aut( c ) , and let L q I : D (Aut( c ) , Ab) ∼ = D ( h c i , Ab) → D S U ( C , Ab) ⊂ D (F un( C ≀ Γ , Ab)) b e its left-deriv ed functor (this is dual to the deﬁn ition of the inﬂation functor Inﬂ c in Sub section 5.1). W e obviously ha v e e Φ c ◦ L q I ∼ = e Ψ c ◦ L q I ∼ = Id , so that it suﬃces to prov e that L q I sends D (Aut( c ) , Ab ) in to the orth ogonal ⊥ D S U ′ ( C , Ab ). As w e hav e already prov ed , this is equiv alent to proving that Φ c ′ ◦ L q I = 0 for an y c ′ ∈ U ′ . Th us by Corollary 6.9, we ha ve to p ro ve that (6.7) ϕ c ′ ◦ q ∗ c ′ ◦ L q I = 0 for an y c ′ ∈ U ′ . By deﬁnition, w e hav e q ∗ c ′ ( T c ⊗ M ) ∼ = s ∗ c ′ j c ! M 71 for any M ∈ F un( h c i , Ab), where j c : h c i → C is th e natural embedd ing. P assing to the d er ived fun ctors, we obtain q ∗ c ′ ◦ L q I ∼ = s ∗ c ′ ◦ L q j c ! , and b y the d eﬁ nition of the functor ϕ c ′ , (6.7 ) is equiv alen t to j c ! R q s c ′ ∗ T c ′ = 0 , where T c ′ ∈ F un( C c ′ , Z -mo d ) is as in (6.5). Since the pro jection s c ′ : C c ′ → C is a ﬁbration, this immediately follo ws b y base change.  6.3 DG mo dels. Corollary 6.12 and Corollary 6.14 show that the cat- egory D S f s ( C , Ab ) admits a ﬁltration by admissible full triangulated sub- catego ries D S U ( C , Ab) in dexed by ﬁ n ite closed subsets U ⊂ [ C ], and th e asso ciated graded quotien ts D S c ( C , Ab) = D S U ( C , Ab) / D S U \{ [ c ] } ( C , Ab ) , c ∈ U maximal of this ﬁltration are iden tiﬁed with the deriv ed ca tegories D (Aut( c ) , Ab ). In the remainder of this section, w e will obtain a full description of the catego ry D S f s ( C , Ab) in terms of th e fu nctors e Φ c . Among other th in gs, this will later allo w us to describ e the gluing data b et w een th ese asso ciated graded quotien ts. The general str ategy is rather straigh tforwa rd – we apply a v ersion of the standard T anakian formalism. Namely , w e treat the collectio n e Φ c as a “ﬁb er fun ctor” for the category D S f s ( C , Ab), and (i) lift th e fu n ctors e Φ c to DG functors e Φ c q , and (ii) presen t a natural C -graded A ∞ -coalg ebra T C q ( − ) w hic h acts on e Φ c q ( − ) (in particular, this will include the Aut( c )-action). The f unctor Φ c : D S f s ( C , Ab ) → D (Ab ) is by deﬁnition left-adjoin t to the functor D (Ab) → D S f s ( C , Ab ) giv en b y M 7→ M ⊗ T c ; the “natur al A ∞ - coalge br a” in (ii) is then an A ∞ mo del for Φ c ( T c ′ ) , c, c ′ ∈ C , with the com ultiplication giv en by adjunction. T he same adjunction giv es a comparison fun ctor f rom D S f s ( C , Ab) to the derive d category of A ∞ - comod ules o v er T C q , and we c hec k that this comparison functor is an equiv- alence (this is Theorem 6.17). 72 W e note that here it is crucial to work with A ∞ -coalg ebras r ather than A ∞ -algebras. In fact, the complex T C q ( f ) is acyclic for an y non-inv ertible map f : c → c ′ . How ev er, as we h a v e n oted in Subs ection 1.5, quasiisomor- phic A ∞ -coalg ebras ma y hav e diﬀeren t categories of A ∞ -comod ules, and in particular, acyclic complexes T C q ( f ) cannot b e replaced with 0. 6.3.1 The complexes. Fix an ob j ect c ∈ C . F or an y n ≥ 1, let C ′ n ( c ) b e the group oid of diagrams c 1 → · · · → c n → c in C and inv er tib le maps b et wee n them, and let C n ( c ) ⊂ C ′ n ( c ) b e the full sub catego ry spanned by suc h diagrams that • th e map c n → c is n ot an isomorphism. As in S ubsection 1.3, let σ n : C ′ n → C b e the fu nctor whic h sends a diagram to c 1 ∈ C . F or any E ∈ F un ( S ( C ≀ Γ) , Ab), let (6.8) Φ c n, q ( E ) = C q ( C n ( c ) , σ ∗ n q ∗ E ) . F or n = 0, set Φ c 0 , 0 ( E ) = E ( c ) and Φ c 0 ,i ( E ) = 0 for i 6 = 1. As in Sub- section 1.3, deﬁne a second diﬀerent ial d : Φ c n +1 , q ( E ) → Φ c n, q ( E ) as the alternating sum of maps d i , 1 ≤ i ≤ n , w here d i remo v es the ob ject c i from the diagram, and let Φ c q ( E ) b e the total complex of the r esulting bicomplex. An y map f : c → c acts on the d iagrams c 1 → · · · → c n → c b y comp osition on th e right-hand side; this turn s Φ c q ( E ) into a complex e Φ c q ( E ) of representa tions of the group Aut( c ). Lemma 6.15. The c omplex e Φ c q ( E ) c omp utes e Φ c ( E ) . Pr o of. By Prop osition 6.5 , it suﬃces to construct a functorial quasiisomor- phism (6.9) RHom q h c i ( e Φ c q ( E ) , M ) ∼ = RHom q ( q ∗ E , ν c ( M )) for any M ∈ F un( h c i , Ab ). Computing the left-hand side of (6.9) by (1.6), w e get (6.10) M 0 ≤ i RHom q − i h c i ( e Φ c i ( E ) , M ) ∼ = ∼ = M 0 ≤ n M 0 ≤ i i are isomorphisms. T h en c 1 ,s → · · · → c i,s → c ′ is a w ell- deﬁned ob ject of the group oid C i ( c ), wh ic h w e denote b y α (1) s . Moreo ve r, comp osing the map f i,s : c i,s → c i with the isomorphism c ′ → c i,s in ve rse to the map c i,s → c ′ , we obtain a diagram c ′ ∼ = c i,s f i ,s − − − − → c i − − − − → . . . − − − − → c n − − − − → c whic h giv es an ob ject in the group oid C n − i ( f ); we denote it by α (2) s . These constructions are fun ctorial, s o that sending a diagram α to the formal union of the pr o ducts α (1) s × α (2) s , s ∈ S 1 giv es a fu nctor β n : C n ( c ) →   a 0 ≤ i ≤ n C i ( c ′ ) × C n − i ( f )   ≀ Γ . F or any E ∈ F un( S ( C ≀ Γ) , Ab), w e hav e a n atural map σ ∗ n q ∗ E → β ∗ n σ ′ ∗ n E , where σ ′ ∗ n is ob tained b y extending the natur al f unctor C i ( c ′ ) × C n − 1 ( f ) − − − − → C n ( c ′ ) σ n − − − − → C to wreath pro du cts. If E is additiv e in the sense of Deﬁnition 4.9, then σ ′ ∗ n E is also additiv e, s o that we h a v e a canonical decomp osition σ ′ ∗ n E  ` s α (1) s × α (2) s  ∼ = M s E ( α (1) s ) . 75 W e can th en comp ose the map C q ( β q ) ind uced by the f unctors β q with the pro jections onto the terms of this decomp osition and obtain a canonical map of complexes (6.12) b f : Φ c q ( E ) → Φ c ′ q ( E ) ⊗ T C q ( f ) . 6.3.3 Comultiplication a nd higher op erat ions. No w assume giv en a third ob ject c ′′ ∈ C and a map g : c ′′ → c ′ , and apply (6.12) to E = T c ′′ . This giv es a com ultiplication map b f ,g : T C q ( f ◦ g ) → T q ( f ) ⊗ T C q ( g ) b y the follo wing pr o cedure. C on s ider agai n a d iagram α = [ c ′′ → c 1 → · · · → c n → c ] ∈ C n ( f ◦ g ), and form the pullback diagram c 1 × c c ′ → · · · → c n × c c ′ → c ′ . In this case, we also ha ve a natural map c ′′ → c 1 × c c ′ ; this distinguishes a comp onen t c ′ 1 in the decomp osition c 1 × c c ′ = ` c 1 ,s , hence also a comp onen t c ′ i in the d ecomp osition c i × c c ′ = ` c i,s for an y 1 ≤ i ≤ n , and w e hav e a commutativ e d iagram c ′′ − − − − → c 1 − − − − → . . . − − − − → c n − − − − → c    x   f 1 x   f n x   f c ′′ − − − − → c ′ 1 − − − − → . . . − − − − → c ′ n − − − − → c ′ . W e again take the sm allest intege r i such th at c ′ i +1 → c ′ is an isomorphism, and obtain a fun ctor β ′ n : C n ( f ◦ g ) → a 0 ≤ i ≤ n C i ( g ) × C n − i ( f ) . This functor indu ces our map b f ,g . Moreo v er, since our f unctors β n , β ′ n are deﬁn ed by p ullbac ks, they are asso ciativ e up to a canonical isomorphism . This means that for an y comp os- able l -tuple of maps f 1 , . . . , f l in the category C l , we ha v e canonical fu nctors β ′ n,l : C n ( f 1 ◦ · · · ◦ f l ) × I l → a n 1 + ··· + n l = n C n 1 ( f 1 ) × · · · × C n l ( f l ) , where I l are the terms of the asymmetric op erad of categories considered in Subsection 1.6, and these fu nctors are compatible with the op er ad structure on I l . T aking the bar complexes and using the op erad map A s s ∞ → C q ( I q , Z ) of S ubsection 1.6, we turn T C q ( f ) int o a C -graded A ∞ -coalg ebra in the sens e of Deﬁnition 1.11. The analogous fu nctors β n,l turn the collecti on Φ c q ( E ), c ∈ C , into an Ab-v alued A ∞ -comod ule o v er T C q ( − ). 76 6.4 The comparison theorem. W e can no w pro ve the comparison th e- orem expressing D S f s ( C , Ab) in terms of the coalgebra T C q ( − ). Deﬁnition 6.16. The category D T ( C , Ab) = D ( T C q , Ab) is the deriv ed cat- egory of Ab -v alued A ∞ -comod ules o v er the C -graded A ∞ -coalg ebra T C q ( − ). An ob ject E q ∈ D T ( C , Ab) is ﬁnitely su pp orte d if E q ( c ) = 0 except for c with isomorphism cla ss in a ﬁnite clo sed subset U ⊂ [ C ]; the full s ub cat- egory spann ed by ﬁnitely su pp orted ob jects is denoted by D T f s ( C , Ab) ⊂ D T ( C , Ab). In the last Subsection, w e ha ve co ns tr ucted a functor Φ q : F un( S ( C ≀ Γ) , Ab) → D T ( C , Ab ). T aking the total complex of a double complex, we extend it to a fun ctor Φ q : DS ( C , Ab ) → D T ( C , Ab ) . (There is the us ual am biguit y in taking the total complex of a p ossibly inﬁnite bicomplex – w e can either tak e the sum of th e terms on a d iagonal, or the pro d u ct of these terms. Here we tak e th e sum.) Theorem 6.17. The functor Φ q induc es an e quivalenc e Φ q : DS f s ( C , Ab ) → D T f s ( C , Ab) . Pr o of. F or any s ubset U ⊂ [ C ], let C U ⊂ C b e the full sub category spanned b y ob j ects w ith isomorp hism classes in C , w ith the emb edding functor ι U : C U → C , and let T C U q ( − ) = ι ∗ U T C ( − ) b e the C U -graded A ∞ -coalg ebra m ad e up of the complexes T C q ( f ) with f in C U . Let D T U ( C , Ab ) b e the triangulated catego ry of A ∞ -comod ules o v er T C U q . W e ha v e the restriction f unctor ι ∗ U : D T ( C , Ab) → D T U ( C , Ab), and for an y t w o subsets U ′ ⊂ U , w e hav e a restriction functor ι ∗ U,U ′ : D T U ( C , Ab ) → D T U ′ ( C , Ab). If U is closed, and either U ′ ⊂ U or the complemen t U \ U ′ ⊂ is also closed, then we hav e an ob vious inv erse functor D T U ′ ( C , Ab ) → D T U ( C , Ab ) giv en by extension by 0 — explicitly , M ∈ DT U ′ ( C , Ab) is sen t to M ′ ∈ D T U ( C , Ab ) s uc h th at M ′ ( c ) = ( M ( c ) , [ c ] ∈ U ′ , 0 , otherwise , with the ob vious structure maps. If it wa s U ′ whic h w as closed, th en this extension functor is righ t-adjoin t to the restriction f unctor i ∗ U,U ′ (this is ob vious for the homotop y categ ories Ho ( T C U q , Ab), Ho ( T C ′ U q , Ab), and since b oth r estriction and extension preserv e acyclic como d ules, the adju nction descends to the derived categories). 77 Lemma 6.18. F or any two ﬁnite close d su b sets U ′ ⊂ U ⊂ [ C ] , the c ate gory D T U ( C , Ab ) admits a semiortho gonal de c omp osition hD T U \ U ′ ( C , Ab) , D T U ′ ( C , Ab) , wher e the emb e ddings D T U \ U ′ ( C , Ab) ⊂ D T U ( C , Ab ) and D T U ′ ( C , Ab ) ⊂ D T U ( C , Ab ) ar e given by e xtension by 0 . Pr o of. By adj unction, DT U ′ ( C , Ab ) ⊂ DT U ( C , Ab ) is left-admissible, with the pro jection giv en by r estriction. On th e other hand , D T U \ U ′ ( C , Ab) by deﬁnition is exactly the k ernel of the restriction fun ctor.  Lemma 6.1 8 implies by indu ction that D T f s ( C , Ab) = ∪ U D T U ( C , Ab ), the union ov er all ﬁnite closed U ⊂ [ C ], wh ere D T U ( C , Ab) is identi ﬁed with the f u ll sub category in D T ( C , Ab ) spanned b y ob jects E q suc h that E q ( c ) = 0 unless [ c ] ∈ U . The comparison fu nctor Φ q ob viously sen ds D S U ( C , Ab) in to D T U ( C , Ab ), and it suﬃces to pr o v e th at it is an equiv alence for any ﬁnite closed U ⊂ [ C ]. Fix suc h a subset U , tak e an ob ject c ∈ C whose isomorphism class [ c ] is a m aximal elemen t in U , and let U ′ = U \ { [ c ] } . By in duction on the cardinalit y of U , we ma y assume that Φ q : D S U ′ ( C , Ab ) → D T U ( C , Ab ) is an equiv alence. Moreo ve r, for any in ve rtible m ap f , the group oids C n ( f ), n ≥ 1 are obviously empty , so that T C q ( f ) is Z placed in d egree 0. Therefore the category D T { [ c ] } ( C , Ab ) is equiv alen t to D ( h c i , Ab ), and the functor Φ q : ⊥ D S U ′ ( C , Ab) ∼ = D ( h c i , Ab) → D T { [ c ] } ( C , Ab) is also an equiv alence. Then by Corollary 6.14, D S U ′ ( C , Ab ) ⊂ D S U ( C , Ab ) is admissible, so that by Lemma 1.3, it suﬃces to p ro v e that Φ q sends D S U ′ ( C , Ab) ⊥ = Inﬂ c ( D ( h c i , Ab)) ⊂ D S U ( C , Ab) in to D T U ′ ( C , Ab ) ⊥ ⊂ D T U ( C , Ab). Indeed, den ote Ho { c } ∼ = Ho ( T C { c } q , Ab), Ho U = Ho ( T C U q , Ab). T hen the restriction functor Ho U → Ho { c } has an obvious right- adjoint I : Ho { [ c ] } → Ho U whic h sends a complex M q of Aut( c )-mo dules in to an A ∞ -comod ule I ( M q ) su c h that (6.13) I ( M q )( c ′ ) =   M f ∈C ( c,c ′ ) T q ( f ) ⊗ M q   Aut( c ) . If th e complex M q is h -injectiv e, then I ( M q ) is also h -injectiv e b y adju nction, so that its image in the quotien t cate gory D T U ( C , Ab ) lies in D T U ′ ( C , Ab ) ⊥ ⊂ 78 D T U ( C , Ab ). Ho wev er, since Ho { c } is equiv alent to the catego ry of un - b ound ed complexes of fu nctors in F un( h c i , Ab ), ev ery ob ject has an h - injectiv e replacemen t. Thus w e can tak e an y ob j ect M ′ = Inﬂ c ( M ) ∈ D S U ′ ( C , Ab ) ⊥ , and c ho ose an h -injectiv e complex M q of Aut( c )-como dules represent ing M = e Φ c ( M ′ ) ∈ D ( h c i , Ab ) ∼ = D (Aut( c ) , Ab). Then b y deﬁn i- tion, M ′ = Inﬂ c ( M ) is repr esen ted by the complex Inﬂ c ( M q ) = ( M q ⊗ T c ) Aut( c ) , and w e see that Φ q ( M q ) ∈ D T U ( C , Ab) is pr ecisely isomorphic to I ( M q ).  Remark 6.19. As the p r o of of Theorem 6.17 sho ws, ev ery ﬁnitely sup- p orted ob ject in the catego ry Ho ( T q , Ab) do es hav e an h -injectiv e replace- men t. As w e hav e men tioned in Su bsubs ection 1.5.3, this pr op erty seems rather sp ecial. 6.5 Induction a nd pro ducts. Assume now that the category C has a terminal ob ject, as in S u bsection 5.2. Then D S ( C , Ab) is a tensor category , and by Prop osition 5.6 all th e ﬁxed p oints functors Φ c are tensor fu nctors, so that the sub category D S f s ( C , Ab) ⊂ D S ( C , Ab) is closed under tensor p r o d- ucts. T o describ e the tensor pr o duct structure on D S f s ( C , Ab) in terms of the equiv alent categ ory D T f s ( C , Ab), one w ould ha ve to introd u ce a p ro du ct on the C -graded A ∞ -coalg ebra T C q – more precisely , an asso ciativ e p ro du ct on the complex T C q ( f ) for any m ap f in C . I n kee ping with the T anakian formalism, T C q then b ecomes a “ C -graded Hopf algebra”, with an indu ced tensor pro duct on the category D T ( C , Ab). Unfortunately , it seems that the particular C -graded A ∞ -coalg ebra T C q that we ha v e used in Theorem 6.17 do es n ot hav e a natural Hopf algebra structure. Possibly this could b e healed b y taking some qu asiisomorphic A ∞ -coalg ebra, or using a Segal-lik e n otion of a “sp ecial A ∞ Γ + -coalg ebra” instead of a commutativ e A ∞ Hopf algebra, b ut I h a v en’t pursued this. Nev ertheless, we hav e a simpler result whic h we will n eed for applications in Section 8. Consider the trivial C -g raded coalg ebra Z C giv en by Z C ( f ) = Z for an y map f in C . T hen since C is assumed to b e Hom-ﬁnite, the derived catego ry D ( Z C , Ab) of Ab-v alued A ∞ -comod ules o ver Z C is equiv alen t to the deriv ed catego ry D ( C opp , Ab). Moreo v er, since T 0 C ( f ) = Z = Z C ( f ) for any f , w e ha v e a natural map λ : Z C = F 0 T C q → T C q , 79 where F 0 stands for the 0-th term of the stupid ﬁltration. This map indu ces a corestriction fun ctor λ ∗ : D ( C opp , Ab) ∼ = D ( Z C , Ab) → D T ( C , Ab ). Explic- itly , for an y Z C -comod ule E q , we hav e λ ∗ E q ( c ) ∼ = E q , c ∈ C , and for an y map f : c ′ → c in C , the como dule structure map of Deﬁnition 1.11 is th e natural map E q ( c ) E q ( f ) − − − − → E q ( c ′ ) ⊗ Z = E q ( c ′ ) ⊗ Z C ( f ) λ ( f ) − − − − → E q ( c ′ ) ⊗ T C q ( f ) , with all the higher maps b eing equal to 0. Moreo v er, for an y complex E q of p oint wise-ﬂat functors in F un( C opp , Ab) and an y A ∞ -comod ule M q o v er T C q , we h a v e an obvi ous A ∞ -comod ule E q ⊗ M q o v er T C q , with ( E q ⊗ M q )( c ) = E q ( c ) ⊗ M q ( c ), and with the como du le structure maps obtained as the p ro du cts of the como d ule structure maps for M q and th e canonical maps E q ( c ) → E q ( c ′ ) for any comp osable n -tuple f 1 , . . . , f n of maps in C with th e comp osition f 1 ◦ · · · ◦ f n b eing a map from c to c ′ . This pr eserves quasiisomorphisms, th us descents to a tensor pro du ct fun ctor (6.14) D ( C opp , Ab) × D T ( C , Ab ) → D T ( C , Ab ) . Lemma 6.20. Under the c omp arison fu nc tor Φ q : D S ( C , Ab) → D T ( C , Ab ) of The or em 6.17, the induction fu nctor q opp ! : D ( C opp , Ab) → D S ( C , Ab) of Subse ction 5.3 go es to the c or estriction functor λ ∗ – that i s, we have a natur al functorial quasiisomorphism (6.15) Φ q ( q opp ! ( E q )) ∼ = λ ∗ E q for any E q ∈ D ( C opp , Ab) . Mor e over, for any obje cts E q ∈ D ( C opp , Ab) , M q ∈ D S ( C , Ab) , we have a natur al isomorph ism (6.16) Φ q (( q opp ! E q ) ⊗ M q ) ∼ = E q ⊗ Φ q ( M q ) , wher e the pr o duct in the right-hand side is the pr o duct (6.14) . Pr o of. Both claims immediately follo w from Prop osition 5.8.  7 T ate homology description. W e will n o w use Theorem 6.17 to obtain an explicit and useful description of the gluing data in the semiorthogonal decomp osition of Corollary 6.12 . 80 7.1 Generalized T ate cohomology . Assu me giv en a ﬁnite group G , and a family { H i } of su b groups H i ⊂ G . F or ev ery H i , w e ha ve the indu ction functor Ind H i G : D b ( H i , Z ) → D b ( G, Z ) , where D b ( G, Z ), resp. D b ( H i , Z ) are the b ounded deriv ed cate gories of Z [ G ], resp. Z [ H i ]-mo dules; th is functor is adjoin t b oth on the left and on the righ t to the natural restriction fu nctor D b ( G, Z ) → D b ( H i , Z ). Let D b { H i } ( G, Z ) ⊂ D b ( G, Z ) b e the full thic k triangulated sub categ ory spann ed by direct sum mands of ob jects of the form Ind H i G ( V q ), V q ∈ D b ( H i , Z ), H i ∈ { H i } , and denote by D ( G, { H i } , Z ) = D b ( G, Ab) / D b { H i } ( G, Z ) the quotien t category . Deﬁnition 7.1. The g ener alize d T ate c ohomolo gy ˇ H q ( G, { H i } , M q ) of the group G with co eﬃcients in an ob ject M q ∈ D b ( G Z ) w ith r esp ect to the family { H i } is giv en by ˇ H q ( G, { H i } , M q ) = Ext q D ( G, { H i } , Ab) ( Z , M q ) , where Z is the trivial repr esentati on of the group G , and Ex t q ( − , − ) is computed in the quotien t catego ry D ( G, { H i } , Z ). By the deﬁnition of the quotien t category , ˇ H q ( G, { H i } , M q ) is expressed as follo ws. Consider the categ ory I ( G, { H i } ) of ob jects V i ∈ D b ( G, Z ) equipp ed w ith a map V q → Z whose cone lies in D b { H i } ( G, Z ) ⊂ D b ( G, Z ). Let I = I ( G, { H i } ) opp b e the opp osite category , and let (7.1) I f g ⊂ I = I ( G, { H i } ) opp b e the sub category spanned by those V q whic h can b e r epresen ted by com- plexes of ﬁn itely generated Z [ G ]-mod ules. Then we ha v e (7.2) ˇ H q ( G, { H i } , M q ) = lim → Ext q ( V q , M q ) , where the limit is tak en o v er the category I f g , and Ext q ( − , − ) is compu ted in the category D ( G, Z ). S ending V q to th e du al complex V ∗ q iden tiﬁes I f g with the category of ob jects V ∗ q in D b ( G, Z ) whic h are r ep resen ted by ﬁnitely 81 generated Z [ G ]-mo dules and equipp ed with a map Z → V ∗ q whose cone lies in D b { H i } ( G, Z ). Then (7.2) can b e rewritten as ˇ H q ( G, { H i } , M q ) = lim → H q ( G, M q ⊗ V q ) , where again the limit is o v er I f g . Since I f g is small and ﬁltered, the limit is we ll-deﬁned, and give s a cohomological fu nctor. W e can u se the same expression to deﬁn e T ate cohomology with co eﬃ- cien ts. Namely , let D b ( G, Ab) b e the derive d category of representa tions of the group G in the ab elian category Ab. Deﬁnition 7.2. The g ener alize d T ate c ohomolo gy ˇ H q ( G, { H i } , M q ) of the group G with co eﬃcien ts in an ob ject M q ∈ D b ( G Ab) with resp ect to th e family { H i } is giv en by ˇ H q ( G, { H i } , M q ) = lim → H q ( G, M q ⊗ V q ) , where the limit is tak en o v er the category I f g ⊂ I = I ( G, { H i } ) of (7.1) Again, since Ab is b y assu mption a Grothend iec k catego ry , and I f g is small and ﬁltered, generaliz ed T ate cohomology is a w ell-deﬁned cohomo- logica l fu nctor from D ( G, Ab ) to Ab . Lemma 7.3. F or any M ∈ D ( G, Ab) and any W q ∈ D { H i } ( G, Z -mo d ) which c an b e r epr esente d by a b ounde d c omplex of ﬁnitely gener ate d Z [ G ] - mo dules, we have ˇ H q ( G, { H i } , M ⊗ W q ) = 0 . Pr o of. Let W ′ q = Z ⊕ W q , and let w : I f g → I f g b e the fun ctor w hic h sends V q ∈ I f g to W ′ q ⊗ V q = V q ⊕ ( V q ⊗ W q ). Then w is obviously coﬁnal in the sense of [KS], so that the natural map Id → w indu ced by Z → W ′ q giv es an isomorphism lim → H q ( G, M ⊗ V q ) ∼ = lim → H q ( G, M ⊗ W ′ q ⊗ V q ) , where b oth limits are o v er V q ∈ I f g . Since ˇ H q is a cohomologica l fu n ctor, this prov es th e claim.  82 7.2 Adapted complexes. In practice, the ind ex category I f g is still to o big. T o b e able to compute generalized T ate cohomology more eﬃcient ly , w e use the follo wing gadget. Lemma 7.4. A ssume given a b ounde d fr om ab ove c omplex P q of ﬁnitely gener ate d Z [ G ] -mo dules. F or any inte ger l , denote by F l P q the ( − l ) -th term of the stupid ﬁltr ation on P q . Then the fol lowing c onditions ar e e q uivalent. (i) F or any of the sub gr oups H i ∈ { H i } and any inte ger l , ther e exists an inte ge r l ′ ≥ l such that the map F l P q → F l ′ P q b e c omes 0 after r estricting to D ( H i , Z ) . (ii) F or any V q ∈ D { H i } ( G, Z ) , we have lim l → H q ( G, V q ⊗ F l P q ) = 0 . (iii) F or any V q ∈ D { H i } ( G, Z ) , any Gr othendie ck ab elian c ate gory Ab , and any M ∈ D ( G, Ab ) , we have lim l → H q ( G, M ⊗ V q ⊗ F l P q ) = 0 . Pr o of. The condition (iii) con tains (ii) as a particular case. By adju n c- tion, (i) implies (iii) for V q of the f orm Ind H i G ( V ′ q ), V ′ q ∈ D ( H i , Z ). Since D { H i } ( G, Z ) consists of direct summands of sums of suc h V q , this condi- tion on V q can b e dropp ed. Finally , (ii) applied to a b ounded complex V q of ﬁn itely generated Z [ G ]-modu les can b e r ewritten as follo w s: for an y l , an y m ap κ : V ∗ q → F l P q b ecomes 0 after comp osing with th e natural m ap F l P q → F l ′ P q for suﬃcien tly large l ′ ≥ l . Ap p lying this to V ∗ q = Ind H i G ( F l P q ) with κ b eing the adju nction map yields (i) .  Deﬁnition 7.5. A complex P q of Z [ G ]-mo du les is said to b e adapte d to the family { H i } if (i) P i = 0 for i < 0, P 0 ∼ = Z , and for an y i ≥ 0, P i is a ﬂ at Z -mo d ules of ﬁnite r an k , and is a direct summand of a sum of Z [ G ]-mo dules induced from one of the subgroups H i , and (ii) the complex P q satisﬁes the equiv alen t conditions of Lemma 7.4. 83 Prop osition 7.6. Assume given a ﬁnite gr oup G , a family { H i } of sub- gr oups H i ⊂ G , and c ompl ex P q of Z [ G ] -mo dules adapte d to the family { H i } . Then for any M q ∈ D b ( G, Ab) , we have an isomorphism (7.3) ˇ H q ( G, { H i } , M q ) ∼ = lim l → H q ( G, M q ⊗ F l P q ) , wher e F l P q is as in Deﬁnition 7.5 (ii) , and this isomorphism is functorial in M q . Pr o of. Denote for the moment ˇ H q o ( G, { H i } , M q ) ∼ = lim l → H q ( G, M q ⊗ F l P q ) . Consider the pr o duct I f g × N of the index category I f g of (7 .1) and the partially ordered s et of non-negativ e in tegers. Computing the dou b le limit ﬁrst in one order, then in another, we obtain an isomorph ism lim l → ˇ H q ( G, { H i } , M q ⊗ F l P q ) ∼ = lim → H q ( G, { H i } M q ⊗ F l P q ⊗ V q ) ∼ = lim → ˇ H q o ( G, { H i } M q ⊗ V q ) , where the last limit is o ve r V q ∈ I f g , and the intermediate limit is ov er V q × l ∈ I f g × N . By Lemma 7.3 and Deﬁnition 7.5 (i), ˇ H q ( G, { H i } M ⊗ F l P q ) do es not dep end on l , so that lim l → ˇ H q ( G, { H i } M q ⊗ F l P q ) ∼ = ˇ H q ( G, { H i } M q ) . By Deﬁnition 7.5 (ii) in the form of Lemma 7.4 (iii) , ˇ H q o ( G, { H i } M q ⊗ V q ) do es not dep end on V q ∈ I f g , so that lim → ˇ H q o ( G, { H i } M q ⊗ V q ) ∼ = ˇ H q o ( G, { H i } M q ) . This ﬁnish es the pr o of.  Example 7.7. T ak e as { H i } th e family consisting only of the trivial sub- group { e } ⊂ G . A Z [ G ]-mod ule is indu ced from { e } if and only if it is free, so that the category D b { H i } ( G, Z ) consists of dir ect su mmands of ﬁ- nite complexes of free Z [ G ]-mod ules – equiv alen tly , these are the p erfect ob jects in D b ( G, Z ). Thus in th is case, ˇ H q ( G, { e } , − ) is the s tand ard T ate 84 (co)homolo gy functors. F ur ther, Deﬁn ition 7.5 (ii) simply means that the complex P q is acyclic (th us con tractible as a complex of ab elian groups), and Deﬁnition 7.5 (i) means that P i is a ﬁnitely generated p r o jectiv e Z [ G ]- mo dule for any i ≥ 1. Th is gives the standard pro cedure for compu ting T ate h omology: tak e a p ro jectiv e r esolution P q of th e trivial represent ation Z and let e P q b e the cone of the augmenta tion map P q → Z ; then ˇ H q ( G, V q ) ∼ = lim l → H q ( G, V q ⊗ F l e P q ) for an y V q ∈ D b ( G, Z -mo d). In general, there are d iﬀeren t wa ys to constru ct an adapted complex for a ﬁ nite group G and a family { H i } ; w e will pr esen t one construction later in Subsection 7.5. 7.3 T ate cohomolog y and ﬁxed p oin ts functors. No w assume give n a small catego ry C of Galois typ e, and consider the category D S f s ( C , Ab ) ∼ = D T f s ( C , Ab ), as in Theorem 6.17. W e w an t to exp r ess th e gluing data b e- t w een the pieces of the semiorthogonal d ecomp osition of Corollary 6.12 in terms of generalized T ate cohomology . The main r esu lt is as follo ws. T ak e a morphism f : c ′ → c in C , and consider the corresp onding complex T C q ( f ) of (6.11). Denote by Aut( f ) ⊂ Aut( c ′ ) the sub group of those automorphisms σ : c ′ → c ′ that commute with f , f ◦ σ = f . This group acts on ev ery s et C n ( f ) b y left comp osition, that is, a diagram c ′ f 1 − → c 1 → · · · → c n → c goes to c ′ f ′ ◦ σ − → c 1 → · · · → c n → c . Th us Aut( f ) also acts on T C q ( f ). Moreo ver, for an y d iagram α = [ c ′ → c 1 → c ] ∈ C 1 ( f ), let Aut( α ) ⊂ Aut( f ) b e its stabilizer in the group Aut( f ). Prop osition 7.8. The c omplex T C q ( f ) e quipp e d with the natur al action of the g r oup Au t( f ) i s adapte d to the family { Aut( α ) | α ∈ C 1 ( f ) } in the sense of Deﬁnition 7.5. Pr o of. The condition (i) of Deﬁnition 7.5 is ob vious: the stabilizer of any diagram c ′ → c 1 → · · · → c n → c is co ntained in the stabilize r of the diagram c ′ → c 1 → c obtained b y f orgetting c i , i ≥ 2, so that it su ﬃces to us e the follo wing trivial observ atio n. Lemma 7.9. A ssume give n a ﬁnite gr oup G with a family of sub gr oups { H i } , H i ⊂ G , and let X b e a ﬁnite G -set such tha t the stabilizer G x of any p oint x ∈ X is c onta ine d in one of the sub gr oups H i . Then the set 85 Z [ X ] with the natur al G -action lies in the sub c ate gory D b { H i } ( G, Z -mo d) ⊂ D b ( G, -mo d) .  T o c hec k the condition (ii) in the form of Lemma 7.4 (ii), consider th e catego ry C c of Corollary 6.9. Let C f b e the category of diagrams c ′ → c 1 → c suc h that the comp osition map c ′ → c is equal to f , and let π f : C f → C c b e th e pro jection fun ctor whic h sen d s an ob j ect [ c ′ → c 1 → c ] of C f to the ob ject [ c 1 → c ] of C c . This is a discrete coﬁbr ation. Let T f = π f ! Z ∈ F un( C c , Z -mo d ). Th e group Aut( f ) acts on T f , and we ha ve T C q ( f ) ∼ = Φ c q ( T f ) , where Φ c q ( − ) is the complex (6.8). F or an y su bgroup H ⊂ Aut( f ) and any h -injectiv e complex V q of Z [ H ]-mod u les, we ha ve lim l →  V q ⊗ F l T C q ( f )  H ∼ = lim l → F l Φ q ( T f ⊗ V q ) H ∼ = Φ q ( T f ⊗ V q ) H , and b y Lemma 6.15, this computes (7.4) Φ c  V q ⊗ T f  H . It suﬃ ces to prov e that this is trivial whenever H = Aut( α ) for some α = [ c ′ → c 1 → c ] ∈ C 1 ( f ). Indeed, ﬁ x s uc h an α , and denote its co mp onen t maps by f 1 : c ′ → c 1 , f 2 : c 1 → c . Then we hav e the category C c 1 and the functor T f 1 ∈ F un( C c 1 , Z -mo d ), and the group Au t( α ) = Aut( f 1 ) acts on T f 1 . Moreo v er, comp osition with f 2 deﬁnes functors ρ : C c 1 → C c , ρ ′ : C f 1 → C f , and w e hav e a comm utativ e diagram C f 1 ρ ′ − − − − → C f π f 1   y   y π f C c 1 ρ − − − − → C c . Since b oth C f 1 and C f ha v e an in itial ob ject, and ρ ′ preserve s them, we ha v e L q ρ ′ ! Z ∼ = ρ ′ ! Z ∼ = Z , so that w e ha ve an isomorph ism L q ρ ! T f 1 ∼ = ρ ! T f 1 ∼ = ρ ! π f 1 ! Z ∼ = π f ! ρ ′ ! Z ∼ = π f ! Z ∼ = T f , and this isomorphism is Au t( α )-equiv arian t. This yields a quasiisomorphism  V q ⊗ T f  Aut( α ) ∼ = L q ρ !  V q ⊗ T f 1  Aut( α ) , 86 so that b y Corollary 6.9, (7.4) is of the form ϕ c L q ρ ! E q for some E q ∈ D ( C f 1 , Z -mo d). This is equal to 0: by deﬁnition of the fun ctor ϕ c , w e ha ve Hom( ϕ c L q ρ ! E q , M q ) ∼ = Hom( L q ρ ! E q , M q ⊗ T ) ∼ = Hom( E q , ρ ∗ T ⊗ M q ) for an y M q ∈ D (Ab), and ρ ∗ T = 0.  Corollary 7.10. F or any Galois-typ e c ate gory C and two obje cts c, c ′ ∈ C , the homolo gy H q (Φ c Inﬂ c ′ ( M q )) of the obje ct Φ c Inﬂ c ′ ( M q ) ∈ D (Ab) is giv e n by (7.5) H q (Φ c Inﬂ c ′ ( M q )) ∼ = M f ∈C ( c ′ ,c ) ˇ H q (Aut( f ) , { Aut( α ) | α ∈ C 1 ( f ) } , M q ) for any M q ∈ D ( h c ′ i , Ab) .  T o w r ite do wn a form ula for e Φ c q Inﬂ c ′ , the gluing fun ctor in the semi- orthogonal d ecomp osition of Corollary 6.12 , one h as to incorp orate the n at- ural Aut( c )-act ion on Φ c in to (7.5) ; we lea ve it to the reader. 7.4 In vertible ob jects. Let us say th at an ob j ect M ∈ D is a triangu- lated tensor category D is invertible if the functor M ⊗ − : D → D giv en b y m ultiplication by M is an equiv alence (for example, a u nit ob ject I ∈ D is in ve rtible, and so are all its shifts I [ l ], l ∈ Z ). F or simp licit y , for the m oment w e restrict our atten tion to th e case Ab = Z -mo d. Keep the assump tions of the previous s u bsection, and assu me also that C has a terminal ob ject, so that D S ( C , Z ) is a tensor triangulated category . In addition, assume th at [ C ] is ﬁn ite, so that DS f s ( C , Z ) = D S ( C , Z ). As an application of Prop o- sition 7.8, let u s p ro v e the follo wing criterion of inv ertibilit y for ob jects in D T ( C , Z ) ∼ = D S ( C , Z ). Prop osition 7.11. Assume giv en an obje ct M ∈ D ( C opp , Z ) such that (i) the r estriction M ( c ) = j c ∗ M ∈ D ( h c i , Z ) with r esp e ct to the inclusion j c : h c i → C is invertible for any c ∈ C , and (ii) for any map f : c ′ → c in C , e quip M ( c ) with the trivial Aut( f ) -action, so that the map M ( f ) : M ( c ) → M ( c ′ ) is Au t( f ) - e q uivariant. Then the c one of this map lies in (7.6) D { Aut( α ) | α ∈C 1 ( f ) } (Aut( f ) , Z -mo d) ⊂ D b (Aut( f ) , Z -mo d) . 87 Then the induc e d obje ct q opp ! M ∈ DS ( C , Z ) is invertible. Pr o of. Let M : D T ( C , Z ) → D T ( C , Z ) b e the functor giv en by m ultiplication b y M , M ( E ) = M ⊗ E . By (6.15) of Lemma 6.20, M p reserv es supp orts: if E ∈ DS ( C , Z ) is su pp orted in some s ubset U ⊂ [ C ], then M ( E ) is also supp orted in U . Th us as in the pro of of Theorem 6.17 , it suﬃces to pro v e that M : D T U ( C , Z ) → D T U ( C , Z ) is an equiv alence for any ﬁ nite closed U ⊂ [ C ]. T ak e a maximal elemen t [ c ] ∈ U and let U ′ = U \ { [ c ] } . Then M sen d s the pieces of the semiorthog- onal decomp osition of Lemma 6.18 into themselv es. By ind uction, M is an auto equiv ale nce of D T U ′ ( C , Z ), and by the condition (i), M is also an au- to equiv alence of the orthogonal ⊥ D T U ′ ( C , Z ) ∼ = D ( h c i , Z ). Again as in the pro of of Theorem 6.17, by Lemma 1.3 it suﬃ ces to pro ve that M sen d s the orthogonal D T U ′ ( C , Z ) ⊥ in to itself. But ev ery ob j ect in this orthogonal is of the form I ( E q ) for some h -injectiv e complex E q of Aut( c )-mo du les, where I is as in (6.13), and by Corollary 7.1 0 and Lemma 7.3, the cond ition (ii) th en implies that M ( I ( E q )) is quasiisomorphic to I ( E q ⊗ M ( c )).  7.5 The case of Mack ey functors. W e now turn to the case of Mac k ey functors – we assume giv en a group G , an d tak e C = O G , the category of ﬁnite G -orbits. First, we f u lﬁl t w o earlier promises – giv e a construction of an adapted complex, and prov e that inﬂ ation is fully faithful. Lemma 7.12. F or any ﬁnite gr oup G with a family of sub gr oups { H i } , ther e exists a c omplex P q of Z [ G ] -mo dules adapte d to the family { H i } in the sense of Deﬁnition 7.5. Pr o of. Note that w e ma y assume that together with any subgroup H i ⊂ G , the family { H i } con tains all the subgroups H ⊂ H i – add ing suc h subgroups to the family do es not change the conditions of Deﬁnition 7.5 (nor of Def- inition 7.2). Ha ving assum ed this, consider th e full sub catego ry C ′ ⊂ O G spanned by the one-p oin t orbit [ G/G ] and all the orb its [ G/H i ], H i ∈ { H i } . F or any su bgroup H 1 , H 2 ⊂ G , every G -orbit in th e pr o duct G/H 1 × G/H 2 of of th e form G/H ′ for some H ′ ⊂ H 1 . Therefore the category C ′ is of Galois t yp e, so that Prop osition 7.8 applies. T ak e as f the u nique map f : [ G/ { e } ] → [ G/G ]; then G = Aut( f ), Aut( α ) ∈ { H i } for any α ∈ C ′ 1 ( f ), and T C ′ q ( f ) is th e r equired adapted complex.  88 Lemma 7.13. Assume given a normal sub gr oup H ⊂ G with the quotient N = G/H . Then the inﬂation functor c Inﬂ N G : D M ( N , Ab) → D M ( G, Ab ) is ful ly faithful. Pr o of. Let U ⊂ [ O G ] b e the set of orbits [ G/H ′ ] suc h [ G/H ] ≥ [ G/H ′ ], that is, there exists a G -equiv ariant map G/H → G/H ′ , that is, H ′ ⊂ G con tains a conjugate of H ⊂ G . Since H ⊂ G is assumed to b e normal, H itself must b e con tained in H ′ . Therefore the orb its G/H ′ , [ G/H ′ ] ∈ U are in one-to-one corresp ondence with orbits N/ ( H ′ /H ) of the qu otien t group N = G/H , and the full su b category in O G spanned by ob jects with classes in U is equiv alen t to the category O N . Moreo ver, it is easy to s ee th at th e em b edd ing O N ⊂ O G is compatible with the graded A ∞ -coalg ebras T q – for an y map f in O N , w e h av e T O N q ( f ) ∼ = T O G q ( f ) . Th us D M ( N , Ab) ∼ = D T U ( O G , Ab) ⊂ D M ( G, Ab), and c Inﬂ N G is the full em b edd ing D T U ( O G , Ab) ⊂ D T ( O G , Ab) = D M ( G, Ab).  No w we mak e the follo wing easy observ ation. Lemma 7.14. Assume given a Galois-typ e c ate gory C with the c orr esp ond- ing C -gr ade d A ∞ -c o algebr a T C q of Deﬁnition 6.16, anot her C -gr ade d A ∞ - c o algebr a T ′ q , and an A ∞ -map ν : T ′ q → T C q such that (i) if f is i nv e rtible, the map ν : T ′ q ( f ) → T C q ( f ) is an isomorphism, (ii) for a non-invertible f : c ′ → c and any Z [Aut( f )] -mo dule V , the map ν induc es an isomorphism (7.7) lim l → H q (Aut( f ) , V q ⊗ F l T ′ q ( f )) ∼ = lim l → H q (Aut( f ) , V q ⊗ F l T C q ( f )) , wher e F q T C q , F q T ′ q ar e the stupid ﬁltr ations. Then the c or estriction functor ν ∗ induc es an e quivalenc e b etwe en the derive d c ate gory DT f s ( C , Ab) of ﬁnitely supp orte d A ∞ -c omo dules over T C q and the c ate gory D T ′ f s ( C , Ab) of ﬁnitely supp orte d A ∞ -c omo dules over T ′ q . 89 Pr o of. The same as Theorem 6.17 and Prop osition 7.11: since C is lattice- lik e, b oth categ ories ha v e ﬁ ltrations b y supp ort and the corresp onding semi- orthogonal decomp ositions, and the corestriction fun ctor ν ∗ is compatible with these decomp ositions. T h en (i) insures that ν ∗ is an equiv alence of the asso ciated graded pieces, and (ii) insu res that ν ∗ is compatible with the gluing.  The condition (ii) of this Lemma is satisﬁed, for example, when T ′ q ( f ) is adapted to { Aut( α ) } for an y f , as in Prop osition 7.5. Ho wev er, it can also b e satisﬁed for other reasons: for instance, if the righ t-hand side of (7.7) is equal to 0, th e left-hand side m a y just b e trivial, T ′ q ( f ) = 0. This is esp ecially usefu l in the Mac ke y fu nctors case, f or the follo wing r eason. Lemma 7.15. L et G b e a ﬁnite gr oup, and let { H i } b e the family of al l pr op er sub gr oups H ⊂ G . (i) The gener alize d T ate c ohomolo gy ˇ H q ( G, { H i } , V ) with c o eﬃcients in any Z [ G ] -mo dule V is trivial unless G is a p -gr oup for some prime p , and in this c ase, it is annihilate d by p . (ii) If G is a cyclic prime gr oup, G = Z /p n for some prime p and i nte ger n ≥ 2 , then ˇ H q ( G, { H i } , V ) = 0 for any Z [ G ] -mo dule V . Pr o of. F or a p rop er subgroup H ⊂ G , we h a v e the natur al induction and coinduction maps Z → Ind H G ( Z ), Ind H G ( Z ) → Z , and their comp osition is m ultiplication by th e in dex | G/H | of the subgroup H . T herefore m ultiplica- tion by | G/H | is trivial in the qu otien t catego ry D ( G, { H i } , Z -mo d), h ence annihilates ge neralized T ate cohomo logy . If G is not a p -group, then the greatest common d enominator of the in dices of its Sylow subgroup s is 1, so that 1 also ann ih ilates T ate cohomology . If G is a p -group, then it con tains a subgroup of index p . This pro ve s (i) . F or (ii), note that for an y n , the s econd cohomology group H 2 ( Z /p n Z , Z ) con tains a canonical p erio dicity element u n represent ed by Y oneda by the exact sequence 0 − − − − → Z − − − − → Z [ Z /p n Z ] 1 − σ − − − − → Z [ Z /p n Z ] − − − − → Z − − − − → 0 , where σ is the generator of the cyclic group Z /p n Z . F or an y p ositiv e in- teger n ′ < n , w e ha ve a natural quotien t map q n,n ′ : Z /p n Z → Z /p n ′ Z , and one easily c hec ks that q ∗ n,n ′ ( u n ′ ) = p n − n ′ u n . In the qu otient category D ( G, { H i } , Z -mo d), all the ob j ects q ∗ n,n ′ Z [ Z /p n ′ Z ] b ecome trivial, s o that 90 all the maps q n,n ′ u n ′ are inv ertible. If n ≥ 2, this imp lies th at p is in vert- ible. Since p annihilates T ate cohomolog y by (i), this is only p ossible if ˇ H q ( G, { H i } , − ) = 0.  Corollary 7.16. Assume that G is a ﬁnite gr oup whose or der is invertible in the c ate gory Ab . Then D M ( G, Ab) ∼ = M H ⊂ G D ( N H /H, Ab) , wher e the sum is over al l the c onjugacy classes of sub gr oups H ⊂ G , and D ( N H /H, Ab) i s the derive d c ate gory of the r epr esentations of the quotient N H /H of the normalizer N H of the sub gr oup H ⊂ G by H itself. Pr o of. Since G is ﬁnite, ev erything has ﬁnite sup p ort, D S ( O G , Ab) ∼ = D S f s ( O G , Ab) ∼ = D T f s ( O G , Ab). Moreo ve r, ev ery map f b etw een t w o G - orbits is isomorphic to the quotien t map [ G/H ] → [ G/ H ′ ] for some sub- groups H ⊂ H ′ ⊂ G , the automorphisms group of an ob ject [ G/H ] is precisely N H /H , an d f or an y map f : [ G/H ] → [ G/ H ′ ], an y p rop er sub - group in the automorphisms group Aut( f ) is of the f orm Aut( α ) for some α ∈ C 1 ( f ). No w tak e T ′ q ( f ) = T C q ( f ) for inv ertible f , T ′ q ( f ) = 0 otherwise, and apply Lemma 7.14. Th e condition (ii) is satisﬁed, since the left-hand side of (7.7) is isomorp hic to ˇ H q (Aut( f ) , { Aut( α ) } , − ), and this is trivial by Lemma 7.15.  In the ordin ary , non-deriv ed Mac ke y f unctor case, this is a Theorem of Thev enaz [T1]. No w take Ab arbitrary , G p ossibly in ﬁnite, C = O G . In this case , we can still use Lemma 7.14 and Lemma 7.15 to simp lify the A ∞ -coalg ebra T C q . Namely , any m ap b et we en t wo ﬁnite G -orbits is isomorph ic to the quotien t map f : [ G/H ] → [ G/H ′ ], where H ⊂ H ′ ⊂ G are s u bgroups of ﬁ nite index. Denote by Ind ( f ) the index of H in H ′ . Then one can rep lace T C q with the C -graded A ∞ -coalg ebra T ′ q deﬁned as T ′ q ( f ) = ( T C q ( f ) , Ind ( f ) = p n for some prime p , 0 , otherwise . Again, by Lemma 7.15 (i) this will not c hange the category D T q ( C , Ab ). 91 7.6 Cyclic group. W e can go even further in the imp ortant sp ecial case G = Z , the inﬁnite cyclic group. F or ev ery int eger n , c ho ose a p ro jectiv e resolution P n q of the trivial Z [ Z /n Z ]-mo dule Z . F or example, w e can tak e the standard p erio d ic r esolution (7.8) id − σ − − − − → Z [ Z /n Z ] id + σ + ··· + σ n − 1 − − − − − − − − − − → Z [ Z /n Z ] id − σ − − − − → Z [ Z /n Z ] , where σ ∈ Z /n Z is the generator. Let e P n q b e the cone of the augmentati on map P n q → Z (that is, e P 0 = Z , and e P n l = P n l − 1 for l ≥ 1). F or an y morp hism f : c ′ → c in O G from c ′ = [ Z /n ′ Z ] to c = [ Z/n Z ], deﬁne the complex e T q ( f ) b y e T q ( f ) =      Z , Ind ( f ) = 1 , e P n ′ q , Ind ( f ) = n ′ n = p is pr ime , 0 , otherwise . T o extend this to an A ∞ -coalg ebra, w e need to deﬁne co-multiplic ation maps, and the only p ossible n on-trivial comultiplica tion m aps are those that enco de Aut( c ′ ) × Aut( c )-acti on on L f ∈ O G ( c ′ ,c ) e T q ( f ) in the case Ind ( f ) = p , n ′ = np . F or any c = [ Z /n Z ] ∈ O G , we h av e Aut( c ) = Z /n Z . More generally , f or an y c ′ = [ Z /n ′ Z ], c = [ Z/n Z ], n ′ divisible by n , w e hav e O G ( c ′ , c ) = Z /n Z , and the group Aut( c ′ ) × Aut( c ) acts trans itivel y on this set; w e hav e O G ( c ′ , c ) = (Aut( c ′ ) × Au t( c )) / Aut( c ′ ) , where th e emb ed ding Au t( c ′ ) ⊂ Aut( c ′ ) × Aut( c ) is the pro d u ct of the iden- tit y map Aut( c ′ ) → Aut( c ′ ) and the natural pro jection Aut( c ′ ) = Z /n ′ Z → Aut( c ) = Z /n Z . In the particular case n ′ = n p , w e tak e the giv en Z /n ′ Z - action on e P n ′ q , and induce the Au t( c ′ ) × Aut( c )-action on L f ∈ O G ( c ′ ,c ) e T q ( f ) via the ident iﬁcation M f ∈ O G ( c ′ ,c ) e T q ( f ) = M f ∈ O G ( c ′ ,c ) e P n ′ q ∼ = Ind Aut( c ′ ) Aut( c ′ ) × Aut( c ) e P n ′ q . All the higher maps b n , n ≥ 3 are set to b e trivial, b n = 0. Prop osition 7.17. With the notat ion ab ove, the c ate g ory D S f s ( O G , Ab) is e quivalent to the derive d c ate gory of ﬁnitely supp orte d Ab -value d A ∞ - c omo dules over the O G -gr ade d A ∞ -c o algebr a e T q . Pr o of. First, w e construct a map of O G -graded A ∞ -coalg ebras ν : e T q → T O G q . If Ind ( f ) = 1, in other w ords f is inv ertible, w e ha v e e T q ( f ) = 92 T O G q ( f ) = Z b y deﬁn ition. If Ind ( f ) is not prime, e T q ( f ) = 0, so th at there is nothing to deﬁne. It remains to d eﬁne an Aut( c ′ ) × Aut( c )-equiv ariant map ν : e T q ( f ) → T O G q ( f ) for any map f : c ′ → c of p rime index Ind ( f ) = p , c = [ Z /n Z ], c ′ = Z /pn Z . W e ha ve Aut( c ′ ) = Z /np Z , and by adjunction, deﬁn ing ν is equiv alen t to deﬁning an Z /pn Z -equiv arian t map ν : e P np q → T q ( c ′ , c ) = M f ∈ O G ( c ′ ,c ) T O G q ( f ) . Both complexes are acyclic. In d egree 0, e P np 0 = Z , T O G 0 ( c ′ , c ) = Z [ O G ( c ′ , c )] is a free Z /np Z -mo dule, and we take as ν the natural m ap which ident iﬁes Z w ith the sub group of Z /np Z -inv arian ts in T O G 0 ( c ′ , c ). In higher degrees, e P np q +1 is a resolution of Z , and T O G q +1 ( c ′ , c ) is a resolution of T O G 0 ( c ′ , c ). But the resolution e P np q +1 is by deﬁnition pro jectiv e; therefore the giv en map ν : e P np 0 → T O G 0 ( c ′ , c ) extends to a m ap of r esolutions ν : e P np q +1 → T O G q +1 ( c ′ , c ). T o ﬁn ish the p ro of, it r emains to c hec k the conditions of Lemma 7.14. The condition (i) is satisﬁed by deﬁ nition. As for (ii) , it is satisﬁed unless Ind ( f ) is prime b y virtue of Lemm a 7.15 (ii). An d if Ind ( f ) = p is prime, b oth T O G q ( f ) and e T q are complexes of Z [Aut( f )]-mo dules adapted to the f amily of Lemma 7.15 (whic h in this case consist of th e unique p rop er su bgroup in Au t( f ) = Z /p Z , namely , the un it subgroup { e } ⊂ Au t( f )). This also implies (ii) of Lemma 7.14.  By Lemma 7.13, this description of the ca tegory D S f s ( O Z , Ab) also yields a description of the catego ry D M ( Z /n Z , Ab ) for ev ery ﬁ nite cyclic group Z /n Z – one just restricts to the orbits [ Z /l Z ] with l a divisor of n (for a ﬁnite group , the “ﬁnite su pp ort” condition b ecomes v acuous). In the simplest p ossible case G = Z /p Z , p prime, this b oils down to the follo wing. Corollary 7.18. L et G = Z /p Z . Then the c ate gory D M ( G, Ab) is obtaine d by inverting quasiisomorp hisms f r om the c ate gory of triples h V q , W q , ρ i of a c omplex W q of obje cts in Ab , a c omplex V q of G -r e pr esentations in Ab , and a map ρ : W q → b C q ( G, V q ) , wher e b C q ( G, V q ) is the standar d 2 -p erio dic c omplex which c omputes T ate c ohomolo gy ˇ H q ( G, V q ) by me ans of the r esolution (7.8) . Pr o of. Clear.  93 8 Relation to stable homotop y . T o ﬁnish the pap er, w e ﬁ x a ﬁnite group G and explain the relation b etw een the category of deriv ed Mac k ey functors D M ( G ) and G -equiv arian t stable homotop y theory . W e will only giv e a sk eleton exp osition so as to p resen t the general principles. In particular, we w ill restrict our atten tion to ﬁn ite CW sp ectra when giving pr o ofs. T o ke ep the text accessible to a p erson without kno wledge of equiv arian t stable homotop y , w e d o recall some of the basics; for fur ther information, the reader s hould construct the original references, whic h are [Br] for S u bsection 8.1 and [LMS ] for Sub s ection 8.2. 8.1 Spaces. The homotopy category Hom( G ) of G -spaces is deﬁned in a straigh tforw ard wa y: ob jects are top ological sp aces equipp ed with a con tin u- ous actio n of the group G , maps b et we en them are contin uous G -equiv arian t maps, a homotop y b et w een t wo maps f , f ′ : X → Y is a con tin uous G - equiv ariant map F : X × I → Y w ith F = f on X × { 0 } and F = f ′ on X × { 1 } , where I is the unit inte rv al [0 , 1] with the standard top ology and trivia l G -actio n, m orp hisms in the homotopy category are homotop y classes of maps. Note that in this category , for an y subgroup H ⊂ G the functor X 7→ X H whic h sends a space X to the space X H of H -in v ariant p oints is w ell-deﬁned. As usual, to get a us eful theory one p asses to the cat- egory of p ointed s p aces, and restricts one’s attent ion to the fu ll sub category spanned by CW complexes. Base p oin ts are assumed to b e G -in v ariant. One also needs some compatibilit y condition b et w een the G -action and th e CW structure; a con v enient notion is th at of a G -CW c omplex — this is a C W complex X equip p ed with a contin uous G -action suc h that f or an y g ∈ G , the ﬁxed p oints subset X g ⊂ X is a sub complex (that is, a union of cells). F rom no w on, we will by ab u se of notation denote the homotop y catego ry of p ointed G -CW complexes by Hom( G ). F or any subgroup H ⊂ G and an y G -CW complex X , the ﬁxed p oints subset X H ⊂ X is obvi ously a CW complex. The geometric r ealizat ion of a simplicial G -set is a G -CW complex; th erefore any G -set whic h is homotop y equiv alent to a G -equiv ariant CW complex is also equiv alen t to a G -CW complex. F or an y tw o p oin ted G -spaces X , Y , their smash pro d uct X ∧ Y is also a p oin ted G -space, and if X and Y are G -CW complexes, then s o is their pro duct X ∧ Y . Equiv alen tly , the ﬁxed p oints subset X H ⊂ X of a G -space X can b e describ ed as follo w s: take the orbit [ G/H ], treat it as a G -space with the dis- crete top ology , and consider the s pace of con tin uous maps Maps([ G/H ] , X ) 94 with a natural top ology on it. Then we ha ve (8.1) X H ∼ = Maps([ G/H ] , X ) . This sho ws th at for an y G -space X , the corresp ondence H 7→ X H is a actually a functor from the category O opp G opp osite to the category of ﬁnite G -orbits to the category of top ological spaces. If X is a G -CW complex, then this is a fu nctor from O opp G to the catego ry of CW complexes and cellular maps b etw een; if the G -CW complex X is ﬁnite, then so are all the X H . No w, for any CW complex X , we ha ve a natural cellular chain complex C q ( X, Z ), and this construction is functorial w ith r esp ect to cellular maps. Therefore for any G -CW complex, we hav e a n atural fu nctor [ G/H ] 7→ C q ( X H , Z ) from the category O opp G to the catego ry of complexes of ab elian groups. Let us denote the corresp ond ing complex of functors in F un( O opp G , Z -mo d) b y C G q ( X, Z ). Th is complex w as constructed and studied b y Bredon [Br]; in his terminology , functors from O opp G to ab elian group s are called c o e ﬃcient systems , so that C G q ( X, Z ) is a complex of co eﬃcien t systems. W e note that up to a qu asiisomorphism, it do es not dep end on the particular C W mo del for an ob j ect X ∈ Hom( G ). The em b edding of the ﬁ xed p oint pt → X indu ces an injectiv e map Z ∼ = C G q ( pt , Z ) → C G q ( X, Z ); let e C G q ( X, Z ) = C G q ( X, Z ) / Z b e the quotien t. Again, up to a quasiisomorph ism, it only dep end s on the class of X in Hom( G ). Deﬁnition 8.1. The complex e C G q ( X, Z ) is called the naive r e duc e d G -e qu- ivariant c el lular chain c omplex of the CW complex X . Th e corresp onding ob ject e C G q ( X, Z ) ∈ D ( O opp G , Z -mo d) is called the naive r e duc e d G -e quivariant homolo gy of X considered as an ob ject in Hom( G ). Lemma 8.2. F or any two obje cts X, Y ∈ Hom( G ) , we have a natur al quasi- isomorph ism e C G q ( X ∧ Y , Z ) ∼ = e C G q ( X, Z ) ⊗ e C G q ( Y , Z ) , wher e the tensor pr o duct i n the right-hand side is the p ointwise tensor pr o d- uct in the c ate gory D ( O opp G , Z -mo d) . Pr o of. The corresp onding s tatemen t “without the group G ” is completely standard. Since the tensor p ro du ct in D ( O opp G , Z -mo d) is p oin t wise, it suﬃces to sho w that ( X × Y ) H = X H × Y H ; this is obvio us fr om (8.1 ).  95 8.2 Stabilization. Recall th at for the usual ﬁnite p ointed CW complexes, the S p anier-Whitehead stable homotop y catego ry SW is d eﬁned as follo ws: ob ject are again ﬁn ite p oin ted CW complexes, maps fr om X to Y are giv en b y (8.2) Hom( X , Y ) = lim i → [Σ i X, Σ i Y ] , where [ − , − ] means the set of h omotop y classes of maps, and Σ is the sus- p ension fu n ctor, so that Σ i X = S i ∧ X , where S i is the i -sphere. By deﬁnition, the su sp ens ion f unctor Σ is fully faithful on the category SW ; in fact the deﬁnition is d esigned exactly to ac hiev e this. The triangu- lated c ate gory of sp e ctr a , denoted S t Hom , is obtained by formally inv erting the susp ension, thus m aking it not only fully faithful but an autoequiv alence. The passage f rom SW to StHom inv olv es a somewhat delicate limiting pro- cedure due to Boa rd man; I do not feel qualiﬁed to re-count it here, and r efer the reader to any of the standard textb o oks (e.g. [Ad]) for the deﬁn ition of StHom and f urther discuss ion. W e hav e a n atural fully f aithfu l em b ed- ding S W ⊂ StHom whic h is compatible with the susp ension and is usu ally denoted by Σ ∞ . The categ ory sthom of ﬁnite CW sp e ctr a is give n b y sthom = [ i Σ − i (Σ ∞ ( SW )) ⊂ StHom . This is the smallest f ull triangulated sub categ ory in StHom cont aining S W . Sometimes, e.g. in [BBD, 1.1. 5], it is this category whic h is called “the stable homotop y category”. It can b e equiv alen tly describ ed as th e category of pairs h X, n i of a ﬁnite CW complex X and an in teger n , with maps from h X, n i to h Y , m i giv en by lim l ≥ max( n,m ) → [Σ l − n X, Σ l − m Y ] . W e should note here that the category sthom is certainly n ot suﬃ cien t for man y app lications in top ology , since almost all sp ectra represent ing inter- esting generalized homology theories are n ot ﬁ nite CW sp ectra. The cellular c hain complex e C q ( − , Z ) is compatible with sus p ensions, th us descends to the homology fu nctor f rom sthom to the derive d category D ( Z -mo d) wh ic h w e denote by h : s thom → D ( Z -mo d); explicitly , h ( h X, n i ) is r epresen ted by the complex e C q ( X )[ n ]. 96 F or G -spaces, one can rep eat the S panier-Whitehead construction lit - erally; this r esu lts in a “naiv e G -equiv ariant Spanier-Whitehead cate gory” SW naive ( G ) – ob jects are G -spaces X ∈ Ho m( G ), maps are giv en b y the same form ula (8.2) as in the n on-equiv arian t case. Ho w ev er, there is a more int eresting and more n atural option. Namely , the i -sph ere S i is the one-p oint compactiﬁcatio n of th e i -dimensional real v ector space V ; in the equiv arian t world, this vect or sp ace should b e allo wed to carry a non -trivial G -action. T o make sense of the d irect limit in (8.2), one ﬁxes a so-called c omplete G -unive rse , that is, an inﬁnite d imensional represent ation U of the group G whic h has an in v arian t inner pro duct and is “large enough” in the sense that ev ery ﬁ nite-dimensional inner-pro duct represent ation V of the group G o ccurs in U coun tably man y times. Then the limit in (8.2) should b e take n o v er all inn er p ro du ct G -in v arian t su b - spaces V ⊂ U : [ X, Y ] U = lim V ⊂ U → [Σ V X, Σ V Y ] , where Σ V is th e V -susp ension fu nctor given by Σ V ( X ) = S V ∧ X , and S V ∈ Hom( G ) is the one-p oin t compactiﬁcation of V , with b ase p oint at inﬁnity . Th is construction works for an arb itrary c omp act top ologic al group G . Ho w ev er, wh en G is a ﬁ n ite group with discrete top ology , there is an ob vious preferred choice of the unive rse: w e can tak e U = R ⊕∞ , th e sum of a counta ble num b er of copies of the regular repr esen tation R = R [ G ]. This results in the follo wing deﬁnition. Deﬁnition 8.3. The “gen uine” G -e quivariant Sp anier-Whitehe ad c ate gory SW ( G ) has ﬁnite p oin ted G -CW complexes as ob j ects, and m aps from X to Y are giv en b y Hom( X, Y ) = lim i → [Σ iR X, Σ iR Y ] , where [ − , − ] stands for the s et of morphisms in Hom( G ), R = R [ G ] is the regular representa tion of the group G , and Σ iR b y abuse of notation denotes (Σ R ) i . Among other representa tions of G , R conta ins the trivial one, so that passing to the limit in Deﬁnition 8.3 in cludes taking the limit (8.2), and we ha v e a natural tautologic al f unctor (8.3) i ! : SW naive ( G ) → SW ( G ) whic h is ident ical on ob jects. How ev er, this is m ost certainly n ot iden tical on morph ism s, thus not an equiv alence. 97 T o pass to the sp ectra, one again inv erts the susp ension, either in the gen uine category SW ( G ) or in th e naiv e category S W naive ( G ); this results in the so-called c ate gory of naive G -sp e ctr a StHom naive ( G ) and the c ate gory of genuine G -sp e ctr a StHom ( G ). Both are triangulated catego ries. F or the precise deﬁnitions, I refer the reader to the b o ok [LMS], the standard ref- erence on the sub ject. The susp ension fu nctor Σ is an auto equ iv alence on StHom naive ( G ), and all the susp ension functors Σ V , V ⊂ R ⊕∞ , are auto equiv ale nces on StHom ( G ). W e again ha v e full embed d ings Σ ∞ naive : SW naive ⊂ StHom naive ( G ), Σ ∞ : SW ⊂ S t Hom ( G ), and we can deﬁne th e triangulated categories of ﬁnite G -sp ectra b y sthom naive ( G ) = [ i Σ − i (Σ ∞ naive ( SW ( G ))) ⊂ StHom naive ( G ) and sthom ( G ) = [ i Σ − iR (Σ ∞ ( SW ( G ))) ⊂ StHom ( G ) , where Σ − iR stands for (Σ R ) − i . As in the n on-equiv ariant case, one can equiv alently describ e these catego ries as the categories of pairs h X, n i , X ∈ Hom( G ), n ∈ Z , w here h X , n i corresp onds to Σ − i ( X ) ∈ StHom naive ( G ) in the naiv e case, and to Σ − iR ( X ) ∈ StHom ( G ) in th e gen uine case. 8.3 Equiv ariant homology . The Bredon equiv arian t homology f u nctor e C G q ( − , Z ) : Hom( G ) → D ( O opp G , Ab) of Deﬁn ition 8.1 ob viously extend s to the naiv e category SW naive ( G ), and th en fu rther to the category of ﬁn ite sp ectra, so that we ha ve a natural functor (8.4) sthom naive ( G ) → D ( O opp G , Ab) whic h we denote by h G naive ( − ). Ho w ev er, a m oment’s reﬂection shows that it do es not extend to the gen uine categ ory sthom ( G ). This is where the Mac k ey functors come in. Deﬁnition 8.4. The G -e q uivariant homolo gy h G (Σ ∞ X, Z ) of a ﬁ nite p oin t- ed G -CW complex X ∈ Hom( G ) is the induced derived Mac k ey fun ctor h G ( X, Z ) = q opp ! e C G q ( X, Z ) ∈ D M ( G ) , where q opp ! : D ( O opp G , Z -mo d) → D S ( O G , Z -mo d ) = D M ( G ) is the induction functor of Sub section 5.3. 98 Lemma 8.5. F or any two ﬁnite p ointe d G -CW c omplexes X , Y , we have a natur al isomorph ism h G ( X ∧ Y , Z ) ∼ = h G ( X, Z ) ⊗ h G ( Y , Z ) ∈ D M ( G ) . Pr o of. A com bination of Lemma 8.2 and (6.16) of Lemma 6.20.  Prop osition 8.6. F or any ﬁnite-dimensional r epr esentation V of the gr oup G , the obje c t h G ( S V , Z ) ∈ D M ( G ) is invertible in the sense of Subse ction 7.4. Pr o of. Since h G ( S V , Z ) = q opp ! e C G q ( S V , Z ), we can use the criterion of Prop o- sition 7.11. T o c heck the condition (i), note th at f or any s ubgroup H ⊂ G , w e hav e e C G q ( S V , Z )([ G/H ]) ∼ = e C q (( S V ) H , Z ) , and s ince ( S V ) H = S V H is a sp here, its r educed homology is Z [dim R V H ]. T o chec k th e condition (ii), ﬁx t wo subgroup s H 1 ⊂ H 2 ⊂ G , let S i = ( S V ) H i = S V H i , i = 1 , 2, let f : [ G/H 1 ] → [ G/H 2 ] b e the quotient map, let e f : S 2 → S 1 b e the corresp ond ing cellular em b edding, and let W = Aut( f ) ⊂ Aut[ G/H 1 ]. Sin ce for an y prop er su b group W ′ ⊂ W , w e hav e a factorizat ion [ G/H 1 ] − − − − → [ G/H 1 ] /W ′ − − − − → [ G/H 2 ] of the map f through some G -orbit [ G/H 1 ] /W ′ , ev ery prop er su bgroup W ′ ⊂ W app ears as Au t( α ) in (7.6). Th e group W itself app ears only if the map [ G/H 1 ] /W → [ G/H 2 ] is not an isomorph ism, or in other words, S 2 ⊂ S W 1 is a prop er inclusion. W e ha v e to show that the relativ e homology ob ject e C q ( S 1 /S 2 , Z ) ∈ D b ( W , Ab) lies in the sub category D { Aut( α ) } ( W , Ab) ⊂ D b ( W , Ab). If S 2 6 = S W 1 , W app ears in the family { Aut( α ) i , so th at D { Aut( α ) } ( W , Ab ) = D b ( W , Ab ) and there is nothing to pro v e; th us w e ma y assume S 2 = S W 1 . But since b y deﬁnition, the stabilizer of an y non-trivial cell in the quotien t S 1 /S W 2 is a prop er subgroup W , the claim then immediately follo ws from Lemma 7.9.  99 Corollary 8.7. The G -e quivariant homolo gy of Deﬁnition 8.4 extends to a wel l-deﬁne d functor h G ( − , Z ) : sthom ( G ) → D M ( G ) . Pr o of. F or any ﬁnite-dimensional representat ion V of the group G , let σ V : D M ( G ) → D M ( G ) b e the fun ctor giv en by σ V ( M ) = M ⊗ h G ( S V , Z ) . Then b y Lemma 8.5, we ha v e h G (Σ V X, Z ) ∼ = σ V ( h G ( X, Z )) for any ﬁ - nite G - CW complex X ∈ Hom( G ), and b y P rop osition 8.6, σ V is an au- to equiv alence. By deﬁnition, ev ery ob ject X ′ ∈ sthom G is of the form X ′ = (Σ V ) − 1 (Σ ∞ ( X )) for some s u c h V and and X ; the d esir ed extension is then giv en by h G ( X ′ , Z ) = ( σ V ) − 1 ( h G ( X, Z )) , and b y Lemma 8.5, this d o es n ot d ep end on the c hoice of the identiﬁca tion X ′ = (Σ V ) − 1 (Σ ∞ ( X )).  8.4 A dictionary . Corollary 8.7 allo ws one to compare notions from the G -equiv ariant stable homotop y theory with those of the theory of Mac k ey functor. W e ﬁnish the section, and indeed the whole pap er, with a sh ort dictionary sa ying what should corresp ond to what. All the m aterial on equiv ariant s table h omotop y is tak en from [LMS]. P ersonally , I also ﬁ nd v ery us efu l the b rief in tro du ction to [HM]. First of all, the tautolog ical functor (8.3) obviously extend s to ﬁnite sp ectra, an d by d eﬁnition, we ha ve a comm utativ e d iagram of triangulated catego ries and triangulated f unctors (8.5) sthom naive ( G ) i ! − − − − → sthom ( G ) h G naive   y   y h G D ( O opp G , Z ) q opp ! − − − − → D M ( G ) . The smash pr o duct on the category Hom( G ) extend s to the categories of ﬁnite sp ectra sthom naive ( G ) and sthom ( G ), so that they b ecome symmetric tensor triangulated cate gories. Unfortunately , the pro d uct do es not com bine w ell w ith the stabilization pro cedure of (8.2), so that this extension is ve ry non-trivial already in the non-equiv arian t case, see e.g. [Ad]. Although in 100 the last ﬁfteen years, new and more s atisfactory ap p roac hes app eared (e.g. [EKMM] or [HSS ]), still, none of them can b e recount ed in a few pages. Nev- ertheless, w hatev er construction one uses, our equiv arian t homology functor h G ough t to b e compatible with the smash pro du cts. Lemma 8.8. F or any two obje cts X , Y ∈ sthom ( G ) , we have h G ( X ∧ Y ) ∼ = h G ( X ) ⊗ h G ( Y ) . Sketch of a pr o of. T o giv e an honest pro of, w e w ould need to use an exact def- inition of the pro duct of sp ectra, wh ich w ould b e b eyo nd the scop e of the p a- p er; instead, we show ho w the statemen t is deduced from the standard prop- erties of the smash pro duct. W rite do wn X = (Σ V ) − 1 ( X ′ ), Y = (Σ W ) − 1 ( y ′ ), x ′ , Y ′ ∈ Hom( G ), V , W ⊂ R ⊕∞ . By Lemma 8.5, w e h a v e h G ( X ′ ∧ Y ′ ) ∼ = h G ( X ′ ) ⊗ h G ( Y ′ ) and h G ( S V ⊕ W ) ∼ = h G ( S V ∧ S W ) ∼ = h G ( S V ) ⊗ h G ( S W ), so that σ V ⊕ W ∼ = σ V ◦ σ W ∼ = σ W ◦ σ V , where σ V is as in the pro of of Corollary 8.7. Then h G ( X ∧ Y ) ∼ = h G ((Σ V ) − 1 X ′ ∧ (Σ W ) − 1 Y ′ ) ∼ = h G ((Σ V ⊕ W ) − 1 ( X ′ ∧ Y ′ )) ∼ = ( σ V ⊕ W ) − 1 h G ( X ′ ) ⊗ h G ( Y ′ ) ∼ = h G ( X ) ∼ = h G ( Y ) , as required.  An analogous statement f or sth om naive ( G ) is also obviously tru e. Mo re- o v er, the fu n ctor i ! of (8.5) is also tensor, and q opp ! is tensor b y (6.16 ) of Lemma 6.20, so that (8.5) is actually a diagram of tensor fu nctors. Next, for any subgroup H ⊂ G , the ﬁ xed p oints fu nctor X 7→ X H from Hom( G ) to the category of CW complexes p reserv es pro d ucts; since the ﬁxed p oin ts subset of a sphere S V is also a sphere, the ﬁxed p oints f u nctor is compatible with (8.2 ) and extend s to the category sth om ( G ) of ﬁn ite G - sp ectra. The result is called the ge ometric ﬁxe d p oints functor and denoted b y Φ H : sthom ( G ) → sthom . It is also co mp atible with the smash p r o duct. Und er our th e equiv arian t homology functor h G , it go es to th e ﬁxed p oints fu nctor on the category D M ( G ): Lemma 8.9. F or any sub gr oup H ⊂ G and any X ∈ sthom ( G ) , we have a quasiisomorph ism (8.6) h ◦ Φ H ( X ) ∼ = Φ [ G/H ] ◦ h G ( X ) 101 Pr o of. By (6.16) of Lemma 6.20, we ha ve Φ [ G/H ] ( σ V ( M )) ∼ = Φ [ G/H ] ( M )[dim V H ] for any V ⊂ R ⊗∞ and any M ∈ D M ( G ). Therefore the statement is compatible with susp ensions, and it suﬃ ces to prov e it for X = Σ ∞ ( X 0 ) for some ﬁn ite G -CW complex X 0 ∈ Hom( G ). Then it imm ediately follo ws from (6.15) of Lemma 6.20.  Consider n o w the full categories of sp ectra StHom naive ( G ), StHom ( G ). The tautologi cal functor i ! of (8.3) extends to all sp ectra, although it b e- comes rather n on-trivial in the pro cess. S o do the smash pro du ct and th e geometric ﬁxed p oin t fun ctors. Just as in the case of the ﬁ n ite sp ectra, the functors Φ H , H ⊂ G , and i ! are tensor f unctors. I t is n atural to exp ect that our equiv arian t homology functor h G also extends to StHom ( G ). Conjecture 8.10. The e quivariant homolo gy functor h G of Cor ol lary 8.7 and the naive homolo gy fu nc tor h G naive of (8.4) extend to the c ate gories of sp e ctr a, so that (8.5) extends to a c ommutative diagr am (8.7) StHom naive ( G ) i ! − − − − → StHom ( G ) h G naive   y   y h G D ( O opp G , Z -mo d ) q opp ! − − − − → DM ( G ) of tensor triangulate d functors. Mor e over, for any sub gr oup H ⊂ G , (8.6) extends to an isomorphism Φ [ G/H ] ◦ h G ( X ) ∼ = h ◦ Φ H ( X ) of functors fr om StHom ( G ) to D ( Z -mo d) . Moreo v er, on the lev el of sp ectra, the ta utological functor i ! acquires a righ t-adjoin t i ∗ : StHom ( G ) → StHom naive ( G ), and we can consider the diagram StHom naive ( G ) i ∗ ← − − − − StHom ( G ) h G naive   y   y h G D ( O opp G , Z -mo d ) q opp ∗ ← − − − − D M ( G ) , where q opp ∗ is the restriction w ith resp ect to the em b edding q opp : O opp G → Q Γ G – that is, the righ t-adjoin t fun ctor to q opp ! . By base c hange, w e ha ve a natural map h G naive ◦ i ∗ → q opp ∗ ◦ h G . 102 Conjecture 8.11. The b ase c hange map h G naive ◦ i ∗ → q opp ∗ ◦ h G is an isomorph ism. Using i ∗ , one can deﬁn e another ﬁxed p oin t functor asso ciated to a sub- group H ⊂ G . Indeed, we h a v e an obvious ﬁxed p oin ts fu nctor sthom naive ( G ) → sthom , and it extends to a f unctor from StHom naive ( G ) to StHom (isomorphic to Φ H ◦ i ! ); comp osing it with i ∗ , w e obtain a triangulated functor StHom ( G ) → StHom . This fun ctor is called the Lewis-Ma y ﬁxed p oin ts fun ctor. T h ere is no stan- dard letter asso ciated to it; usu ally it is denoted simply by X 7→ X H . If Conjecture 8.10 and Conjecture 8.11 are kno wn, then the follo wing is immediate. Corollary 8.12. F or any X ∈ StHom ( G ) , we have a functorial isomor- phism Ψ [ G/H ] ( h G ( X )) ∼ = h ( X H ) . Pr o of. By Conjecture 8.11 and Conjecture 8.10, we hav e h ( X H ) ∼ = h (Φ H ( i ! i ∗ ( X ))) ∼ = Φ [ G/H ] ( q opp ! q opp ∗ h G ( X )) , and b y Lemma 6.20, this is isomorp hic to q opp ∗ ( h G ( X ))([ G/H ]) ∼ = Ψ [ G/H ] ( h G ( X )) , as required.  Let me conclude the pap er with the follo wing remark. As we ha v e n oted in Remark 3.9, the catego ries QC ( − , − ) used in S ection 3 in our deﬁ nition of deriv ed Mac k ey fu nctors are naturally symmetric monoidal, so that the group completions Ω B |QC ( − , − ) | of th eir classifying s p aces are inﬁnite lo op spaces. Th us one ca n deﬁne a catego ry B G “enric hed in sp ectra” with ob jects [ G/H ] and morphisms giv en by Ω B |Q O G ( − , − ) | . If one does this with enough p recision, one can then d eﬁne the category of “enric hed fun ctors” from B G to S tHom in such a wa y th at it b ecomes a triangulated catego ry . As I und erstand, this construction is w ell-kno wn in top ology , an d it is well- kno wn th at the resulting category is StHom ( G ). It seems that this h as not b een written do wn (probably b ecause the tec hnology n eeded to make 103 this precise app eared n ot long ago and is still rather cum b ersome, and the diﬃculties of working out all the details out w eigh the b en eﬁts of giving y et another description of a we ll-studied ob ject) . Be it as it ma y , if this sk etc h is assumed to work, then the comparison w ith the present pap er b ecomes quite transparent : all w e do is replace sp ectra with complexes wh ich compute their homology , and replace “enriched fun ctors” w ith A ∞ -functors. Wh at I don’t kno w is whether our third d escription of D M ( G ), n amely the one giv en in Section 6, has an y counterpart with sp ectra instead of complexes. It seems it would b e very useful, since for Mac k ey fu n ctors, this description is b y f ar the m ost eﬀectiv e. O n the other hand, on the lev el of complexes on e has to w ork with an A ∞ -coalg ebra T C q , not an A ∞ -algebra, and v arious ﬁ niteness phenomena b ecome crucially imp ortan t; it is not clear whether th is can b e made to w ork in StHom . References [Ad] J.F. Adams, Stable Homotopy and Gener alize d Homolo gy , Univ. of Chicago Press, 1974. [BBD] A. Beilinson, J. Bernstein, and P . Deligne, F aisc aux p ervers , Ast´ erisque 100 , So c. 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Steklov Ma t h Institut e Moscow, USSR E-mail addr ess : kale din@mccm e.ru 106

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