The de Rham homotopy theory and differential graded category

The de Rham Homotop y Theory and Diﬀeren tial Graded Categor y Syunji Moriy a ∗ Departmen t of Mathematics, F acult y of Science, Ky ot o Univ ersity . Abstract This pap er is a ge neralization o f [30]. W e develop the de Rham homotopy theory of not neces sarily nilpo tent spa ces. W e use tw o alge br aic ob jects: close d dg-c ate gories and e quivariant dg-algebr as . W e see these t wo ob jects corr esp ond in a certain way (Pro p.3.3.4, Thm.3 .4 .5). W e prove an equiv a lence betw een the homoto py catego ry o f schematic homotopy types [22] and a homo topy categor y of closed dg-catego ries (Thm.1.0.1). W e give a description of homotopy in v a riants of spa ces in terms of minimal mo dels (Thm.1.0.2). The minimal mo del in this context behav es muc h like the Sulliv a n’s minimal mo del. W e also provide some examples. W e prove an e q uiv alence betw een ﬁber wise rationaliza tions [5] and closed dg-catego ries with subsidia r y da ta (Thm.1.0.4). Keywor ds: r ational homotop y the ory, non-simply c onne cte d sp ac e, dg-c ate gory, schematic homotop y typ e. 1 In tro duction In [8], Sulliv an constructed the corresp ondence b etw een the rational homotopy t yp es of nilp oten t spaces and commuta tiv e dg-algebras ov er rationals, wh ich asso ciates p olynomial de Rham algebras to sp aces. In particular, he sho w ed that homotop y in v arian ts of nilp oten t spaces, suc h as rational homotop y groups, can b e deriv ed from the algebras. This Sulliv an’s theory is called th e de Rham homotop y theory . In the n on-nilp otent case, as a generalization of the rationalization, the ﬁb erw ise rationalizat ion was prop osed by Bousﬁ eld and Kan ([5], s ee also in tro ductions of [16, 30]). F or this notion, A.G ´ omez-T ato, S.Halp erin and D.T anr ´ e [16] generalized the Su lliv an’s result to non-nilp otent spaces. Recen tly , as another n on-nilp otent generalization of the ratio nalizatio n, the sc hematization [22] w as in tr o duced by T o ¨ en. While the rationalizatio n is the lo calizatio n with resp ect to the rational h omology groups, the sc hematization is a candidate for a lo calization w ith resp ect to all cohomology groups with co ef- ﬁcien ts in ﬁnite rank lo cal sy s tems. In this pap er, w e generalize the Su lliv an’s result for the s c hematization o ve r a ﬁeld of c haracteristic 0. W e u se tw o algebraic ob jects whic h are generalizations of comm utativ e dg-algebras: clo se d dg-c ate gories [30] and e quivariant c ommutative dg-algebr as . Th ese tw o al gebraic ob- jects ha ve diﬀerent adv antag es. W e establish an equiv alence b et w een the h omotop y category of sc hematic homotop y types and a homotop y catego ry of closed dg-categorie s (Thm.1.0.1). W e giv e a d escription of homotop y in v ariants of not necessarily nilp otent spaces in terms of the minimal mo dels of equiv arian t dg- algebras (Thm.1.0.2). Let k b e a ﬁeld of c haracteristic 0. A closed dg-catego ry is a k -linear dg-catego ry wh ic h is equipp ed with a closed tensor stru ctur e consistent with th e diﬀeren tial graded stru cture (see Def.2.1.1 ). A t yp ical ∗ Corresponding address: Departmen t of Mathematics, F acult y of Science, Kyoto Univers it y , Kyoto, 606-850 2, Japan. E- mail adress: moriyasy@math.kyot o-u.ac.jp T elephone number: 81-075-753-3700 F AX n umber: 81-075-753-3711 1 example of a closed tensor str u cture is tensors and in ternal h oms on the category of representat ions of a group. If one views a dg-algebra as a d g-categ ory w ith only one ob ject, a tensor structur e on a dg-categ ory is a natur al generalization of commutat ivit y of a dg-algebra. W e also n eed to consider internal homs. Th e existence of a mo del category structur e on the catego ry of small closed dg-categories was pr o ved in [30]. In [30], a closed dg-category T PL ( K ) is d eﬁned for a simplicial set (or triangulated space) K . Its ob jects are ﬁ nite r an k k -linear lo cal systems on K and complexes of morp hisms are p olynomial de Rh am complexes with lo cal co eﬃcien ts. This is an analogue of th e dg-category of ﬂ at bund les on a m an if old, whic h w as deﬁned by S impson [12, section 3]. W e consider T PL ( K ) as a generalization of the p olynomia l de Rham algebra which cont ains inform ation of ﬁn ite d imensional representat ion of the fund amen tal group . As we see in [30 ], when the fund amen tal group of K is a ﬁnite group and k = Q , the constru ction K 7→ T PL ( K ) is equiv alen t to the ﬁb er w ise r ationaliza tion of K . But in general, it is diﬀerent as we consider only ﬁnite r ank lo cal systems. W e introdu ce a sp ecia l class of closed d g-categ ories which we call T annakian d g-categories (Def.3.1.1). It is c haracterized by conditions abstracted fr om th e closed dg-category T PL ( K ) of connected K . These conditions are stated in terms of T annakian th eory [11] and the completeness of a dg-category [12, section 3]. T annakian theory concerns a dualit y b etw een aﬃne group sc hemes and certain closed tensor k -linear catego ries (see App en dix A.2 ). The completeness of th e dg-cat egory means that information of exact sequences in the category of 0-th co cycles determines and is determined b y the ﬁ rst cohomology group s of morphisms (see su bsection 2.2). T annakian dg-catego ries are exactly those wh ic h corresp ond to sc hematic homotop y t yp es. While closed dg-categorie s h a ve go o d functorial and homotopical p r op erties, they are not suitable for compu tations. T o co v er this incon venience, we use a π 1 -equiv ariant comm utativ e d g-algebra A red ( K ) deﬁned as follo w s ( π 1 = π 1 ( K )). Let π red 1 b e the pro-reductiv e completion of π 1 o ve r k (see s u bsection 1.1) and O ( π red 1 ) b e its co ordinate r ing. It has t wo actions of π 1 : th e left and right translations. With the right translation, w e regard it as a lo cal system on K . 1. As a complex, A red ( K ) is the p olynomial de Rham complex with co eﬃcient s in the lo cal system O ( π red 1 ). 2. Th e multiplicatio n is d eﬁned from those of p olynomia l forms and the co ordinate r in g. 3. Th e action of π 1 is deﬁned from the left tr anslation on O ( π red 1 ). The imp ortance of this d g-alge bra ﬁrst seemed to b e recog nized by Deligne and it h as b een studied b y Katzark o v, P an tev, T o ¨ en [22, 24, 25] and Pridh am [27, 28, 29]. W e will p ro ve a corresp ondence b e- t ween T annakian dg-ca tegories and equiv ariant dg-algebras, w here T PL ( K ) corresp onds to A red ( K ) (see Prop.3.3.4, Th m.3.4.5). The pr o of is n ot diﬃ cu lt but th is is v er y u s eful. F or example, w e cannot ﬁnd th e natural rep r esen tations of the fun dament al group on h igher homotop y group s if w e see only the dg-algebra, but these repr esentati ons app ear as ob jects of the corresp ondin g d g-categ ory . Results in the follo wing are obtained by using th is corresp ondence. 1.0.1 Main results W e sh all state main results. Let SHT ∗ b e the category of p oin ted schematic homotop y typ es (Def.5.1.1). It is a full sub cat egory of the category of ∞ -stac ks o ve r k , wh ic h is c haracterized b y a certain k -linearity of homotop y sheav es. The schemati zation ( K ⊗ k ) sc h of a simplicial set K is a univ ersal schematic homotop y t yp e for K . Let T an ∗ b e the category of T annakian dg-categories with a ﬁb er functor. A ﬁb er functor of a T annakian dg-categ ory C is a d g-fu n ctor from C to the categ ory of ﬁnite dimensional v ector spaces, whic h preserve s closed tensor stru ctur es (Def.2.1.3). Let s Set c ∗ denote the category of p oin ted connected simp licial sets. 2 Theorem 1.0.1 (Thm.5.2.1) . Ther e exists an e quivalenc e of c ate gories Ho ( SHT ∗ ) − → Ho ( T an ∗ ) op such that the fol lowing diagr am is c ommutative up to natur al isomorph isms: Ho ( SHT ∗ ) ∼ / / Ho ( T an ∗ ) op Ho (s Set c ∗ ) . ( −⊗ k ) sc h O O T PL 7 7 o o o o o o o o o o o Her e, Ho ( − ) denotes the c orr esp onding homotopy c ate gory (se e subse ction 1.1). The e qu ivalenc e i s induc e d by a Q uil len p air b etwe en lar ger mo del c ate gories. It is known that the sc hematization comes fr om A red ( K ). In fact, it is realized as the homotop y qu o- tien t of top ological realization of A red ( K ) b y π red 1 (see [25, Cor.3.3]). But this construction is not natural with resp ec t to m ap s b etw een simplicial s ets as the constru ction K 7→ π 1 ( K ) red is not natural with re- sp ect to th ose w h ic h d o not p reserv e semi-simp le r epresen tations of f undamental groups (see Rem.3.3.2, the naturalit y part of the statemen t of [25, Cor.3.3] is wrong. the naturalit y folds only for the su b category whose morp hisms are those wh ic h preserve semi-simple ones). The theorem sa ys the sc hematization an d the construction K 7→ T PL ( K ) are n aturally equiv alen t. W e prov e this u s ing the mo del structure on the catego ry of closed d g-categ ories and r esults of [22]. W e deduce a similar equiv alence in the unp oin ted case from Th m.1.0.1, see Cor.5.2.7. A feature of the Sulliv an’s theory is that the m in imal m o del describ es the rational homotop y th eory of a space in a very transparent wa y (see [8, P38, Analogy to top ol ogy] or [10, Thm.11.5]). F or a simp ly connected sp ace, the indecomp osable mo du les are dual to the rational h omotop y group s and the diﬀeren tials corresp ond to rational k-in v ariants of the Po stnik ov to wer. The corresp ondence b et w een d g-categ ories and dg-algebras enables us to translate homotop y inv ariants of a sp ace into in v ariants of minimal algebras and we ob tain an analogous description in the non-nilp ote n t case, as follo ws. A red ( K ) h as a minimal mo d el (in the usu al sense) with a semi-simple π 1 -action. Let M b e suc h a minimal mo d el of A red ( K ). Let V 1 , V 2 , . . . , V i , . . . , V i ⊂ M i b e a sequence of semisimple π 1 -mo dules generating M f reely as a commutativ e graded algebra. O f course, V i is isomorphic to the i -th indecomp osa ble mo d ule. As M is min imal, d ( V i ) ⊂ M 1 ⊗ k V i ⊕ M ( i − 1) i +1 , where d is the diﬀerent ial of M and M ( i − 1) is the dg-subalgebra of M generated by L j ≤ i − 1 M j . S o there is a unique decomp osition d | V i = d M 1 ⊗ k V i ⊕ d M ( i − 1) consisting of t wo homomorphisms of π 1 -mo dules d M 1 ⊗ k V i : V i → M 1 ⊗ k V i and d M ( i − 1) : V i → M ( i − 1) i +1 . Theorem 1.0.2 (Lem.3.3.9, Thm.4.1.3) . We use the ab ove notations. Put π i = π i ( K ) for i ≥ 1 . F or i ≥ 2 , we c onsider π i as a π 1 -mo dule b y the c anonic al action. L et n ≥ 2 . Supp ose π 1 is algebr aic al ly g o o d (Def.4.1.1) and π i is of ﬁnite r ank as an Ab elian gr oup for e ach 2 ≤ i ≤ n . Then, for e ach 2 ≤ i ≤ n , 1. the dual ( V i ) ∨ is isomorphic to ( π i ⊗ Z k ) ss , the semisimpliﬁc ation of π 1 -mo dule π i ⊗ Z k , 2. the map d M 1 ⊗ k V i enc o des the inform ation of a pr esentation of π i ⊗ Z k as suc c essive extensions of irr e ducible c omp onents of ( π i ⊗ Z k ) ss , and 3. the map d M ( i − 1) c orr esp onds to the k-invariant (tensor e d with k ) of the i - th level of the Postnikov tower of K . 3 In p articular, the two data V i and d M 1 ⊗ V i determine the π 1 -mo dule π i ⊗ Z k up to isomorphisms. As examples of algebraically goo d groups, ﬁnitely generated free group s, ﬁnitely generated Ab elian groups, fund amen tal groups of Riemann su rfaces are kno wn (see [25, su bsection 4.3]). F or other examples, see [25, Thm.4.16] and Thm.4.2.8. W e pro ve Thm .1.0.2 by an argumen t similar to the pro of of Hirsc h Lemma [10, Thm.11.1]. W e can also reco ver cohomology group s of an y ﬁnite r ank k -lo cal co eﬃcient s easily f r om the minimal mo del. F or a semisimple lo cal system or equ iv alen tly , a semi-simple represent ation V of π 1 , it is H ∗ (( M ⊗ V ) π 1 ), the cohomolog y group of the complex of in v arian ts of M ⊗ V . F or a general ﬁn ite dimensional repr esen tation V , we need to t wist the complex ( M ⊗ V ss ) π 1 b y some Maurer-Cartan elemen t b efore we tak e cohomolog y ( V ss denotes the semi-simpliﬁcation of V ). 1.0.2 Examples W e shall provide a simple example dedu ced fr om T hm.1.0.2. Example 1.0.3 (Example 4.2.5) . Supp ose k is algebr aic al ly close d. L et n ≥ 2 b e an inte ger and M b e a ﬁnite r ank ab elian gr oup with Z - action. L et K = K ( Z , M , n ) := K ( Z ⋉ K ( M , n − 1) , 1) . Her e, K ( M , n − 1) is an Eilenb er g M aclane sp ac e r e alize d as simplicial ab elian gr oup with the induc e d action of Z . L et g ∈ GL( M ⊗ Z k ) b e the action of the gener ater 1 of Z and g = g s + g n b e a Jor dan de c omp osition, wher e g s is semi-simple and g n is nilp otent. Then, V i =    k · s ( i = 1) (( M ⊗ Z k ) ss ) ∨ ( i = n ) 0 ( other w ise ) and d ( s ) = 0 , d ( x ) = t g n ( x ) · s for x ∈ V n . Her e, ( M ⊗ Z k ) ss is the ve ctor sp ac e M ⊗ Z k on which 1 ∈ Z acts by the semisimple p art g s , and Z acts on V 1 trivial ly. W e giv e an explicit description of the mo d el of comp onents of a free lo op space (Prop.4.2.9) under the assumption that the component con tains loops whic h represent an elemen t of the cen ter of the fun damen tal group. W e also present a mo del of cell attac hment (Example 4.2.6 ). Mo dels of these top ological constru c- tions are not kno wn for the formulat ion of [16]. By a metho d similar to the p ro of of Th m .1.0.2, w e will pro v e that for a nilp otent simplicial s et K of ﬁn ite t yp e, th e m in imal mo d el of A red ( K ) is isomorp hic to the Sulliv an’s minimal mo del with trivial π 1 -action (Th m.4.2.3). W e also provi de a description of min imal mo del of a classifying sp ace of a group wh ic h is an extension of give n group by an ab elian group (Thm.4.2.8). 1.0.3 Equiv alence with the ﬁb erw ise rat ionalization In general, it is imp ossible to reco v er the fun damen tal group fr om the closed dg-category . Th e b est thin g we can obtain is th e pro-alge braic completion ([2], see also s ubsection 1.1). S o we cannot exp ect the closed d g- catego ry corresp onds to th e ﬁb erwise rationaliza tion of a space. Instead, we pro v e an equiv alence b et w een the ﬁb erwise rationaliz ation and the closed dg-categ ory w ith sub sidiary data. W e s a y a p ointed connected simplicial set is algebraical ly go o d if its fu ndamenta l group is algebraically go o d (see Def.4.1.1) and eac h higher homotop y groups are ﬁnite dimensional Q -vect or spaces. Let Rep(Γ) denotes the category of ﬁn ite dimensional k -linear representati on of a d iscr ete group Γ. Theorem 1.0.4 (Thm.4.3.4) . Ther e exists a c ate gory T an + g d ∗ whose obje cts ar e triples ( T , Γ , φ ) c onsisting of an obje ct T ∈ T an ∗ , an algebr aic al ly g o o d gr oup Γ , and an e q uivalenc e φ : Z 0 T → Rep(Γ) of close d k -c ate gories with a ﬁb er functor. T o a simplicial set K whose fundamental gr oup is algebr aic al ly go o d and 4 whose higher homotopy gr oups ar e of ﬁnite r ank, we c an assign an obje ct (T PL ( K ) , π 1 ( K ) , φ K ) ∈ T an + g d ∗ . This c onstruction induc es an e quivalenc e b etwe en the homotopy c ate gory of algebr aic al ly go o d sp ac es and the homoto py c ate gory of ( T an + g d ∗ ) op . Under this e quiv alenc e (T PL ( K ) , π 1 ( K ) , φ K ) c orr esp onds to the ﬁb e rwise r ationalization of K . 1.0.4 Relation with other wo rks W e shall mention the r elation with other w orks: [22, 25] and [27]. The sc hematic homoto p y typ es and sc hematization are d eﬁned o v er an y ﬁeld of any characte ristic. W e use resu lts of [22, 25] in this p ap er. In [27], another kind of algebraic mo d els of s paces is p rop osed. They are called pr o-algebraic homotopy t yp es and realized as certain simplicial aﬃne group sc hemes. Sc hematic homotop y t yp es and p ro-algebraic homotop y t yp es are closely related. A pro-algberaic homotop y t y p es are considered as a relativ e pro- unip ot en t completion of a schematic homotop y t y p e and these t wo ob jects are equiv alen t on those which come from sp aces (see [27, Cor.3.57]). Some of the results of this pap er were prov ed earlier b y T o ¨ en and Pridham (see [22, 25] and [27]). F or example, in the n otation of Thm.1.0.2, it was prov ed in [27] that ( V i ) ∨ is isomorphic to π i ⊗ k as v ector spaces und er a bit stronger assumption, see [27, Thm.1.58, Rem.4.43]. A feature of our approac h is that the closed dg-categories are nearer to the equiv ariant dg-algebras than other mo dels. In fact, the corresp ondence b et w een dg-catego ries and dg-algebras is v er y clear and so we can obtain d escriptions of the action of fund amental group on homotop y groups and k-in v arian ts as in Thm.1.0.2 and pro v e fundamental theorems 4.2.3 and 4.2.8. These are new resu lts. 1.0.5 Organization of the pap e r In section 2, w e mainly gather d eﬁnitions and r esults from other pap ers. W e recall basic p rop erties of the completion of dg-categories fr om [12]. W e see th at the completion, sligh tly mo d iﬁ ed, ﬁts in the context of closed dg-catego ries. In section 3, w e pr epare some tec h n ical results to pro v e results in sections 4 and 5. In 3.1 we int ro- duce the notion of T ann akian dg-cateo gries. In 3.3, we compare T annakian d g-categ ories and equ iv arian t dg-algebras and dedu ce some lemmas. In the s tatement s and p ro ofs of these lemmas, we use the mo del catego ry structure on the catego ry of closed dg-categ ories (see Th m.2.1.5). W e also use internal homs. W e in tr o duce a n otion of iterated Hirs c h extensions of dg-algebras. In 3.4, w e deﬁne the functor T PL , whic h w e call the generalized de Rham f unctor, and pro ve that T PL ( K ) comes from A red ( K ). All argumen ts in this section are elemen tary except for the languages of mo del categories. In s ection 4, w e recall the n otion of algebraica lly go o d ness of a d iscrete group and p ro ve Thm.1.0.2 and 1.0.4. W e also provide some examples. F or the p ro of of Thm .1.0.2, w e mainly f ollo w the metho d of [10] and ju stify a tec h nical p art by using the mo del catego ry structure on cubical sets in [20], see App en d ix A.1. In the p ro ofs of some results in this section, we u se results of the next section 5. In section 5 , w e prov e Thm.1.0.1. In 5.1, we r ecall the notions of sc hematic homotopy t y p es and s c hema- tizatio n from [22] and d eﬁ ne a fu nctor from the catego ry of simplicial preshea v es to the category of closed dg-categ ories. Th is is an analogue of the generalized de Rham f unctor (and denoted b y T PL , to o). As for logica l ord er , section 5 is previous to section 4. In App endix, we sho w some v arian ts of the p olynomial d e Rh am theorem and summarize th e T annakian theory of [11]. 1.1 Notations and ter minologies W e ﬁx a ﬁ eld k of c haracteristic 0. Q den otes the ﬁ eld of rational num b ers. All d iﬀeren tial graded ob jects are assumed to b e deﬁned o ver k and non-ne gatively c ohomolo gic al ly gr ade d . W e denote by C ≥ 0 ( k ) th e 5 catego ry of n on-negativ ely cohomologically graded complexes and chain maps . By d g-algebra, w e mean comm utativ e d g-algebra (w ith non-negativ e cohomologica l gradin g). Dg-a lgebra and dg-catego ry are ab- breviated to dga and dgc, r esp ectiv ely . W e denote b y V ect the category of ﬁnite dimens ional k -ve ctor spaces and k -linear maps. More precisely , it denotes a suitably small fu ll su b category , see the paragraph whic h precedes Def.2.1.3 for deﬁnition. W e us e the notations and terminologies in [30, 1.1]. In p articular, for a category (resp. a dg-categ ory) C , Ob( C ) denotes the set of ob jects of C and Hom C ( c 0 , c 1 ) denotes the set of m orphisms (resp. the complex of m orp hisms) b et we en tw o ob jecs c 0 and c 1 ∈ Ob( C ). dg Ca t ≥ 0 denotes the catego ry of small non-negativ ely cohomologi cally graded dg-catego ries and dg-functors. F or a dg-categ ory C , Z 0 C denotes th e catego ry of 0-th co cycles of C . Its ob jects are the same as ob jects of C , and its morp hisms are morph isms of C whic h are co cycles of degree 0. F or t w o d gc’s C and D , C ⊠ D denotes a dgc w h ose ob jects are pairs ( c, d ) of c ∈ Ob( C ) and d ∈ Ob( D ), and whose complexes of morphism s are tensors of those of C and D (see [30]). If a dga A is free as a graded comm utativ e algebra, i.e., it is the tensor of the symm etric algebra generated by ev en degree generators and the w ed ge algebra of o d d d egree ones, we sometimes write A = V ( V i ) where V i denotes a mo du le of generators of degree i . All schemes are assumed to b e deﬁned o v er k . F or an aﬃne sc heme X , we denote by O ( X ) the co ordinate ring of X . W e alwa ys ident ify a ﬁ nite dimensional k -v ector space w ith an aﬃne add itiv e group sc heme in the obvious w ay . Let Γ b e a d iscrete group and G b e an aﬃne group sc h eme. The term, Γ-mo du le or Γ-representa tion represent s the same thing. W e alw a ys iden tify G -mo du les (or G -representat ions) with O ( G )-comod ules (see [11, P .126]). Rep ∞ (Γ) (resp. Rep ∞ ( G )) denotes th e category of p ossibly inﬁnite d imen tional Γ-mo dules (resp. G -mod ules) o ver k and Rep(Γ) (resp. Rep( G )) denotes the fu ll sub cat egory of Rep ∞ (Γ) (resp. Rep ∞ ( G )) consisting of ﬁn ite dimensional ob jects. F or a tec hn ical r eason, w e need to make the categories Rep(Γ) and Rep( G ) suitably s m all. See see th e paragraph whic h precedes Def.2.1.3 for the precise d eﬁnition. O ( G ) has tw o natural G -action: the r igh t and left translations. W e d enote by O ( G ) r (resp. O ( G ) l ) O ( G ) considered as a G -mo dule b y the righ t (resp . left) translation. W e sa y a represen tation (of a discrete group or a group scheme) is semisimple if it can b e decomp osed into a direct s u m of irreducible r epresen tations. W e sa y an aﬃne group s c heme is pr o-r e ductive (or simply , r e ductive ) if any of its representati ons is semi-simple. An y aﬃn e group sc heme G has a decomp osition: G ∼ = R u ( G ) ⋊ G red , where R u ( G ) is the pro-unip oten t r adical of G , and G red = G/ R u ( G ) is p ro-reductiv e (see [26] for details). Represen tations of G red are in one to one corresp o ndence with semi-simple r epresent ations of G via the pullbac k by the pro jection G → G red . The pr o-algebr aic c ompletion of Γ ([2]), d enoted by Γ alg , is an aﬃne group scheme ov er k with a grou p homomorphism ψ Γ : Γ → Γ alg ( k ), w here Γ alg ( k ) denotes the group of k -v alued p oints of Γ alg , such that ﬁnite dimen sional k -linear representati ons of Γ are in one to one corresp ond ence with ﬁnite d imensional represent ations of Γ alg via the pullbac k of action by ψ Γ . W e p ut Γ red := (Γ alg ) red and call it the pr o- r e ductive c ompletion of Γ. Finite dimensional rep resen tation of Γ red are in one to one corresp ondence with ﬁnite dimensional semi-simple k -representati on of Γ via the pullbac k by Γ ψ Γ → Γ alg ( k ) → Γ red ( k ). Our notion of mo del categ ories is that of [14]. Ho ( M ) denotes the homotop y categ ory of a mo del catego ry M . If M ′ is a full sub ca tegory of M whic h is stable under wea k equ iv alences, Ho ( M ′ ) denotes the fu ll sub category of Ho ( M ) spanned b y M ′ . This is isomorphic to the lo calization of M ′ b y w eak equiv alences. [ − , − ] M ′ denotes the set of morphisms of Ho ( M ′ ). s Set (resp. s Set ∗ denote the catego ry of simplicial sets (resp. p oin ted simplicial sets). F or a group Γ, B Γ or K (Γ , 1) denotes the s implicial nerve of Γ. 6 2 Preliminaries 2.1 Closed dg-categories The follo wing is a r ewrite of [30, Def.2.1.1], wh ere w e call th e same ob jects closed tensor dg-categ ories. Deﬁnition 2.1.1 (closed dg-categorie s, dg Cat cl , Cat cl ) . (1) L et C b e an obje ct of dg Cat ≥ 0 . A closed tensor stru cture on C is a 11-tuple (( − ⊗ − ) , 1 , a, τ , u, Hom , φ, ( − ⊕ − ) , s 1 , s 2 , 0 ) c onsisting of 1. a morphism ( − ⊗ − ) : C ⊠ C − → C ∈ dg Cat ≥ 0 , 2. a distinguishe d obje ct 1 ∈ C , 3. natur al isomorphisms a : (( − ⊗ − ) ⊗ − ) = ⇒ ( − ⊗ ( − ⊗ − )) : ( C ⊠ C ) ⊠ C ∼ = C ⊠ ( C ⊠ C ) − → C, τ : ( − ⊗ − ) = ⇒ ( − ⊗ − ) ◦ T C,C : C ⊠ C − → C, u : ( − ⊗ 1 ) = ⇒ id C : C − → C satisfying usual c oher enc e c onditions on asso ciativity, c ommutativity and unity, se e [3, pp.251], 4. a morphism Hom : C op ⊠ C − → C ∈ dg Cat ≥ 0 , 5. a natur al isomorphism φ : Hom C ( − ⊗ − , − ) = ⇒ Hom C ( − , Hom ( − , − )) : C op ⊠ C op ⊠ C − → C ≥ 0 ( k ) , 6. a morphism ( − ⊕ − ) : C × C − → C ∈ dg Cat ≥ 0 , 7. two natur al tr ansformations P 1 s 1 + 3 ( − ⊕ − ) P 2 : C × C s 2 k s / / C, wher e P i : C × C − → C is the i -th pr oje ction, such that the induc e d morphism s ∗ 1 × s ∗ 2 : Hom C ( c 0 ⊕ c 1 , c ′ ) − → Hom C ( c 0 , c ′ ) × Hom C ( c 1 , c ′ ) is an i somorphism (i.e., c 0 ⊕ c 1 is a c opr o duct), and 8. a distinguishe d obje ct 0 ∈ Ob( C ) su c h that Hom C ( 0 , c ) = 0 for any c ∈ Ob( C ) . We c al l ( − ⊗ − ) a tensor f unctor and Hom a in ternal h om functor . (2) A closed dg-category is an obje ct C of dg Cat ≥ 0 e quipp e d with a c lose d tensor structur e. F or two close d dg-c ate gories C, D , a morphism of closed d g-categ ories is a mor phism F : C → D of dgc’s wh ich pr eserves al l of the ab ove structur es. F or example, F ( c ⊗ d ) = F ( c ) ⊗ F ( d ) (not only natur al ly isomorphic), F ( τ c,c ′ ) = τ F c ,F c ′ and F ( 1 ) = 1 . We denote by dg Cat cl the c ate gory of smal l c lose d dgc’s. (3) A closed k -category is a close d dg-c ate gory whose c omplexes of morphisms ar e c onc e ntr ate d in de gr e e 0 and a morphism of closed k -categories i s the same as a morphism of close d dg-c ate gories. W e denote by Cat cl the c ate gory of smal l c lose d k - c ate gories. 7 The imp orta n t part of this d eﬁnition is th e data concerning ( − ⊗ − ) and Hom . W e assume the existence of copro d ucts in order to make the initial ob ject of dg Ca t cl b e (equiv alen t to) th e closed k -categro y of ﬁnite dimensional k -v ector sp aces Notation. W e s et C ( c ) := Hom C ( 1 , c ) for a closed dgc C and an ob ject c ∈ Ob( C ). Example 2.1.2. (1) L et Γ b e a descr ete g r oup. The k - line ar c ate gory Rep(Γ) of ﬁnite dimensional k - line ar r epr esentations has a close d tensor structur e. The tensor of two r epr esentations is, as usual, the tensor of ve ctor sp ac es with the diagonal action and the internal hom is similar. (2)(The dg-categ ory of ﬂat b undles [12]) L et X b e a C ∞ -manifold. A ﬂat b u nd le ( V , D ) on X is a p air of a C ∞ -ve ctor bund le V and a ﬂat c onne ction D : V → A 1 X ⊗ V , wher e A 1 X is the she af of C ∞ one-forms on X . The tensor of two ﬂat bund les ( V , D ) , ( V ′ , D ′ ) is the p air of the tensor of ve ctor b u nd les V ⊗ V ′ and the ﬂat c onne ction D ⊗ id + id ⊗ D ′ . The internal hom is similar. The dg-categ ory C dR of ﬂat bu ndles on X is deﬁne d as fol lows. Its obje cts ar e ﬂat bu nd les on X and its c omplex of morphisms b etwe en ( V , D ) and ( V ′ , D ′ ) i s the twiste d de Rham c omplex of forms with c o eﬃcients in the internal hom Hom (( V , D ) , ( V ′ , D ′ )) . It is e asy to se e the tensor and the internal hom deﬁne d ab ove ar e e xtende d to a close d tensor structur e on C dR . Se e [ 12 ] for details. (3) The r e ader may fe el the deﬁnition of morphism of close d dgc’s is non-natur al as it do es not r e qu ir e natur al isomor phisms but e qualities. The motiva tion of th is deﬁnitio n is to ensur e that dg Cat cl is close d under limits and c olimits. An example of a morphism of dg Cat cl is the fu nctor Rep(Γ ′ ) → Rep(Γ) induc e d by a gr oup homomo rphim Γ → Γ ′ . W e apply the n otions of an equ iv alence and a q u asi-equiv alence to morp hisms of dg Cat cl via the forgetful functor dg Ca t cl − → dg Cat ≥ 0 . F or example, w e sa y a morp hism in dg Cat cl is an equiv alence if it induces an equiv alence of u nderlying categories. Note th at an equiv alence in dg Cat cl do es n ot alw ays hav e a q u asi- in verse whic h is a morphism of dg Cat cl . W e sa y t wo ob ject s of dg Cat cl are equiv alent if th ey can b e connected b y a ﬁnite c hain of equiv alences in dg Cat cl . W e denote the initial ob ject of d g Ca t cl b y V ect. V ect is equiv alen t to the closed k -catego ry of all ﬁn ite dimensional k -v ector spaces and k -linea r maps. In fact, V ect is id en tiﬁed with the smallest full sub cate gory whic h includes the distinguished ob jects 1 and 0 and is closed un der ⊗ , Hom , and ⊕ . In the r est of the pap er, we assume a vec tor space wh ich und erlies a ﬁnite dimensional repr esen tation of a discrete group or an aﬃne group scheme b elongs to V ect. Deﬁnition 2.1.3 (closed dg-categorie s with a ﬁb er functor, dg Ca t cl ∗ , Cat cl ∗ ) . (1) The category of closed dg-categories with a ﬁb er functor is the over c ate gory dg Cat cl / V ect and denote d b y dg Cat cl ∗ . A n obje ct ( C, ω C ) of d g Cat cl ∗ c onsists of a close d dg-c ate gory C and a morphism ω C : C → V ect ∈ d g Cat cl . We c al l ω C the ﬁb er f unctor of C . (2) A closed k -category with a ﬁb er fun ctor is a close d dg-c ate gory with a ﬁb er fu nctor whose c omplexes of morphism s ar e c onc entr ate d in de gr e e 0 and morph isms of closed k -ca tegories with a ﬁb er fu nctor ar e the same as morphisms of close d tensor dg-c ate gories with a ﬁb er functor. We denote by Cat cl ∗ the c ate gory of smal l close d k -c ate gories with a ﬁb er functor. Example 2.1.4. L et Γ (r esp. G ) b e a discr ete g r oup (r esp. an aﬃne gr oup scheme). We r e gar d Rep(Γ) (r esp. Rep( G ) ) as an obje ct of Cat cl ∗ with the for getful functor to V ect . The follo wing is prov ed in [30]. Theorem 2.1.5 (Thm.2.3.2 of [30]) . (1) The c ate gory dg Cat cl has a c oﬁbr antly gener ate d mo del c ate gory structur e wher e we ak e quivalenc es and ﬁbr ations ar e deﬁne d as fol lows. 8 • A morphism F : C → D ∈ dg Cat cl is a we ak e quivalenc e if and only if it is a quasi-e quivalenc e. • A morphism F : C → D ∈ d g Cat cl is a ﬁbr ation if and only if it satisﬁes the fol lowing two c onditions. – F or c, c ′ ∈ O b( C ) the morp hism F ( c,c ′ ) : Hom C ( c, c ′ ) → Hom D ( F c, F c ′ ) is a levelwise e pimor- phism. – F or any c ∈ Ob ( C ) and an y i somorphism f : F c → d ′ ∈ Z 0 ( D ) , ther e exists an i somorph ism g : c → c ′ ∈ Z 0 ( C ) such that F ( g ) = f . (2) d g Cat cl ∗ has a mo del c ate gory structur e induc e d by that of dg Ca t cl . 2.2 Completeness of dg-category W e sh all recall the notion of completeness of a dg-category from [12, section 3]. Let C b e a dg-catego ry . An extension in C is a pair of morphisms c 0 a → c 2 b → c 1 with a, b ∈ Hom 0 , b ◦ a = 0 and d ( a ) = 0, d ( b ) = 0, suc h that a splitting exists: a splitting is a pair of morphisms of degree 0 c 0 g ← c 2 h ← c 1 suc h that g a = id c 0 , bh = id c 1 and ag + hb = id c 2 . W e deﬁn e a morphism δ ∈ Hom 1 ( c 1 , c 0 ) by δ = g d ( h ). d ( δ ) = 0 and δ deﬁnes a class [ δ ] ∈ H 1 (Hom( c 1 , c 0 )). It is easy to c heck that this class is indep enden t of a c hoice of sp littings. W e call [ δ ] the class of the extension c 0 → c 2 → c 1 . Deﬁnition 2.2.1. We say a dg-c ate gory C is c omplete if f or e ach c 0 , c 1 ∈ O b( C ) any class in H 1 (Hom( c 1 , c 0 )) is a class of some extension. Example 2.2.2. L et C dR b e the c ate gory of ﬂat bund les deﬁne d in Example 2.1.2,(2). As in [12] , C dR is c omplete. In fact, for a c o cycle δ ∈ Hom 1 C dR (( V , D ) , ( V ′ , D ′ )) , we deﬁne a ﬂat bund le ( V ′′ , D ′′ ) by V ′′ = V ⊕ V ′ and D ′′ =  D 0 δ D ′  . The se quenc e ( V ′ , D ′ ) − → ( V ′′ , D ′′ ) − → ( V , D ) is an extension c orr esp onding to δ . W e will construct the completion of a dgc C w hic h has ﬁnite copro du cts. W e deﬁn e a dgc C as follo ws. The ob jects of C are pairs ( c, η ) with c ∈ Ob( C ) and η ∈ Hom 1 ( c, c ) such that d ( η ) + η 2 = 0 . ( W e call an elemen t η satisfying the ab o v e equation a Maur er-Cartan (M C) element on c .) W e set Hom n (( c, η ) , ( c ′ , η ′ )) := Hom n ( c, c ′ ) and d C ( f ) := d C ( f ) + η ′ ◦ f − ( − 1) deg f f ◦ η . W e identify C with the f ull sub -dg-categ ory of C consisting of ob jects of the form ( c, 0), c ∈ C and d eﬁne the completion b C to b e th e smallest f ull sub -dg-catego ry of C including C and closed u nder extensions (and isomorphisms). The follo wing is a rewrite of [12, Lemma 3.1, Lemma 3.3]. 9 Lemma 2.2.3 ([12]) . (1) L et C b e a dgc close d under ﬁnite c opr o ducts. b C is c omplete. F or two obje cts ( c 0 , η 0 ) , ( c 1 , η 1 ) ∈ b C , a MC-element c orr esp onding to a c o cycle α ∈ Z 1 (Hom b C (( c 1 , η 1 ) , ( c 0 , η 2 ))) is given by  η 0 α 0 η 1  . (2) The functor C → b C , c 7→ ( c, 0) is initial up to natur al isomorphims among dg-functors C → D with D c omplete. Mor e pr e cisely, for such functor F : C → D ther e exists a fu nctor b F : b C → D factorizing F and such b F is unique up to u ni q ue natur al isomorph isms. (3) L et F : C → D b e a quasi-e quivalenc e. The induc e d functor b F : b C → b D is also a quasi-e quiv alenc e. 2.2.1 Completion and closed dg-categories A closed tens or structure on C indu ces a closed tensor structure on b C as f ollo ws. ( c 1 , η 1 ) ⊗ ( c 2 , η 2 ) = ( c 1 ⊗ c 2 , η 1 ⊗ id + id ⊗ η 2 ) , Hom (( c 1 , η 1 ) , ( c 2 , η 2 )) = ( Hom ( c 1 , c 2 ) , Hom (id , η 2 ) − Hom ( η 1 , id)) . The other stru ctur es such as h omomorphisms b etw een complexes of morph ism s and natural isomorphisms are the same as those of C . If C is a closed dgc, w e consider b C as a closed dgc with this in d uced structure. Note that this closed dgc do es n ot ha ve a unive rsalit y lik e L em.2.2.3 in dg Cat cl . W e s hall mo dify b C . The mo diﬁcation mak es the tensor and the internal hom ”free”. F or example, w e w an t to av oid t w o ob jects ( c 0 ⊗ c 1 ) ⊗ c 2 and c 0 ⊗ ( c 1 ⊗ c 2 ) h app en to b e equal. Let D b e a closed d gc. Let W cl (Ob( D ) ⊔ { 1 , 0 } ) denote the set of the words generated by Ob( D ) ⊔ { 1 , 0 } with op eratio ns ⊗ ′ , Hom ′ and ⊕ ′ ( 1 and 0 are f ormal s y mb ols, see [30, sub -su bsection 2.2.2] for an explicit deﬁnition). W e regard Ob( D ) ⊔ { 1 , 0 } as a s ubset of W cl (Ob( D ) ⊔ { 1 , 0 } ). Let R : W cl (Ob( D ) ⊔ { 1 , 0 } ) − → Ob( D ) ⊔ { 1 , 0 } b e the function giv en by 1. R X = X for X ∈ Ob( D ) ⊔ { 1 , 0 } . 2. R ( X ⊗ ′ Y ) = RX ⊗ RY , R ( Hom ′ ( X, Y )) = Hom ( R X , RY ) an d R ( X ⊕ ′ Y ) = R X ⊕ R Y , inductiv ely . W e deﬁne a closed dgc D c as a ”pullbac k” of D by R : Ob( D c ) = W cl (Ob( D ) ⊔ { 1 , 0 } ) , Hom D c ( X, Y ) = Hom D ( RX , RY ) . Clearly the construction dg Ca t cl ∋ D 7− → D c ∈ dg Ca t cl is f unctorial and R ind uces a n atural equiv alence R D : D c − → D ∈ dg Cat cl . Lemma 2.2.4. L et E ∈ dg Cat cl . L et C , D ∈ dg Cat cl /E . Supp ose D is c omplete. (1) Supp ose D is ﬁbr ant in dg Cat cl /E . We r e gar d C c as an obje ct over E whose augmentation a C c : C c → E is th e c omp osition C c R C / / C / / E . Supp ose an augmentation a b C c : ( b C ) c → E ∈ dg Cat cl such that a b C c ◦ ( I C ) c = a C c ( I C : C → b C is the c anonic al inclusion) is given. L et F : C c − → D ∈ dg Cat cl /E b e a morphism. Ther e e xists a morphism e F : ( b C ) c − → D ∈ dg Ca t cl /E such that the fol lowing diagr am is c ommutative. C c F / / ( I C ) c   D b C c e F > > } } } } } } } 10 L et e F ′ : ( b C ) c − → D ∈ dg Cat cl /E b e another morphism with F = e F ′ ◦ ( I C ) c . Then ther e exists a unique natur al i somorphism ϕ : e F ⇒ e F ′ pr eserving tensors, such that ϕ | C c and ( a D ) ∗ ( ϕ ) ar e the identities ( a D : D → E is the augmentation of D ). (2) Assume the inclusion E → b E i s an isomorphism (i.e., M C-elements in E ar e al l zer os). The pul lb ack by the c anonic al functor C → b C gives a bije ction: [ b C , D ] dg Cat cl /E ∼ = [ C, D ] dg Cat cl /E . Her e, the augmentation b C − → E is g i ven by b C → b E ∼ = E . Pr o of. (1) W e ﬁr st consider the case of E = ∗ . A s in [12], one can c h o ose a f unctor F 1 : b C c → D s uc h that F = F 1 ◦ ( I C ) c and F 1 preserve s tensors and internal hom’s u p to natural isomorph isms wh ic h are compatible with coherency isomorphisms. Since the ob jects of b C c are freely generated by ob jects of b C and { 0 , 1 } , one can mo dify F 1 so that it b ecomes a morphism of d g Cat cl . T hus we get e F . Th e latter part is similar to [12]. F or general E , we ﬁ rst tak e a morph ism F 2 : b C c → D ∈ dg Cat cl suc h that F = F 2 ◦ ( I C ) c . By the latter claim in the case of E = ∗ , we h av e a un iqu e natural isomorphism ϕ : a D ◦ F 2 ⇒ a b C c . Let x ∈ Ob( b C ) ⊔ { 1 , 0 } . As the augmen tation a D : D → E ∈ dg Ca t cl is a ﬁbration, one can lift ϕ x to an isomorphism f x : F 2 ( x ) → ∃ e F ( x ) ∈ D . Using f x ’s, one can mo dify F 2 so that a D ◦ F 2 = a b C c . This mod iﬁed F 2 is th e required e F . Th e latter claim follo ws from the case of E = ∗ (2) W e ma y assume C is coﬁbran t and D is ﬁbrant in dg Cat cl /E . Note that C c and b C c are coﬁbr an t. Indeed, let Q ( C c ) b e a coﬁbrant replacemen t of C c with a trivial ﬁbration Q ( C c ) → C c . As the comp osition Q ( C c ) → C c → C is a trivial ﬁb ration, we can tak e a right inv erse C → Q ( C c ). Using this morp hism, one can see the map Q ( C c ) → C c has a r ight inv erse so C c is coﬁb ran t. F or b C c , one can ﬁ nd a r ight inv erse of a trivial ﬁbration Q ( b C c ) → b C c using (1). W e only ha v e to c h ec k the map (( I C ) c ) ∗ : Hom dg Cat cl /E ( b C c , D ) − → Hom dg Cat cl /E ( C c , D ) induces a bijection b et w een th e sets of righ t homotop y classes. T h e surjectivit y follo ws from (1). Let e F 1 , e F 2 : b C c − → D ∈ dg Cat cl /E b e t wo morp hisms su c h that e F 1 | C c and e F 2 | C c are r ight homotopic. Let P D b e a path ob ject of D . W e ma y regard C c , b C c and P D as ob jects of dg Cat cl /D × E D . Then by (1), there exists a m orphism e H : b C c − → P D ∈ dg Cat cl /E such that the follo wing d iagram commutes. C c H / /   P D   b C c e H : : u u u u u u u u u e F 1 × e F 2 / / D × E D Th us, e F 1 and e F 2 are righ t homotopic. 3 T annakian dg-categories and reductiv e equiv arian t dg-algebras 3.1 T annakian dg-categories W e shall deﬁne T annakian dg-catego ries. F or the deﬁnition of neutral T annakian catego ries, see App endix A.2. Deﬁnition 3.1.1 (T annakian dgc’s, T an , T an ∗ ) . L et C b e a close d dg-c ate gory. We say C is a T annakian dg-c ate gory if the fol low ing c onditions ar e satisﬁe d. 11 1. Z 0 ( C ) is a neu tr al T annakian c ate g ory with r esp e ct to the close d tensor structur e induc e d fr om that of C or e qu ivalently, ther e exist an aﬃne gr oup sc heme G over k and a ﬁnite chain of morphisms of close d k -c ate gories: Z 0 ( C ) → C 1 ← C 2 → · · · ← C n → Rep( G ) , wher e al l arr ows ar e e qui valenc es of underlying c ate gories. In p articular, Z 0 ( C ) is an Ab elian c ate gory. 2. C is c omplete (se e Def.2.2.1). 3. If 0 → c 0 → c 1 → c 2 → 0 is a short exact se quenc e in Z 0 ( C ) then c 0 → c 1 → c 2 is an extension in C in the sense explaine d in the b e ginning of subse ction 2.2 . We denote by T an the ful l sub c ate gory of dg Cat cl c onsisting of T annakian dg-c ate gories. We denote by T an ∗ the ful l sub c ate gory of dg Ca t cl ∗ c onsisting of obje cts whose underlying close d dgc b elongs to T an . The thir d condition means that extensions in terms of represen tations and extensions in terms of a dgc coincide. Let C b e a T annakian dg-category and C ss denote fu ll sub dg-category of C consisting of semisimple ob ject s of Z 0 C . The third one is equiv alen t to the one that the m orphism c C ss c → C indu ced b y the natural inclusion is a quasi-equiv alence. So T an and T a n ∗ are s table un der wea k equiv alences of dg Cat cl and dg Cat cl ∗ , resp ectiv ely . Note that for a T annakian d g-categ ory with a ﬁ b er fu nctor ( C, ω C ) ∈ T an ∗ , the functor Z 0 ω C : Z 0 C → V ect is automatically exact and faithful. 3.2 Equiv arian t dg-algebras Let G b e an aﬃne group sc h eme. W e denote b y dg Alg ( G ) the c ate gory of G -e quivariant dg-algebr as . An ob ject of d g A l g ( G ) is a comm u tativ e dg-algebra with a O ( G )-comod ule structure whic h is compatible with the grading, the diﬀerentia l and the algebra structure. A morphism of dg Al g ( G ) is a morphism of d ga’s whic h is compatible with the O ( G )-comod u le structures. G -equiv arian t dg-alge bra is abb reviated to G -dga. W e sa y a G -dga A is c onne cte d if H 0 A ∼ = k . dg A lg ( G ) 0 denotes the f u ll sub categ ory of d g A lg ( G ) consisting of conn ected ob jects. The follo wing can b e prov ed b y an argument similar to the non-equiv ariant case. Prop osition 3.2.1. L et G b e a r e ductive aﬃne gr oup scheme. The c ate gory dg Al g ( G ) admits a mo del c ate gory structur e such that 1. a morphism f : A − → B ∈ dg Al g ( G ) is a we ak e qui valenc e if and only if it is a quasi-isomorphism (of underlying c omplexes), and 2. a morphism f : A − → B ∈ dg Alg ( G ) i s a ﬁbr ation if and only if it is a levelwise epimorphism. Deﬁnition 3.2.2 (minimal G -dga’s) . L et G b e an aﬃne gr oup scheme. We say a c onne cte d G -dga is minimal if its underlying dga is minimal i n the usual sense (se e [6, 10]). L et A b e a c onne cte d G -dga. A minimal mo del of A is a minimal G -dga M such that ther e exists a we ak e quivalenc e M → A of G -dga’s. F or a minimal G -dga M , the G -mo dule [ M / ( M ≥ 1 · M ≥ 1 )] i is c al le d the i -th indecomp osable mo dule of M . Her e, M ≥ 1 · M ≥ 1 is the submo dule of M ge ner ate d by { x · y | d eg x ≥ 1 , deg y ≥ 1 } . The follo wing is w ell-kno wn. 12 Prop osition 3.2.3. L et G b e a pr o-r e ductive aﬃne gr oup scheme. Then, any c onne cte d G -dga has a minimal mo del. such minimal mo del is unique up to non-uni q ue natur al isomorph isms. Pr o of. The pr o of is similar to the u sual trivial group case. See [6, 10, 8]. Deﬁnition 3.2.4 (r ed uctiv e dga’s, dg A lg red 0 , dg Alg red 0 , ∗ ) . 1. A redu ctiv e equiv ariant dg-algebra i n short, a reductiv e dga i s a p air ( G, A ) of a pr o-r e ductiv e aﬃne gr oup scheme G and a G -dga A . 2. A morphism of reductiv e dga’s f : ( G, A ) → ( H, B ) i s a p air of a morphism of gr oup schemes f gr : H → G and a morphism of H - dga’s f : ( f gr ) ∗ A. − → B . (Note that f g r deﬁnes a functor ( f g r ) ∗ : dg Alg ( G ) − → dg Alg ( H ) by pul ling b ack the g r oup action.) 3. A morph ism of r e ductive dga’s f : ( G, A ) → ( H , B ) is said to b e a quasi-isomorphism if f gr is an isomorph ism and f : ( f gr ) ∗ A. − → B is a quasi-isomorphism. The c ate gory of r e ductive dga’s is denote d by d g Alg red . We say a r e ductiv e dga ( G, A ) is c onne cte d if H 0 ( A ) ∼ = k . We denote by dg Al g red 0 the f ul l sub c ate gory of dg A lg red c onsisting of c onne cte d obje cts and by dg Alg red 0 , ∗ the over c ate gory dg Al g red 0 / ( e, k ) , wher e e is the trivial gr oup. We always identify the c ate gory dg Alg ( G ) with a sub c ate gory of d g Alg red in the obvious way. F or an obje ct ( G, A ) ∈ dg Alg red 0 , a minimal mo del of ( G, A ) is a minimal mo del of A as a G -dga in the sense of Def.3.2.2 3.3 Comparison In th is s ubsection, w e show a corresp ondence b et w een reductiv e equ iv ariant dga’s an d T annakian d gc’s. The direction from dga to d gc is functorial b ut the other direction is not so and w e on ly hav e a f unction A red : Ob( T an ∗ ) → Ob(dg A lg red 0 , ∗ ), see Rem.3.3.2. W e shall deﬁne t wo functors T ss : dg Alg red 0 − → d g Cat cl , T : dg Alg red 0 − → T an . Let A = ( G, A ) ∈ d g Alg red 0 . F or a como dule M ∈ Rep ∞ ( G ), w e deﬁne the mo du le of inv ariant s M G b y M G := Ker(id M ⊗ k − ρ M : M − → M ⊗ O ( G ) ) , where k : k → O ( G ) is the unit map and ρ M : M → M ⊗ O ( G ) is the coactio n. W e set Ob( T ss A ) := Ob(Rep( G )) , Hom T ss A ( V , W ) := ( Hom ( V , W ) ⊗ A ) G . Here, Hom is the int ernal hom of Rep( G ), and Hom ( V , W ) ⊗ A is considered as a complex of comod ules. ( Hom ( V , W ) ⊗ A ) G is deﬁned by taking in v ariants in th e d egreewise manner. W e deﬁne the comp osition and closed tensor structure of T ss ( A ) u sing corr esp onding structures of Rep( G ) and th e multiplica tion of A . A morp hism f : ( G, A ) → ( H , B ) of dg Alg red 0 giv es a functor ( f g r ) ∗ : Rep( G ) → Rep( H ) and f induces a morp hism T ss f : T ss A → T ss B of closed dg-categories. W e set T A := [ T ss A. The right hand sid e is the completion (see sub s ection 2.2). As T ( e, k ) ∼ = T ss ( e, k ) ∼ = V ect, a morphism ( G, A ) → ( e, k ) deﬁnes morphisms T ss ( G, A ) → V ect , T ( G, A ) → V ect so we obtain fu nctors b et ween augmen ted catego ries: T ss : dg Alg red 0 , ∗ − → d g Cat cl ∗ , T : dg Alg red 0 , ∗ − → T an ∗ . W e omitt the pro of of the follo wing. See [26, Lem.1.3].the reductivit y of G is necessary for (2). 13 Lemma 3.3.1. (1) F or V ∈ Ob(Rep ∞ ( G )) the c o action V → V ⊗ k O ( G ) induc es an i somorph ism of ve ctor sp ac es: V → ( V ⊗ O ( G ) r ) G . Similarly, the c o action also i nduc es an isomorphism V → V ⊗ G O ( G ) l . Her e, V ⊗ G O ( G ) l = Ker((id V ⊗ τ ) ◦ ( ρ V ⊗ id O ( G ) ) − id V ⊗ λ G : V ⊗ O ( G ) l → V ⊗ O ( G ) l ⊗ O ( G )) , wher e τ : O ( G ) ⊗ O ( G ) → O ( G ) ⊗ O ( G ) is the isomorph ism given by τ ( x ⊗ y ) = y ⊗ x and λ G is the c o action of the left tr anslation.(In short, V ⊗ G O ( G ) l = { Σ i v i ⊗ l i ; Σ i g · v i ⊗ l i = Σ i v i ⊗ g · l l i ∀ g ∈ G } ) In p articular, as su bmo dules of V ⊗ O ( G ) , ( V ⊗ O ( G ) r ) G = V ⊗ G O ( G ) l . (2) If f : A → B ∈ dg A lg ( G ) 0 is a quasi-isomorphism, the induc e d morphism T f : T A → T B ∈ dg Cat cl is a quasi-e quivalenc e. (3) T A is a T annakian dgc for any A ∈ d g Alg red 0 . Remark 3.3.2. If we deﬁne the homotop y catego ry Ho (d g Alg red 0 , ∗ ) of dg Alg red 0 , ∗ as the lo c alization of dg Alg red 0 , ∗ obtaine d by inv e rting quasi-isomorph isms, by L emma 3.3.1 T induc es a functor b etwe en homotopy c ate- gories: T : Ho (dg Alg red 0 , ∗ ) − → Ho ( T an ) . Unlike the ﬁnite gr oup c ase [30], this is not an e qu i valenc e simply b e c ause morphisms b etwe en T annakian dgc’s do not always pr eserve semisimple obje cts. By the same r e ason the c onstruction T an ∋ T 7− → A red T ∈ dg Alg red 0 , ∗ is not functorial. If we r estrict morphisms of T an ∗ to those which pr eserves semi-simple obje cts, this function is extende d to a functor which induc es an e quivalenc e of homotopy c ate gories, se e Pr op.3.3.4 and L em.3.3.7. Deﬁnition 3.3.3 (the reductiv e dga asso ciated to T ; A red T ) . L et T = ( T , ω T ) b e a T annakian dg-c ate gory with a ﬁb er functor. L et π 1 (Z 0 T ) denote the aﬃne gr oup scheme which r epr esents Aut ⊗ (Z 0 ω T ) (se e A pp endix A.2). We r e gar d O ( π 1 (Z 0 T ) red ) as a π 1 (Z 0 T ) red -r epr esentation by the right tr anslatio n. We deﬁne an augmente d π 1 (Z 0 T ) red -e quivariant dg-algebr a A red ( T ) ∈ dg A lg red 0 , ∗ as fol low s. L et O ( π 1 (Z 0 T ) red ) = [ λ V λ b e the pr esentation as the union of ﬁnite dimensional π 1 (Z 0 T ) red -subr epr ese ntations. We r e gar d V λ ’s as obje cts of T via the e quivalenc e ] Z 0 ω T : Z 0 T → Rep( π 1 ( T ) red ) (se e A pp endix A.2). A s a c omplex, we set A red ( T ) = coli m λ T ( V λ ) (se e the notation under Def.2.1.1). The c olimit is taken i n the c ate gory C ≥ 0 ( k ) (in th is c ase, this is the set-the or etic union). The k -algebr a structur e on O ( π 1 (Z 0 T ) red ) deﬁnes a structur e of dga on A red ( T ) , the left tr anslation of π 1 (Z 0 T ) red on O ( π 1 (Z 0 T ) red ) deﬁnes an actio n on A red ( T ) and the c ounit map of O ( π 1 (Z 0 T ) red ) and the ﬁb er functor of T deﬁnes an augmentation of A red ( T ) . Note that we do n ot say the c onstruction T 7− → A red ( T ) is functorial. Prop osition 3.3.4. (1) F or an augmente d 0-c onne cte d r e ductive dga A , A red T ( A ) i s isomorphic to A in dg Alg red 0 , ∗ . (2) F or a T annakian dgc ( T , ω T ) with a ﬁb er functor, T A red ( T ) is e quivalent to T in d g Cat cl ∗ . (3) F or a T annakian dgc T ∈ T an , ther e exists a c onne cte d r e ductive dga A ∈ dg A lg red 0 such that T A i s e quivalent to T as a close d dg-c ate gory. 14 By this pr op osition, the functor T : d g A l g red 0 , ∗ → T an ∗ and the fu nction A red : Ob( T an ∗ ) → Ob(dg Alg red 0 , ∗ ) induce a bijection b et w een the isomorphism classes (resp. quasi-isomorphism classes) of dga’s and the equiv alence classes (resp. quasi-equiv alence classes) of closed d gc’s. Pr o of. (1) This follo ws from Lem.3.3.1 ,(1) and the fact that ﬁnite limit and ﬁltered colimit (in the cate gory of m o dules) commutes. (2) Let A = A red T . W e ma y assu me (Z 0 T ) ss = Rep ( G ) where G = π 1 (Z 0 T ) red . In the follo win g we deal with O ( G ) r as if it is an ob ject of Rep( G ). As th e action of G on O ( G ) r is lo cally ﬁn ite, this do es n ot matter. F or an ob ject V ∈ O b(Rep( G )) w e ha ve an exact sequ ence of G -mo dules: 0 / / V ρ V / / V u ⊗ O ( G ) r φ / / V u ⊗ O ( G ) u ⊗ O ( G ) r , where ( − ) u denotes the corresp ond ing trivial mo d ule and φ := (id V ⊗ τ ) ◦ ( ρ V ⊗ id O ( G ) ) − id V ⊗ λ G , see Lem.3.3.1. W e ha ve a m ap b et w een sequences of complexes 0 / / ( V ⊗ A ) G / /          ( V u ⊗ O ( G ) r ⊗ A ) G φ ⊗ id A / / ( V u ⊗ O ( G ) u ⊗ O ( G ) r ⊗ A ) G V u ⊗ A / /   ρ A O O V u ⊗ O ( G ) u ⊗ A   ρ A O O 0 / / Hom T ( 1 , V ) / / Hom T ( 1 , V u ⊗ O ( G ) r ) φ ∗ / / Hom T ( 1 , V u ⊗ O ( G ) u ⊗ O ( G ) r ) . As the b oth horizon tal sequ ences are lev elwise exact and the v ertical arrows are isomorph isms by L em.3.3.1, the dotted arrow uniquely exists. W e deﬁne a d g-functor F : T ss A − → T by F ( V ) = V for V ∈ Ob(Rep( G )) and F ( V ,W ) : Hom T ss A ( V , W ) − → Hom C ( F V , F W ) b eing the comp ositio n Hom T ss A ( V , W ) = ( Hom Rep( G ) ( V , W ) ⊗ A ) G − → Hom T ( 1 , Hom T ( V , W )) ∼ = Hom T ( V , W ) . One can c h ec k this is a morph ism of dg Ca t cl ∗ . By deﬁnition of T annakian dgc’s, F is extended to an equiv alence F : T A c − → T ∈ d g Cat cl ∗ (see Lem.2.2.4). The pro of of (3) is similar to th at of (2). The follo wing lemmas are used later Lemma 3.3.5. (1) L et G b e a r e ductive aﬃne gr oup scheme. Then, T ss ( G, k ) c is c oﬁbr ant in dg Cat cl ∗ . If f : A → B ∈ dg Alg ( G ) b e a c oﬁbr ation b etwwen 0-c onne cte d G -dga’s, then T ss ( f ) c : T ss ( G, A ) c → T ss ( G, B ) c is a c oﬁbr ation in d g Cat cl . (2) L et ( G, A ) ∈ dg Al g red 0 , ∗ . Ther e exists a c ommutative squ ar e T ss ( G, k ) c   / / V ect   T ss ( G, A ) c / / T ss ( e, A ) wher e e denotes the trivial gr oup, the left vertic al morphism is induc e d by the unit k → A and the b ottom horizonta l morphism is induc e d by the morphism ( G, A ) → ( e, A ) which is the identity on dg-algebr as. This diagr am i s a pushout squar e in dg Cat cl ∗ and a homotopy pushout squar e. 15 Pr o of. (1) Th e pro of is similar to that of [30, Th m.3.2.10]. (2) W e pro v e the ﬁr st assertion. Let T ss ( G, k ) c   ω / / V ect   T ss ( G, A ) c F / / C b e a comm utativ e square in dg Cat cl . F or V , W ∈ Ob( T ss ( e, A )) w e deﬁne a morph ism of complex e F ( X,Y ) : Hom T ss ( e,A ) ( V , W ) − → Hom C ( V , W ) as the follo wing comp osition. Hom( V , W ) ⊗ k A ∼ = ( Hom ( V , W ) ⊗ O ( G ) ⊗ A ) G ∼ = Hom T ss A c ( V , W ⊗ O ( G )) F → Hom C ( V , W ⊗ F ( O ( G ))) F u → Hom C ( V , W ) . Here, u : O ( G ) − → 1 ∈ V ect denotes the counit map. It is easy to c hec k F ( V ,W ) ’s form a morph ism e F : T ss ( e, A ) − → C ∈ dg Cat cl ∗ and this is the uniqu e m orp hism making appropr iate diagram commutat iv e. By (1) and [14 , Lem.5.2.6], this is a h omotop y p ushout square so the former one is. Remark 3.3.6. We c an give an explicit c oﬁbr ant mo del f or any T annakian dg-c ate gory. In fact, by L em.2.2.4 and L em.3.3.5, for a c oﬁbr ant c onne cte d G -dga A , T A c is c oﬁbr ant in dg Cat cl (se e the pr o of of L em.2.2.4 , (2)). The follo wing giv es a description of hom-sets of closed dg-categorie s in terms of dg-algebras and closed k -categorie s. This is u sed to translate a prop ert y of schematic h omotop y types into a prop ert y of equiv arian t dg-algebras, s ee the pro of of Prop.4.2.9. Lemma 3.3.7. (1) L et E b e a close d dgc and C ∈ dg Cat cl /E b e a close d dgc over E . L et A = ( G, A ) ∈ dg Alg red 0 b e an obje ct and T ss A c − → E ∈ dg Cat cl b e a morphism . We c onsider T ss A c as an obje ct of dg Cat cl /E . F or a morphism α : Z 0 T ss A c → Z 0 C ∈ Cat cl We deﬁne a dg-algebr a C α ∈ dg Alg ( G ) as f ol low s. W e ﬁx a pr esentation of O ( G ) r as a union of ﬁnite dimensional G -r epr e se ntations: O ( G ) r ∼ = S λ V λ . We set C α := colim λ C ( α ( V λ )) , se e the notation under Def. 2.1.1. The algebr a structur e is deﬁne d fr om that of O ( G ) and the action of G is deﬁne d fr om the left tr anslation on O ( G ) . Then, ther e e xists a bije ction: Hom dg Cat cl /E ( T ss ( A ) c , C ) ∼ =  ( α, β )     α : Z 0 T ss A c → Z 0 C ∈ Cat cl / Z 0 E , β : A → C α ∈ dg A lg ( G ) /E α  If the augmentation a C : C → E ∈ d g Cat cl is a ﬁbr ation, this bije ction induc es a bije ction: [ T ss A, C ] dg Cat cl /E ∼ = { ( α, β ) | α ∈ [Z 0 T ss A, Z 0 C ] Cat cl / Z 0 E , β ∈ [ A, C α ] ′ dg Alg ( G ) /E α } . Her e, 1. [Z 0 T ss A, Z 0 C ] Cat cl / Z 0 E := Hom Cat cl (Z 0 T ss A c , Z 0 C ) / ∼ , wher e α 1 ∼ α 2 if and only if ther e exists a natur al isomor phism t : α 1 ⇒ α 2 such that t pr eserves tensors, i.e., t ( X ⊗ Y ) = t ( X ) ⊗ t ( Y ) , and ( a C ) ∗ t : a C ◦ α 1 ⇒ a C ◦ α 2 : Z 0 T ss A c → Z 0 E is the identity tr ansformation. 16 2. [ A, C α ] ′ dg Alg ( G ) /E α := [ A, C α ] dg Alg ( G ) /E α / ∼ , wher e f 1 ∼ f 2 if and only if ther e exists a tensor pr eserv- ing natur al automorph ism t : α ⇒ α over E such that t α ( O ( G )) ◦ f 1 = f 2 . (2) We use the nota tion of the pr evious p art. L et ( G, A ) , ( H , B ) ∈ dg Alg red 0 , ∗ . L et α : Z 0 T ss ( G, A ) c → Z 0 T ( H , B ) ∈ Cat cl ∗ b e a morphism. Supp ose α pr eserves semi- simple obje cts so that it i nduc es a morphism α ∗ : H → G . Then, in the c ase E = V ect , [ A, T ( H, B ) α ] dg Alg ( G ) / V ect α ∼ = [ αA, B ] dg Alg ( H ) /k . Her e, αA denotes the pul lb ack of A by α ∗ : H → G . In p articular, if H is the one-element gr oup, [ T ( G, A ) , T ( H, B )] dg Cat cl ∗ ∼ = [ A, B ] dg Alg /k . In the c ase E = ∗ (the terminal obje ct of dg Cat cl ), [ A, T ( H, B ) α ] ′ dg Alg ( G ) ∼ = [ αA, B ] dg Alg ( H ) /f ∼ f ∗ g (=: [ αA, B ] ′ dg Alg ( H ) ) . Her e f ∗ g denotes the morphism A ∋ a 7→ g · a 7→ f ( g · a ) ∈ B and g runs thr ough C G ( k ) ( α ∗ H ( k )) , the c entr alizer of the image α ∗ ( H ( k )) in G ( k ) . Pr o of. (1) The bijection is deﬁned b y ( F : T ss ( A ) c → C ) 7− → (Z 0 F , colim λ F ( 1 ,V λ ) ). Here colim λ F ( 1 ,V λ ) denotes th e f ollo wing comp ositio n: A ∼ = ( O ( G ) r ⊗ A ) G ∼ = colim λ ( V λ ⊗ A ) G ∼ = colim Hom T ss A c ( 1 , V λ ) F V λ → colim λ Hom C ( 1 , α ( V λ )) = C α . T he pro o f of the former claim is similar to Prop.3.3.4. The latter part f ollo ws from an explicit d escrip tion of path ob jects in d g Cat cl /E (see [30, 2.3.2]) and Lem.3.3.5, (1). In fact, right homotopies of closed dg-catego ries of the ab ov e forms can b e decomp osed in to n atural transformations of Z 0 and right homotopies of dg-algebras. Note that if α 1 ∼ α 2 , E α 1 = E α 2 and C α 1 ∼ = C α 2 in d g Al g ( G ) /E α 1 . (2) W e shall show morph isms F : ( α ∗ ) ∗ A → T ( H , B ) α ∈ dg Al g ( G ) are in one-to-one corresp ond ence with morphisms f : ( α ∗ ) ∗ A → B ∈ dg Alg ( H ). F or giv en F , we set f = O ( α ∗ ) ∗ ◦ F , where O ( α ∗ ) : α ( O ( G )) → O ( H ) is the ind uced H -mo dule morphism and O ( α ∗ ) ∗ : T ( H, B ) α → B is corresp onding pu sh-forward. In the other d irection, for giv en f , W e set F = T f : A ∼ = T A ( O ( G )) → T B ( α ( O ( G )) = T B α . The constru ction f 7→ F 7→ f is clearly th e identit y . W e shall sho w F 7→ f 7→ F is the identit y . Let a ∈ A and put F ( a ) = P i F i ( a ) ⊗ b i ( a ), F i ( a ) ∈ O ( G ) , b i ( a ) ∈ B . Put ρ A ( a ) = P j a j ⊗ l j and λ G ( F i ( a )) = P k F i,k ( a ) ⊗ m i,k ( a ), where λ G denotes th e left translatio n, a j ∈ A , l j , F i,k ( a ) , and m i,k ( a ) ∈ O ( G ). As F is G -equiv arian t, P i,j F i ( a j ) ⊗ b i ( a j ) ⊗ l j = P i,k F i,k ( a ) ⊗ b i ( a ) ⊗ m i,k ( a ). Th e constru cted F is P i,j u ◦ F i ( a j ) · b i ( a j ) ⊗ l j , wh er e u : O ( G ) → k is the counit map, and this is equal to P i,k u ◦ F i,k ( a ) · b i ( a ) ⊗ m i,k ( a ) = P i F i ( a ) ⊗ b i ( a ). Th is corresp onden ce clearly pr eserv es r ight homotop y relation so the former p oin ted claims follo w fr om Lem.2.2.4. F or th e unp ointe d ( E = ∗ ) claim, similarly to the ab o v e, w e h a ve a b ijection: [ A, T ( H , B ) α ] dg Alg ( G ) ∼ = [ αA, B ] dg Alg ( H ) . Under T annakian du alit y , a (tensor-preservin g) natural isomorphism t : α ⇒ α corresp onds an elemen t g of C G ( k ) ( α ∗ H ( k )) and t α ( O ( G )) ◦ f do es f ∗ g u nder the ab o v e bijection. 3.3.1 Iterated Hirsc h ext ensions W e sh all in tro duce th e notion of iterated Hirsch extensions of dg-algebras. The corresp onding notion in the trivial group case was consider ed by Su lliv an, see [8, P .279-28 0]. In fact, an iterated Hirsc h extension is an iteration of Hirsc h extensions as w e s ee b elo w, bu t this notion is usefu l b ecause its classifying data directly corresp ond to homotop y in v ariants of a sp ace. Let l ≥ 1. Let G b e a reductive aﬃne group sc heme and A ∈ dg Alg ( G ) 0 b e a connected G -equiv arian t dg-algebra. Let ( W , η ) ∈ T A b e an ob ject, wh ere W ∈ Ob(Rep( G )) and η is a MC elemen t on W (see subs ection 2.2), and α ∈ Z l +1 ( T A ( W , η )) b e a co cycle of degree l + 1 (see the n otation u nder Def.2.1.1 ). W e deﬁ ne a G -dga A ⊗ ( α,η ) V ( W ∨ , l ) ∈ d g Al g red 0 as follo w s 17 1. As an equiv arian t commuta tiv e gaded algebra, A ⊗ ( α,η ) ^ ( W ∨ , l ) = A ⊗ k ^ ( W ∨ , l ) , where V ( W ∨ , l ) is the fr ee commutat iv e graded algebra generated b y the du al sp ace W ∨ with degree l and r egarded as an G -equiv ariant algebra with the action ind uced by that on W . 2. Th e diﬀeren tial d is d etermined by its restrictions to A and W ∨ . d | A is equal to the diﬀeren tial of A . d | W ∨ is given by α ⊕ ( − t η ) : W ∨ − → A l +1 ⊕ A 1 ⊗ k W ∨ . Explicitly , if we express α an d η as η = X i f i ⊗ a i f i ∈ Hom ( W, W ) , a i ∈ A 1 , α = X j v j ⊗ b j v j ∈ W , b j ∈ A l +1 , w e set [ α ⊕ ( − t η )]( u ) = ( P j u ( v j ) · b j , − P i a i ⊗ ( u ◦ f i )). Maurer-Cartan condition on η and the condition d η α = 0 ensures d 2 = 0. W e alw a ys id en tify A with a subalgebra of A ⊗ ( α,η ) V ( W ∨ , l ) b y a 7→ a ⊗ 1. Deﬁnition 3.3.8. We use the ab ove notations. L et f : A → B b e a morphism of G -dga’s. We say f is an iterated Hirsc h extension of A i f ther e exist data ( W , η ) ∈ Ob( T A ) , α ∈ Z l +1 ( T A ( W , η )) , and an isomorph ism ϕ : B → A ⊗ ( α,η ) V ( W ∨ , l ) such that the fol lowing triangle i s c ommutative. A f / / i ' ' O O O O O O O O O O O O B ϕ   A ⊗ ( α,η ) V ( W ∨ , l ) , wher e i is the natur al inclusion. We say a p air { ( W , η ) , [ α ] } of an obje ct ( W, η ) ∈ O b( T A ) and a class [ α ] ∈ H l +1 ( T A ( W , η )) is the classifying data of f (or B ) if the ab ove c ommutative triangle exists f or ( W , η ) , α , and for some isomorphism ϕ . Classifying da ta is wel l- deﬁne d up to isomorphisms by L em.3.3.9 b elow. If η c an b e chosen as 0 , we say f is a Hirsc h extension of A . An iterated Hirsc h extension extension A → A ⊗ ( α,η ) V ( W ∨ , l ) is decomp osa ble into a sequen ce of Hirsch extensions. Precisely , there exists a ﬁn ite sequen ce of G -dga’s A = A 0 → A 1 → · · · → A m = A ⊗ ( α,η ) ^ ( W ∨ , l ) suc h that A i is a Hirsch extension of A i − 1 for 1 ≤ i ≤ m . This is b ecause the MC element η can b e replaced with a MC-element of the upp er triangular form with zero diagonal, up to isomorp hism (see Lem.2.2.3). In p articular, if A is minimal, A ⊗ ( α,η ) V ( W ∨ , l ) is also minimal. The follo wing is a v arian t of [8 , Thm.2.1] or [10, Lem.9.3]. 18 Lemma 3.3.9. L et G b e a r e ductive aﬃne g r oup scheme. (1) L et A b e a c onne c te d G -dga. L et A ⊗ ( α i ,η i ) V ( W ∨ i , l ) , i = 0 , 1 b e two iter ate d Hirsch extensions of A . Ther e exi sts an isomorphism of G -dga’s ϕ : A ⊗ ( α 0 ,η 0 ) V ( W ∨ 0 , l ) ∼ = A ⊗ ( α 1 ,η 1 ) V ( W ∨ 1 , l ) ﬁxing A if and only if ther e e xists an isomorphism φ : ( W 1 , η 1 ) ∼ = ( W 0 , η 0 ) ∈ Z 0 T A suc h that [ φ ◦ α 1 ] = [ α 0 ] ∈ H l +1 ( T A ( W 0 , η 0 )) . (2) L et M b e a minimal G -dga gener ate d by elements of de gr e e ≤ l . If the l - th inde c omp osable mo dule V of M is ﬁnite dimensional, M is an iter ate d Hirsch extension of the dg- sub algebr a M ( l − 1) gener ate d b y elements of de gr e e ≤ l − 1 . L et { ( W , η ) , [ α ] } b e its classifying data. We identify V with a submo dule of M l . As M is minimal, its diﬀer ential d | V , r estricte d to V , has a u nique de c omp osition d | V = d M ( l − 1) ⊕ d M 1 ⊗ V c onsisting of maps d M ( l − 1) : V − → M ( l − 1) l +1 , d M 1 ⊗ V : V − → M 1 ⊗ V ∈ Rep ∞ ( G ) . Then, 1. the dual V ∨ is isomorphic to W , 2. d M 1 ⊗ V c orr esp onds to the MC element η , and 3. d M ( l − 1) c orr esp onds to a c o cycle α which gives the class [ α ] . Pr o of. See [8, Th m.2.1]. 3.4 de Rham functor In this su bsection, we deﬁne a Quillen adjoint p air T PL : s Set / / (dg Cat cl ) op : h−i o o and p ro ve that T PL ( K ) is a T annakian d g-category if K is connected. W e ﬁrst recall the n otion of standard simplicial comm utativ e dga ∇ ( ∗ , ∗ ) o ver k from [6 , Section 1]. Let p ≥ 0 and ∇ ( p, ∗ ) b e the comm utative graded algebra o v er k generated by indeterminates t 0 , . . . , t p of degree 0 and dt 0 , . . . , dt p of d egree 1 with relations t 0 + · · · + t p = 1 , dt 0 + · · · + dt p = 0 . W e regard ∇ ( p, ∗ ) as a dga with th e diﬀeren tial giv en b y d ( t i ) := dt i . W e can deﬁne simplicial op er ators d i : ∇ ( p, ∗ ) → ∇ ( p − 1 , ∗ ) , s i : ∇ ( p, ∗ ) → ∇ ( p + 1 , ∗ ) , 0 ≤ i ≤ p (see [6]) and w e also regard ∇ ( ∗ , ∗ ) as a simplicial comm utativ e dga. The follo wing d eﬁnition is adopted in [16] Deﬁnition 3.4.1 ([16]) . L et V ect iso b e the sub c ate gory of V ect c onsisting of al l obje cts and isomorph isms. L et K b e a simplicial set. 1. A lo cal system L on K i s a functor (∆ K ) op → V ect iso . 2. A m orphism of local s y s tems L → L ′ is a natur al tr ansformation I ◦ L ⇒ I ◦ L ′ : (∆ K ) op → V ect , wher e I : V ect iso → V ect is the natur al inclusion fu nctor. 19 We deﬁne the tensor L ⊗ L ′ , the inte rnal hom ob ject Hom ( L , L ′ ) and the copro duct L ⊕ L ′ of two lo c al systems L , L ′ by using those of V ect . F or example, ( L ⊗ L ′ )( σ ) = L ( σ ) ⊗ L ′ ( σ ) . We denote by Lo c( K ) the close d k - c ate gory of lo c al systems on K . If K is p ointe d, Loc( K ) is r e gar de d as a c lose d k -c ate gory with a ﬁb er functor. The ﬁb er functor Lo c( K ) → V ect is given by the evaluat ion at the b ase p oint. It is w ell-kno wn th at for a p oi n ted connected simplicial set K , there exists an equiv alence of closed k - catego ries Lo c( K ) ∼ → Rep( π 1 ( K )) wh ich is fun ctorial in K . In the f ollo wing, we s ometimes identify k -lo cal systems with representa tions of the fund amen tal group , ﬁxing suc h an equiv alence. Deﬁnition 3.4.2. L et K b e a simplicial set and L b e a lo c al system on K . The d e Rham complex of L -v alued p olynomial forms C PL ( K, L ) ∈ C ≥ 0 ( R ) i s deﬁne d as fol lo ws. F or e ach q ≥ 0 , the de gr e e q p art is gi ven b y C q PL ( K, L ) = lim ∆ K op ∇ ( ∗ , q ) ⊗ k L . Her e ∇ ( ∗ , q ) is r e gar de d as a fu nctor fr om ∆ K op to the c ate gory of k -ve ctor sp ac es by c omp ose d with the functor ∆ K op → ∆ op , the limit is taken in the c ate gory of p ossibly inﬁnite dimensional k -ve ctor sp ac es. F or q ≤ − 1 , we set C q PL ( K, L ) = 0 . The diﬀer ential d : C q PL ( K, L ) → C q +1 PL ( K, L ) is deﬁne d fr om that of ∇ ( ∗ , q ) . W e shall deﬁne the generalized de Rham fu nctor T PL : s Set − → (dg Cat cl ) op . This is a natural generalization of the de Rham fun ctor of [6 , Deﬁnition 2.1]. F or K ∈ s S et w e d eﬁne a closed dgc T PL ( K ) as follo ws. An ob ject is a lo cal system on K and Hom T PL ( K ) ( L , L ′ ) = C PL ( K, Hom ( L , L ′ )). The comp ositi on is d eﬁ ned f r om that of V ect and the multiplica tion of ∇ ( ∗ , ∗ ), i.e., ( η · b ) ◦ ( ω · a ) := ( η · ω ) · ( b ◦ a ) for ω , η ∈ ∇ ( ∗ , ∗ ), a ∈ Hom ( L , L ′ ) and b ∈ Hom ( L ′ , L ′′ ). T he additional s tructures ⊗ , Hom a nd ⊕ are deﬁ ned similarly . (W e agree th at) T PL ( ∅ ) is a terminal ob ject of dg Cat cl . F or eac h morp h ism f : K → L ∈ s S et w e asso ciate a morphism of closed dgc’s f ∗ : T P L ( L ) → T P L ( K ) by (∆( L ) op L → V ect iso ) 7− → (∆( K ) op ∆ f op → ∆( L ) op L → V ect iso ) . Th us we h a ve deﬁ n ed a f unctor T PL : s Set − → (dg Cat cl ) op . Let C ∈ d g Cat cl . W e deﬁne a fun ctor h−i : (dg Cat cl ) op → s Set by h C i n = Hom dg Cat cl ( C, T PL (∆ n )) with ob vious sim p licial op erators. Clearly h−i is a righ t adjoin t of T PL . W e deﬁne an adjoin t pair b etw een p o in ted categories: T PL : s Set ∗ / / (dg Ca t cl ∗ ) op : h−i . o o F or K ∈ s S et ∗ , we d eﬁne T PL ( K ) ∈ dg Cat cl ∗ b y th e follo wing pullback square: T PL ( K ) / /   T PL ( K u ) T PL ( pt )   V ect / / T PL ( ∗ ) , where K u is the unp ointed simplicial set underlyin g K . F or ( C , ω C ) ∈ d g Cat cl ∗ , w e set h C, ω C i := h C i whose base p oin t is giv en by ∗ ∼ = h V ect i h ω C i → h C i . 20 Lemma 3.4.3 ([30]) . The ab ove two adjoint p airs (T PL , h−i ) : s S et → dg Cat cl and (T PL , h−i ) : s Set ∗ → dg Cat cl ∗ ar e Q uil len p airs. Deﬁnition 3.4.4 (A red ( K )) . L et K b e a p ointe d c onne cte d simplicial set. We set A red ( K ) := A red (T PL ( K )) ∈ dg Alg red 0 , ∗ . We always identify the aﬃne gr oup scheme of A red ( K ) with π 1 ( K ) red (se e A pp endix A .2). When we want to clarify the ﬁeld of deﬁnition, we write A red ( K, k ) . W e shall sho w the closed d g-categ ory T PL ( K ) of a connected simplicial set K is a T annakian dg-category . Recall that there is a natural equiv alence T A c → T A of closed dg-categories for eac h equiv arian t dga A (see sub-sub section 2.2.1). Theorem 3.4.5. (1) L et K ∈ s S et (r esp. s Set ∗ ). 1. T PL K is c omplete. 2. 0 → L 0 → L 2 → L 1 → 0 is a sho rt exact se quenc e in Lo c( K ) ∼ = Z 0 T PL ( K ) if and only if L 0 → L 2 → L 1 is an extension in T PL K in the sense of subse ction 2.2. In p articular, if K is c onne cte d, T PL K ∈ T an (r esp. T a n ∗ ). (2) L et K b e a p ointe d c onne cte d simplicial se t. Ther e e xists a morphism of dg Ca t cl ∗ which is an e quivalenc e b etwe en underlying c ate gories: T A red ( K ) c − → T PL K. In p articular, if f : A → A red ( K ) ∈ d g Alg ( π red 1 ) is a quasi-isomorphism ( π 1 = π 1 ( K ) ), ﬁnite dimensional r epr esentations V of π 1 determine and ar e determine d by obje cts ( W, η ) ∈ Ob( T A ) up to isomorphisms via the fol low ing e quivalenc es: Rep( π 1 ) ≃ Lo c( K ) ≃ Z 0 T PL ( K ) ≃ Z 0 T A red ( K ) c ≃ Z 0 T A red ( K ) f ∗ ≃ Z 0 T A. In this c orr esp ondenc e, W is isomorph ic to the semi-simpliﬁc ation of V . Pr o of. First note that th e follo wing facts. 1. Th e completeness is stable und er homotopy pu llbac ks and h omotop y limits of to wers. 2. F or a dgc C , a sequ ence of c hain morphisms c 0 a → c 2 b → c 1 in C is an extension if and only if for any ob ject c ∈ Ob( C ) the t w o sequences of complexes 0 → Hom C ( c, c 0 ) a ∗ → Hom C ( c, c 2 ) b ∗ → Hom C ( c, c 1 ) → 0 , 0 → Hom C ( c 1 , c ) b ∗ → Hom C ( c 2 , c ) a ∗ → Hom C ( c 0 , c ) → 0 are b oth lev elwise exact. By these f acts and small ob ject argum en t, all w e ha ve to do is to p ro ve the prop ositon for X = ∆ n , ∂ ∆ n for n ≥ 1. W e only show the case of T PL ( ∂ ∆ 2 ) and th e others are clear. W e ma y replace ∂ ∆ 2 b y S 1 = ∆ 1 / 0 ∼ 1 and w e iden tify lo cal systems on S 1 with represent ation of the fr ee group Z . Let ( V , g ) , ( V ′ , g ′ ) ∈ Rep( Z ). Let Σ i P i ( t ) dt · f i ∈ Hom 1 T PL S 1 (( V , g ) , ( V ′ , g ′ )) where t = t 0 , P i ( t ) is a p olynomial of t , and f i ∈ Hom( V , V ′ ). Put g 0 =  g ′ Σ i R 1 0 P i ( t ) dt · f i g 0 g  . Then the obvious sequence ( V ′ , g ′ ) → ( V ′ ⊕ V , g 0 ) → ( V , g ) is an extension whose class is equal to [Σ i P i ( t ) dt · f i ]. So T PL ( S 1 ) is complete and the third condition of Def.3.1.1 is p ro ved b y a similar argument . (2) f ollo ws from (1) and Prop.3.3.4. 21 The follo wing lemma is used in the pro of of Thm.4.3.4. Lemma 3.4.6 (cf. Rem.4.43 of [27]) . L et G b e a r e ductive aﬃne gr oup scheme and M b e a minimal G -dga. L et V i denote the i -th inde c omp osable mo dule of M . F or i ≥ 2 , ther e exists an isomorphism of gr oups π i ( R h T Mi ) ∼ = ( V i ) ∨ . This isomorphism is functorial ab out morphisms b etwe en minimal e quivariant dga. Pr o of. By Lem.2.2.4 and 3.3.7, π i ( R h T Mi ) ∼ = [ T M , T PL S i ] dg Cat cl ∗ ∼ = [ M , A P L S i ] dg Alg /k ∼ = ( V i ) ∨ . 4 The de Rham homotop y theory for general spaces In this section, we see ho w the minimal m o dels describ e algebraic top olo gical inv ariant s of sp aces and pro vide s ome examples of minimal mo dels. W e also pro v e an equiv alence b et w een T annakian dg-catego ries with subsidiary data and ﬁb erwise rationalizations. In the pro ofs of results of this section, w e us e the corresp onden ce b et w een T annakian dg-categories and schemat ic homotop y typ es (Th m.5.2.1 and Cor.5.2.7). As f or logica l order, section 5 is pr evious to this section. 4.1 Homotop y inv arian ts W e shall recall the notion of algebraically go o d ness in tro duced by T o ¨ en [22]. Let Γ b e a discrete group. Let H i (Γ , − ) b e the i -th deriv ed functor of inv arian ts Rep ∞ k (Γ) − → k − Mo d , V 7− → V Γ , and H i (Γ alg , − ) b e the i -th deriv ed functor of the functor Rep ∞ k (Γ alg ) − → k − Mo d , V 7− → V Γ alg . An y Γ alg -mo dule can b e regarded as a Γ-mo d ule by pulling bac k by the canonical m ap Γ → Γ alg ( k ) so there exists a canonical natural transformation H i (Γ alg , − ) = ⇒ H i (Γ , − ) : Rep ∞ k (Γ alg ) − → k − Mo d . Deﬁnition 4.1.1 (algebraically go o d, [22, 25, 27]) . U nder ab ove notations, we say Γ is algebr aic al ly go o d over k if for e ach i ≥ 0 and e ach ﬁnite dimensional r epr esentation V ∈ Rep k (Γ alg ) , the c anonic al map H i (Γ alg , V ) − → H i (Γ , V ) is an isomorphism. The follo wing was p ro v ed b y P r idham [27]. Theorem 4.1.2 ([27]) . L et Γ b e a discr ete gr oup. Γ is algebr aic al ly go o d over k if and only if the minimal mo del of A red ( K (Γ , 1) , k ) is gener ate d by elements of de gr e e 1. Pr o of. Clearly , Γ is algebraically go o d if and only if the canonical map A red ( K (Γ alg , 1)) → A red ( K (Γ , 1)) is a quasi-isomorphism (see subsection 5.2). But b y [25, Prop.4.12], for an y discrete group Γ the map H i (A red ( K (Γ alg , 1)) → H i (A red ( K (Γ , 1))) is an isomorphism for i = 0 , 1 and a monomorphism for i = 2 and b y C or.5.2.6, the minimal m o del of A red ( K (Γ alg , 1)) is generated b y degree one elemen ts so the claim follo ws. W e shall sho w ho w the minimal mo d el d escrib es homotop y theory of a sp ace. 22 Theorem 4.1.3. L et K b e a p ointe d c onne cte d simplicial set. Put π i := π i ( K ) . F or i ≥ 2 , we r e gar d π i as a π 1 -mo dule by the c anonic al action. L et 1. K → · · · p i → K ( i − 1) p i − 1 → · · · p 1 → K (1) b e the Postnikov tower of K , 2. M b e the minimal mo del of A red ( K ) , 3. M ( i ) b e the dg- su b algebr a of M gener ate d by ⊕ j ≤ i M j , and 4. n ≥ 2 b e an inte ger. Supp ose π 1 is algebr aic al ly go o d over k and π i is of ﬁnite r ank as an Ab elian gr oup for e ach 2 ≤ i ≤ n . ( 1) Ther e exists a c ommutative diagr am in T an ∗ T M (1) c T l c 1 / / q 1   T M (2) c T l c 2 / / q 2   · · · T l c n − 1 / / T M ( n ) c q n   T PL K (1) p ∗ 1 / / T PL K (2) p ∗ 2 / / · · · p ∗ n − 1 / / T PL K ( n ) such that l i : M ( i ) → M ( i + 1) i s the inclusion and al l the vertic al arr ows ar e quasi- e quivalenc es. (2) F or e ach 2 ≤ i ≤ n , the inclusion l i − 1 : M ( i − 1) → M ( i ) is an iter ate d Hirsch extension (se e Def.3.3.8). L et { ( W i , η i ) , [ α i ] } b e its classifying data. Then, 1. The obje ct ( W i , η i ) ∈ O b( T M ( i − 1)) = Ob ( T M ) is isomor phic to the π 1 -r epr esentation π i ⊗ Z k under the c orr esp ondenc e of Thm.3.4.5,(2). 2. The class [ α i ] ∈ H i +1 ( T M ( i − 1)( W i , η i )) c orr esp onds to the k-invariant tensor e d with k via the isomorph ism: H i +1 [ T M ( i − 1)( W i , η i )] ∼ = H i +1 [T PL ( K ( i − 1) )( π i ⊗ Z k )] ∼ = H i +1 ( K ( i − 1) ; π i ⊗ Z k ) . induc e d b y q i − 1 in the ab ove diagr am. Her e, π i ⊗ Z k is c onsider e d as a lo c al system on K ( i − 1) . See Lem.3.3.9,(2). See also [27 , Thm.1.58, Rem.4.43]. Before w e b egin the pro of, we state a corollary w h ic h may b e usefu l for computations. Corollary 4.1.4. Under the assumption of Thm.4.1.3, supp ose the action of π 1 on π n ⊗ Z k is se misimple. Then, M ( n ) is a Hirsch extension of M ( n − 1) . (Note that we do not assume M 1 = 0 .) T o pr o ve the theorem, w e need to sh o w a v ariant of the Hirsc h Lemma [10, Thm .11.1] formulated in the f ollo wing. Let m ≥ 2. W e shall consider a ﬁb ration p : E − → B b et w een p ointed connected simplicial sets w hose ﬁb er F satisﬁes π 0 ( F ) ∼ = ∗ , π i ( F ) = 0 f or i ≤ m − 1, and π m ( F ) is an Ab el ian group of ﬁnite rank. Put π 1 := π 1 ( B ) ∼ = π 1 ( E ). Let π denote the lo cal system of the n -th k -tensored homotop y group s of ﬁb ers of p or the corresp onding representa tion of π 1 ( B ). Then H m +1 ( B , E ; π ) ∼ = Hom Rep( π 1 ( B )) ( π m +1 ( B , E ) , π ) ∼ = Hom Rep( π 1 ( B )) ( π , π ) . See [15 , P .344] or [9, P .289]. T ake the element ˜ k of H m +1 ( B , E ; π ) corresp ond ing to the id en tity on π via this isomorph ism. By deﬁ n ition, th e image of ˜ k in H m +1 ( B ; π ) is the k-inv arian t of p tensored with k . Let 23 M B b e a minimal mo del of A red ( B ). W e ma y replace T PL ( B ) by T M B and we ha ve H m +1 ( B , E ; π ) ∼ = H m +1 (Cone( p ∗ : C PL ( B ; π ) → C PL ( E ; π )) ∼ = H m +1 (Cone( p ∗ : T M B ( π ss , η ) → C PL ( E ; π ))). Here, ( π ss , η ) is a pair of th e semisimpliﬁcation of π and a MC-elemen t η , see Thm.3.4.5. W e tak e a co cycle ( α, β ) ∈ Cone m +1 ( p ∗ : T M B ( π ss , η ) → C PL ( E ; π )), wh ere α ∈ T M B ( π ss , η ) m +1 and β ∈ C m PL ( E ; π ) whic h r epresen ts ˜ k . T ak e the iterated Hirsc h extension M B ⊗ ( α,η ) V (( π ss ) ∨ , m ). p and β deﬁne a morph ism of π red 1 -dga’s:  : M B ⊗ ( α,η ) ^ (( π ss ) ∨ , m ) − → A red ( E ) b y  | M B = p ∗ and ρ | ( π ss ) ∨ = β (see Def.3.3.8). Here β is considered as a π red 1 -mo dule homomorphism ( π ss ) ∨ − → A red E . The essence of the pro of of the f ollo wing is the same as th at of [10, Thm.11.1]. Lemma 4.1.5. We use the ab ove notations. (1)  induc es an i somorphism b etwe en i -th c ohomolo gy gr oups for e ach i ≤ m and a monomorphism b etwe en m + 1 - th c ohomolo gy gr oups. (2) If a ﬁb er F of p satisfy π i ( F ) = 0 for i ≥ m + 1 , the map  is a quasi-isomorphism. Pr o of. W e p ro ve (2). The pro of of (1) is similar and easier. In the follo w ing, we use cub ical sets instead of simplicial sets. F or details ab out cubical de Rham theory , see Ap p end ix A.1. W e use the same notation as the case of simplicial sets for the corresp onding notio n in the cubical case. Pu t A B := M B ⊗ ( α,η ) V (( π ss ) ∨ , m ). W e deﬁne a descendin g ﬁ ltration A B = F 0 B ⊃ F 1 B ⊃ · · · ⊃ F p B ⊃ · · · b y F p B = ⊕ i ≥ p M p B ⊗ k V (( π ss ) ∨ , n ) i.e., F p B is the ideal of A B generated by ⊕ i ≥ p M p B . T his ﬁ ltration in d uces a ﬁltration of closed dg-catego ry consisting of ideals closed u nder tensor and inte rnal hom: T A B = T F 0 B ⊃ T F 1 B ⊃ · · · ⊃ T F p B ⊃ · · · Note that T F p B ◦ T F q B , T F p B ⊗ T F q B , Hom ( T F p B , T F q B ) ⊂ T F p + q . O n the other h and, w e deﬁn e a ﬁ ltration { F p (T PL E ) } p ≥ 0 of T PL ( E ) by F p (T PL E ) b eing the ( ⊗ , Hom )-closed ideal generated b y images of homogeneous morph ism in T M B of degree ≥ p , by p ∗ : T PL B − → T PL E . W e may describ e F p (T PL E ) as follo ws. Let σ ∈ E b e a n on-degenerate l -cub e. W e iden tify a cub e with the sub-cubical set generated by it. F or notational simp licit y , w e assume p | σ : σ → p ( σ ) is the pro jection to the ﬁrst k comp onent s q k :  l →  k . Indeed, after c han ge of co ordinate, p | σ is isomorphic to q k for some k . Then ω ∈ Hom F p (T PL E ) ( L , L ′ ) ⇐ ⇒ ∀ σ ω | σ ∈  ( k , ≥ p ) ⊗ k  ( l − k , ∗ ) ⊗ k Hom ( L , L ′ )( σ ) . Here, w e identify  ( l , ∗ ) with  ( k , ∗ ) ⊗ k  ( l − k , ∗ ). Note that F p (T PL E ) ◦ F p ′ (T PL E ), F p (T PL E ) ⊗ F p ′ (T PL E ), Hom ( F p (T PL E ) , F p ′ (T PL E )) ⊂ F p + p ′ (T PL E ). Let T  ( E ) ∈ d g Cat ≥ 0 b e a dgc deﬁn ed as follo ws. 1. Ob(T  ( E )) = Loc ( E ). 2. Hom T  ( E ) ( L , L ′ ) = C  ( E ; Hom ( L , L ′ )) (see App endix A.1) and th e comp osition is giv en by the cup-pro d uct. W e deﬁne a ﬁltration { F p (T  ( E )) } on T  ( E ) as u sual, by Hom F p (T  ( E )) ( L , L ′ ) = Ker ( i ∗ : Hom T  E ( L , L ′ ) → Hom T  E ( p − 1) ( L , L ′ )) , where E ( p − 1) = p − 1 ( B p − 1 ) and i : E ( p − 1) → E is the inclusion ( B p − 1 is the p − 1-th skelet on of B ).  induces ﬁ ltration-preserving morph ism T  : T A B − → T A red E ≃ T PL E . 24 Stok es m ap ρ : T PL E → T  E (this is a morphism of dg-graphs) also preserves ﬁ ltration. In the follo wing, for a (closed) dgc C with a ﬁltration { F p ( C ) } , E r ( C ) denotes a (closed) dgc deﬁned b y 1. Ob E r ( C ) = Ob C . 2. Hom E r ( C ) ( − , − ) = E r (Hom C ( − , − ); Hom F • ( C ) ( − , − )), th e E r -term of the sp ectral sequen ce, with th e diﬀeren tial d r . T o pr ov e th e lemma, it is enough to pr o ve E 2 ( ρ ◦ T  ) in d uces isomorph isms b et ween eac h hom-complexes. Note that there is a diagram E 1 T PL ( E ) ϕ / / E 1 ρ ' ' P P P P P P P P P P P P T PL ( B ; H F ) ρ ′   E 1 T  E Here, 1. T PL ( B ; H F ) is a closed dgc deﬁn ed by (a) ObT PL ( B ; H F ) = Ob(Lo c( B )), (b) Hom T PL ( B ; H F ) ( − , − ) = L p + q = n C p PL ( B ; H q F ⊗ Hom ( − , − )), wher e H q F is a lo cal system on B giv en by τ 7→ H q ( p − 1 ( τ (0 , . . . , 0) , Q ), 2. ρ ′ is the Stok es map w ith H q F ⊗ Hom ( − , − )-co eﬃcien ts ( Here, w e us ed the wel l-kno wn id en tiﬁ cation Hom p,q E 1 T  E ( − , − ) ∼ = C p  ( B ; H q F ⊗ Hom ( − , − )), and 3. ϕ is a morph ism of closed dgc’s deﬁned as follo ws. An elemen t x ∈ Ho m p,q E 0 (T PL E ) ( − , − ) deﬁnes a form ω on E suc h that ω | σ ∈  ( k , p ) ⊗ k  ( n − k , q ) (with the ab o ve notations). Let τ ∈ B b e a non-degenerate k -cub e. W e tak e a lift ˜ τ of the follo win g diagram F τ / /   E    k × F τ ˜ τ ; ; w w w w w τ / / B Here F τ = p − 1 ( τ (0 , . . . , 0)). Fixing basis of  ( k , p ), α 1 , . . . , α N , w e can write ˜ τ ∗ ( ω ) = α 1 ⊗ β 1 + · · · + α N ⊗ β N , β i ∈ C q PL ( F τ ; Hom ( − , − ) | τ ). So if x is a co cycle, it gives an element ϕ ( x ) ∈ C p PL ( B ; H q F ⊗ Hom ( − , − )) deﬁn ed as α 1 ⊗ [ β 1 ] + · · · + α N ⊗ [ β N ] on τ . The F ubini’s theorem ensu res the diagram is comm u tativ e. It is easy to see H ∗ ρ ′ : H ∗ T PL ( B ; H F ) → E 2 T  E induces bijections of hom-sets, so all we hav e to do is to prov e H ∗ ( ϕ ◦ E 1 T  ) = H ∗ ( ϕ ) ◦ E 2 ( T  ) : E 2 ( T A B ) → H ∗ T PL ( B ; H F ) is an equiv alence. W e pro v e this by using the fact that the map H ∗ ( ϕ ) ◦ E 2 ( T  ) is a morph ism of closed d gc’s. (This is analogous to the fact that in simply connected case, the corresp ondin g claim w as pro v ed by usin g the f act that the corresp on d ing map is a morp hism of algebras.) W e need the follo wing sub -lemma. W e u se the n otation that E p,q r ( C )( X ) := Hom p,q E r ( C ) ( 1 , X ). Sub-lemma 4.1.6. L et U q ∈ Ob T A B b e H q F r e gar de d as an obje ct of T A B . 25 1. Ther e exists a b ije ction E p,q 2 ( T A B )( V ) ∼ = H p [ T M B (( V π ∨ ) q ⊗ V )] for e ach V ∈ Ob( T A B ) . 2. H ∗ ( ϕ ) ◦ E 2 ( T  ) induc es isomorphisms on ( p, 0) -terms of e ach hom-c omplex f or p ≥ 0 . 3. e ∈ E 2 ( T A B ) 0 ,m (( U m ) ∨ ) ∼ = Hom Rep( π 1 ) ( U m , π ∨ ) b e an element c orr esp onding to an isomor ophism, and ev U q ∈ Hom 0 , 0 E 2 ( T A B ) ( U q ∨ ⊗ U q , 1 ) b e the evaluation. Then the maps E p, 0 2 T A B ( U k m ⊗ V ) − → E p,k m 2 T A B ( V ) x 7− → ev U km ◦ ( e k ⊗ x ) H p [T PL ( B ; H 0 F )( ψ ( U k m ) ⊗ L )] − → H p [T PL ( B ; H k m F )( L )] y 7− → ev ψ ( U km ) ◦ ( ψ ( e k ) ⊗ y ) ar e b ije ctions, wher e k ≥ 1 , e k ∈ E 2 ( T A B ) 0 ,k m (( U k m ) ∨ ) is the k -times e , and ψ = H ∗ ( ϕ ) ◦ E 2 ( T  ) . Pr o of of Sub - lemma. E p,q 0 ( T A B )( V ) and E p,q 1 ( T A B )( V ) are n aturally isomorphic to [(( V π ∨ ) q ⊗ V ) ss ⊗ M p B ] π red 1 and d 1 is equal to d M B + η ( V π ∨ ) q ⊗ V ( η ( V π ∨ ) q ⊗ V is the MC-elemen t of ( V π ∨ ) q ⊗ V ) s o the ﬁr st part follo w s. The second part is clear. F or the th ird part, the ﬁrst map is identiﬁed with the pu shforward e k ∗ : H p [ T M B (( V π ∨ ) k m ⊗ V )] → H p [ T M B ( U k m ⊗ V )] via the bijection of part 1, so this is a b ijection. T o see the second map, note that T  ( e ) = t e ◦ β ∈ C m PL ( E ; ( H m F ) ∨ ). Th e r estriction of this element to a ﬁb er is a co cycle in C m PL ( F ; ( H m F | F ) ∨ ) and it represents an isomorph ism in Hom Rep( π 1 ) ( H m F | F , H m ( F )) ⊂ H m ( F ; ( H m F | F ) ∨ ). Th us ψ ( e ) corresp onds to an isomorphism v ia the identiﬁcatio n H 0 [T PL ( B ; H m F )(( H m F ) ∨ )] ∼ = H 0 ( B ; H m F ⊗ ( H m F ) ∨ ) ∼ = Hom Loc( B ) ( H m F , H m F ) so w e can see the second map is a bijection similarly to th e ﬁrst one. As H ∗ ( ϕ ) ◦ E 2 ( T  ) is a morphism of closed dgc’s, the follo w ing d iagram is comm utativ e. E p, 0 2 T A B ( U k m ⊗ V ) / / ψ   E p,k m 2 T A B ( V ) ψ   H p [T PL ( B ; H 0 F )( ψ ( U mk ) ⊗ ψ V )] / / H p [T PL ( B ; H mk F )( ψ V )] , where the horizon tal maps are the ones in the sub -lemma. This implies that ψ is an equiv alence of catego ries. Pr o of of Thm.4.1.3 . The pro of is successiv e applications of Th m.4.1.2 an d Lem.4.1.5. L em.4.1.5,(1) is necessary to p ro ve M ( n ) is isomorphic to the min im al mo d el of A red ( K ( n ) ), see [6, Prop.7.10]. F or the ﬁrst inﬁn ite higher homotopy group, we hav e the follo w in g. Theorem 4.1.7. L et K b e a p ointe d c onne cte d simplicial set with π 1 ( K ) algebr aic al ly g o o d. We use the notation of Thm.4.1.3. L et n ≥ 2 and supp ose π i ( K ) is of ﬁnite r ank as an ab elian gr oup for 2 ≤ i ≤ n − 1 . As the action of π 1 ( K ) red on V n is lo c al ly ﬁnite and as M is minimal, ther e exist ﬁnite dimensional π 1 ( K ) red -submo dules { V n λ } λ of V n such that S λ V λ = V n , d M 1 ⊗ V n ( V n λ ) ⊂ M 1 ⊗ V n λ , and ( { V n λ } λ , ⊂ ) forms a ﬁlter e d system. The r estriction of d M 1 ⊗ V n to V n λ deﬁnes a M C-element η λ on ( V n λ ) ∨ in T ss M (se e L em.3.3.9). Thus we obtain an inverse system of ﬁnite dimensional π 1 ( K ) -r epr ese ntations c orr esp onding to { (( V n λ ) ∨ , η λ ) } . Then, the limit of the inverse system is isomorp hic to the pr o-ﬁnite dimensional c ompletion of π n ( K ) ⊗ Z k . Her e, the pr o-ﬁnite dimensional c ompletion of (p ossibly inﬁnite dimensional) π 1 ( K ) -r epr ese ntation X is the limit of the inve rse system { W µ ∈ Rep( π 1 ( K )) | µ : X → W µ ∈ Rep ∞ ( π 1 ( K )) } taken in Rep ∞ ( π 1 ( K )) . Pr o of. The pr o of is similar to that of T hm.4.1.3 s o w e omit. 26 4.2 Examples Deﬁnition 4.2.1. We denote by M red ( K ) (r esp. M PL ( K ) ) the minimal mo del of A red ( K ) (r esp. A P L ( K ) ). Her e A P L ( K ) is the usual p olynomial de Rham algebr a over k . When we want to clarify the ﬁeld of deﬁnition, we write A red ( K, k ) , M red ( K, k ) and M PL ( K, k ) . Example 4.2.2. L et X b e a p ointe d c onne cte d simplicial set with π 1 ( X ) ﬁnite. Then M red ( X ) ∼ = M PL ( e X ) , wher e e X denotes the universal c overing of X and the action of π 1 on M PL ( e X ) is induc e d fr om the one on e X (se e [ 30]). The metho d of pro of of Lem.4.1.5 pro vid e some examples. Theorem 4.2.3. L et K ∈ s Set ∗ b e a nilp otent simplicial se t of ﬁnite typ e. Then M red ( K ) ∼ = M PL ( K ) . Her e M PL ( K ) is c onsider e d as π 1 ( K ) red -dga with the trivial action. Pr o of. It is enough to sho w M red ( K ( N , 1)) ∼ = M PL ( K ( N , 1)) for a nilp oten t group N . The pro of is similar to that of Lem.4.1.5 and we use the to wer of ﬁ brations asso cia ted to nilp ote n t extensions instead of the P ostniko v to wer. A sligh t diﬀerence is that π 1 ( B ) 6 = π 1 ( E ) in this case. But in fact, f or a ﬁnite dimensional π 1 ( E )-modu le V , the E 2 -terms concerning V is n aturally isomorphic to those concerning th e π 1 ( B )-modu le V π 1 ( F ) of π 1 ( F )-inv arian ts, w here F denotes the ﬁb er , so the pro of of Lem.4.1.5 still wo rks. Remark 4.2.4. If we expr e ss Thm.4.2.3 in the language of schematic homo topy typ es, ther e exists an isomorph ism of schematic homotopy typ es: ( L ⊗ k ) sc h ∼ = ( L ⊗ k ) uni × K ( π 1 ( L ) red , 1) ∈ Ho ( SHT ) for nilp otent L of ﬁnite typ e (se e [22] for notation). Example 4.2.5. Supp ose k is algebr aic al ly close d. L et n ≥ 2 b e an inte ger and N b e the fr e e ab elian gr oup of r ank l and L et M b e a ﬁnite r ank ab elian g r oup with N - action. L et K = K ( N , M , n ) := K ( N ⋉ K ( M , n − 1) , 1) . Her e, K ( M , n − 1) is an E i lenb er g Maclane sp ac e r e alize d as simplicial ab elian gr oup with the induc e d action of N . L et g j ∈ GL ( M ⊗ Z k ) b e the action of j -th gener ater of N and g j = g s j + g n j b e a Jor dan de c omp osition c ommutative with e ach other, wher e g n j is nilp otent and g s j is semisimple. Note that P j =1 ,...,l g n j · s j is a MC-element c orr esp onding to M ⊗ Z k and the k-invariant is zer o as the se ction K ( N , 1) → K ( N , M , n ) exists. W hen we denote the mo dule of i -dimensional gener ators of M red ( K, k ) by V i , b y Thm.4.1.3, V i =    L j =1 ,...,l k · s j ( i = 1) (( M ⊗ Z k ) ss ) ∨ ( i = n ) 0 ( other w ise ) and d ( s j ) = 0 , d ( x ) = P j =1 ,...,l t g n j ( x ) · s j for x ∈ V n . He r e , ( M ⊗ Z k ) ss is the semisimpliﬁc ation of N -r epr esentation M ⊗ Z k , i.e., the j -th gener ator acts on it by g s j . N acts on V 1 trivial ly. Example 4.2.6 (cell attac hment) . We shal l give an explicit mo del of c el l attachment which is a natur al gener alization of [17, Pr op.13.12]. L et X b e a p ointe d c onne cte d CW c omplex. L et π i := π i ( X ) ( i ≥ 1 ), M red ( X ) = M = V ( V i , d M ) and n ≥ 2 . T ake a ∈ π n ( X ) . We also denote by a : V n → k the c orr esp onding image by the ma p π n ( X ) → [T PL ( X ) , T PL ( S n )] ∼ = ( V n ) ∨ (se e L em.3.4.6). L et X ∪ a D n +1 b e the sp ac e obtaine d by attaching a n + 1 -c e l l to X along a . A mo del of X ∪ a D n +1 (i.e. a π red 1 -dga quasi-i somorphic to A red ( X ∪ a D n +1 ) ), V ( V i ) ⊕ a O ( π red 1 ) l u is given as fol low s. 1. A s a gr ade d mo dule, it is V ( V i ) ⊕ O ( π red 1 ) l u , wher e O ( π red 1 ) l u is a c opy of O ( π red 1 ) l whose de gr e e is n + 1 . 27 2. The algebr a structur e is determine d by that of M and M · ( O ( π red 1 ) l u ) = ( O ( π red 1 ) l u ) 2 = 0 3. The diﬀe r e ntial d i s determine d by the derivation pr op erty fr om the formula dx =    0 if x ∈ O ( π red 1 ) l u d M x, if x ∈ V i , i 6 = n d M x + a ∗ ( x ) u if x ∈ V n , wher e a ∗ : V n → O ( π red 1 ) l is the c omp osition: V n c o action of V n / / V n ⊗ O ( π red 1 ) l a ⊗ id / / O ( π red 1 ) l . . In fact, as π 1 ( X ∪ a D n +1 ) ∼ = π 1 ( X ) , A red ( X ∪ a D n +1 ) is isomorphic to A red ( X ) × A P L ( S n ) ⊗O ( π red 1 ) l A P L ( D n +1 ) ⊗ O ( π red 1 ) l . So by an ar gument similar to [17, Pr op.13.12] we se e the latter is q uasi-isomorphic to the ab ove mo del (se e also L em.3.3.7 ). We shal l pr e sent some c oncr ete examples. Su pp ose k is algebr aic al ly close d. 1) L et X 0 = S 1 × S 2 and a 0 b e a gener ator of π 3 ( X 0 ) ∼ = Z . L et X 1 = X 0 ∪ a 0 D 4 . Using the ab ove mo del, we c an c ompute the minimal mo del M 1 = M red ( X 1 ) . Note that O ( Z red ) is isomorph ic to the gr oup ring k h k ∗ i of the discr ete gr oup k ∗ = k − { 0 } . the ﬁfth stage of M 1 is pr e se nte d as M 1 (5) = ^ ( t, s, v α,i , w α ) α ∈ k ∗ ,i ≥ 1 , deg t = 1 , deg s = 2 , deg v α,i = 4 , and deg w α = 5 with dv α, 1 =  ts 2 for α = 1 0 otherwise , dw α =  t 3 for α = 1 tv α, 1 otherwise dv α,i = tv α,i − 1 for al l α ∈ k ∗ and i ≥ 2 Her e, the ﬁxe d gener ator of π 1 ∼ = Z acts as v α,i 7→ αv α,i and w α 7→ αw α . In p articular, we se e the fourth k- invariant of X 1 is non-zer o. Indep endently, it is e asy to se e π 4 ( X 1 ∪ a 1 D n +1 ) i s the fr e e π 1 -mo dule ge ner ate d by one element, and the pr o-ﬁnite dimensional c ompletion of this has the form ( ⊕ α ∈ k ∗ U α ) ∨ wher e π 1 acts on the inﬁnite dimensional ve ctor sp ac e U α = k h u α, 1 , u α, 2 , . . . i by the inﬁnite size Jor dan blo ck of eige n-value α . 2)Next, we take an element a 1 ∈ π 4 ( X 1 ) and put X 2 = X 1 ∪ a 1 D 5 . We c an i dentify π 4 ( X 1 ) with the L aur ent p olynomial ring Z [ x, x − 1 ] , wher e the action of the ﬁxe d gener ater of π 1 c orr esp onds the multiplic ation of x . we r e g ar d a 1 as a L aur ent p olynomial P ( x ) . L et φ : π 4 ( X 1 ) → ( ⊕ α ∈ k ∗ U α ) ∨ b e the structur e map of the c ompletion and put φ α,i = φ (1)( u α,i ) wher e 1 ∈ Z [ x, x − 1 ] . The diﬀer ential of M 1 (5) ⊕ a 1 O ( π red 1 ) u has the fol low ing expr e ssion. d ( N X i =1 c i v α,i ) = δ α, 1 c 1 ts 2 + N X i =2 c i tv α,i − 1 + ( φ α, 1 , . . . , φ α,N ) P ( A − 1 α,N )    c 1 . . . c N    · α u. Her e, δ α, 1 is the Kr one c ker delta, A α,N is the Jor dan blo ck of size N and α is the element of k h k ∗ i c orr esp onding to α . So if we let R b e the set of non-zer o distinct r o ots of P ( x ) and p α b e the multiplicity of α ∈ R , M 2 (5) = V ( t, s, v α,i , w β ) α − 1 ∈ R, 1 ≤ i ≤ p α , β − 1 ∈ R ∪{ 1 } with the de g r e e and diﬀer ential g i ven by the same formula as M 1 (5) . It is e asy to se e π 4 ( X 2 ) ∼ = Z [ x, x − 1 ] / ( P ( x )) (ﬁnite r ank) so we c an say the pr oﬁnite dimensional c ompletion of π 5 ( X 2 ) is the dual of k h w β i β − 1 ∈ R ∪{ 1 } (Thm.4.1.7). Lemma 4.2.7. L et Γ b e a c ommutative divisible gr oup. Then the Q -pr o-algebr aic c ompletion Γ alg Q is pr o- unip otent. In p articular, if L is a p ointe d c onne cte d simplicial set with π 1 ( L ) c ommutative and divisible, M red ( L ; Q ) ∼ = M PL ( L ; Q ) . Pr o of. Left to the r eader. 28 Theorem 4.2.8. L et 0 − → M − → e Γ − → Γ − → 1 b e an extension of gr oups such that M is a ﬁnitely gener ate d ab elian gr oup or a ﬁnite dimensional Q -ve ctor sp ac e. We r e gar d M as a Γ -mo dule with the action i nduc e d by the extension. Then M red ( K ( e Γ , 1); k ) ∼ = M red ( K (Γ , 1); k ) ⊗ ( α,η ) ^ [(( M ⊗ k ss )) ∨ , 1] . Se e Def.3.3.8 for the notation. Her e, η is the MC-element c orr esp onding to the Γ -mo dule M ⊗ k and α ∈ Z 2 T M red ( K (Γ , 1))(( M ⊗ k ) ss , η ) is a c o cycle which r epr esents the class [ e Γ] ⊗ k ∈ H 2 (Γ , M ⊗ k ) . In p articular, if Γ is algebr aic al ly go o d over k , e Γ is also algebr aic al ly go o d over k . (In these statements, when M satisﬁes the se c ond c ondition, we assume k = Q .) Pr o of. When we use Thm .4.2.3 or Lem.4.2.7, the pro of is similar to that of Thm.4.2.3. 4.2.1 comp onen ts of free lo op spaces Let K b e a connected ﬁb ran t simp licial set and π 1 b e the fund amen tal group of K with resp ect to a ﬁxed base p oin t ∗ . Let Λ K b e the free lo op space of K , i.e., the in ternal hom-ob ject Hom ( S 1 , K ) in the cate gory of unp oin ted simplicial sets. Cho ose a p oin t b ∈ S 1 . The ev aluation at b deﬁn es a ﬁbration p : Λ K → K whose ﬁb er is the p ointed lo op s p ace Ω K of K . Let γ ∈ π 1 . W e let γ denote its representa tiv e lo op and Λ γ K do the connected comp onen t of Λ K cont aining γ . p induces a ﬁb er sequ en ce Ω γ K → Λ γ K → K . Let ∂ : π i ( K ) → π i − 1 (Ω γ K ) ∼ = π i ( K ) b e the b oundary m ap of the long exact sequence of this ﬁb er sequence. One can easily see for α ∈ π i ( K ), ∂ ( α ) v anishes if and only if the Whitehead pr o duct [ γ , α ] v anish es. So there exist exact sequences of groups 0 → π 2 ( K ) γ → π 1 (Λ γ K ) → C π 1 ( γ ) → 1 , 0 → π i +1 ( K ) γ → π i (Λ γ K ) → π i ( K ) γ → 0 for i ≥ 2, where π i ( K ) γ = Ker(id − γ ), π i ( K ) γ = Cok er (id − γ ) ( γ means th e action on the homotop y groups), and C π 1 ( γ ) denotes the cen tralizer of γ in π 1 . T h us, as is w ell-kno wn, diﬀeren t comp onent s of Λ K ha ve diﬀeren t homotop y types. In this s u b-subs ection, w e give a mo del of Λ γ K under the assu mption that γ is in the cente r of π 1 . F or the nilp oten t case, the r esult here is already included in [13] and w e use their argumen t in the p ro of. The d iﬀerence form the nilp otent one is that the category of un p ointed schematic homotop y t yp es is n ot equiv alen t to that of unaugment ed equiv arian t d ga’s. Let M = M red ( K, ∗ ). L et Λ M b e a π red 1 -dga deﬁn ed as follo ws . 1. Λ M = V ( V i , ¯ V i ) = V ( V i ) ⊗ V ( ¯ V i ), where for i ≥ 1, ¯ V i is a cop y of V i with d eg ¯ V i = i − 1. W e iden tify M with a subalgebra of Λ M via the natur al identiﬁcatio n : M ⊃ V i = V i ⊂ Λ M . 2. W e deﬁn e a deriv ation i : M → Λ M of d egree − 1 by i ( x ) = ¯ x ( x ∈ V i , ¯ x ∈ ¯ V i is the copy of x ), i ( xy ) = i ( x ) y + ( − 1) deg x xi ( y ). Then, the diﬀerentia l d Λ M on Λ M is deﬁned b y i ◦ d M + d Λ M ◦ i = 0. Supp ose γ ∈ π 1 b e in the cen ter of π 1 . Let ¯ γ b e th e image of γ un der the map π 1 → π alg 1 ( k ) and ¯ γ = ( u, s ) b e the decomp osition su c h that u ∈ R u ( π alg 1 )( k ), s ∈ π red 1 ( k ). Under the ident iﬁcation of Cor.5.2.6, w e regard u as a linear map ¯ V 1 ∼ = V 1 → k Let I ⊂ Λ M b e the homogeneous ideal generated b y { x − u ( x ) , dx | x ∈ ¯ V 1 } ∪ { y − s · y | y ∈ Λ M} . This id eal is closed und er π red 1 -action and diﬀerenti al. The ﬁrst part of th e ab o v e set is the same as K u in [13, P .4946]. W e deﬁn e a π 1 (Λ γ K, γ ) red -dga Λ γ M by Λ γ M := Λ M /I , 29 where the action of π 1 (Λ γ K, γ ) red is giv en by the pullbac k of th e action of π 1 b y the ev aluation Λ γ K → K at some ﬁxed p oin t of S 1 . Λ γ M is not n ecessarily min imal. Th e follo w ing is a n on-simply connected v ersion of [7 , T heorem]. Prop osition 4.2.9. Supp ose k = Q . L e t K b e a c onne c te d simplicial set suc h that π 1 ( K ) is algebr aic al ly go o d, π 2 ( K ) is a ﬁnitely gener ate d ab elian g r oup or ﬁnite dimensional Q -ve ctor sp ac e, and π i ( K ) is of ﬁnite r ank for e ach i ≥ 3 . L et γ b e an e lement of the c enter of π 1 ( K ) . Then, under the ab ove notations, ther e exists a qu asi- isomorph ism M red (Λ γ K, γ ) ∼ → Λ γ M . Pr o of. See also Rem.4.2.10 b elo w. In the f ollo wing, ( − ⊗ Q ) sc h is abbreviated to ( − ) sc h . Let ( K sc h ) S 1 ∈ s Pr ( Q ) b e an ob ject giv en by ( K sc h ) S 1 ( R ) = ( K sc h ( R )) S 1 for R ∈ Q − Alg , where the righ t hand side is th e exp onen tial in s S et and K sc h is tak en to b e ﬁbran t in s Pr ( Q ) lo c . Let ( K sc h ) S 1 γ b e the co nnected comp onent of ( K sc h ) S 1 con taining γ . In general, w e h a ve a map f : (Λ γ K ) sc h → ( K sc h ) S 1 γ induced by the map K → K sc h ( k ). W e ﬁrst consid er th e case that V = π 2 ( K ) is a Q -vect or space. In this case, as V alg ∼ = V b y Lem.4.2.7, b y comparing the long exact s equences asso ciated to the ﬁbr ations p sc h : (Λ γ K ) sc h → K sc h , q : ( K sc h ) S 1 γ → K sc h , the ev aluation at the base p oin t of S 1 , w e can see f is a weak equiv alence and π 1 (Λ γ K ) red ∼ = π red 1 . Let G = π red 1 . L et ( H , B , a B ) ∈ dg Alg red 0 , ∗ b e a r eductiv e d ga and ψ : H → G b e a h omomorphism. W e ﬁx three s ets H 1 , H 2 , and H 3 as follo ws. H 1 = { β ∈ [ ψ ′∗ M , ∧ ξ ⊗ B ] dg Alg ( Z red × H ) | a B ◦ β = u ∈ [ s ∗ M , ∧ ξ ] dg Alg ( Z red ) } H 2 = { β ′ ∈ [ ψ ′∗ M , ∧ ξ ⊗ B ] ′ dg Alg ( Z red × H ) | a B ◦ β ′ = u ∈ [ s ∗ M , ∧ ξ ] ′ dg Alg ( Z red ) } H 3 = ( β ′ ∈ [ ψ ′∗ M , ∧ ξ ⊗ B ] ′ dg Alg ( Z red × H )      for a representa tiv e ˜ β ′ ∈ [ ψ ′∗ M , ∧ ξ ⊗ B ] of β ′ , a B ◦ ˜ β ′ = u ∈ [ s ∗ M , ∧ ξ ] ) F or th e notati on [ − , − ] ′ , see Lem.3.3.7 and b elo w. (w e pu t E = ∗ ). Here, ( Z red , ∧ ξ ) is the minimal mo del of A red ( S 1 ) and ξ denotes a generator of degree 1, ψ ′ = s × ψ , and u is considered as a morphism s ∗ M → ∧ ξ ∈ dg A lg ( Z red ). By an argument similar to the p ro of of [13, Thm.6.1], w e s ee th at there exists a n atur al bijection: [ ψ ∗ Λ γ M , B ] dg Alg ( H ) ∼ = H 1 (*) (Here w e use H -equiv arian t aﬃne stac ks [25] instead of rational simplicial sets in [13]. and n ote that there is a natural bijection [ A, U ( C )] dg Alg ( H × Z red ) ∼ = [ A Z , C ] dg Alg ( H ) ,where for an H -dga A , and an H × Z red -dga C , A Z denotes Z -coin v arian ts of A , and U ( C ) is C considered as H × Z red -dga with trivial Z red -action.) W e shall see what universal prop ert y M red (( K sc h ) S 1 γ ) ha v e, using L em.3.3.7. W e w ork on the in ter- mediate ca tegory w hose ob jects are p oin ted sc hematic homotop y t yp es but whose morp hisms are those of Ho ( SHT ), the unp oin ted homotop y categ ory b et w een un derlying un p ointed sc hematic homotop y types. ( K sc h ) S 1 γ ha ve a univ ersal prop ert y as follo ws . Let ( Y , y ) b e a p oin ted sc h ematic homotop y typ e. The follo wing tw o sets of morp hisms in Ho ( SHT ) is naturally bijective . 1. morp hisms φ : Y → ( K sc h ) S 1 γ ∈ Ho ( SHT ) suc h that π 0 ( φ ( Q )) : π 0 ( Y ( Q )) → π 0 (( K sc h ) S 1 γ ( Q )) maps [ y ] to [ γ ]. 2. morp hisms ϕ : Y × ( S 1 ) sc h → K sc h suc h th at the comp osition y × S 1 → Y × ( S 1 ) sc h ( Q ) → K sc h ( Q ) is f reely homotopic to γ . 30 As γ is in the cen ter, this bijection giv es the follo wing bijection. { α ∈ [Rep( G ) c , Rep( H ′ )] | ω H ′ ◦ α = ω G ∈ [Rep( G ) c , V ect ] } ∼ = { α ′ ∈ [Rep( G ) c , Rep( H ′ × Z alg )] | i ∗ Z ◦ α ′ = s ∈ [Rep( G ) c , Rep( Z alg )] } , (**) see L em.3.3.7. Here [ − , − ] means [ − , − ] Cat cl so ”=” m eans n aturally isomorphic, resp ec ting tensors , and in the left h and side G is iden tiﬁ ed w ith π 1 (Λ γ K ) red while it is identiﬁed with π 1 ( K ) red in the righ t han d side ( H ′ = π 1 ( Y )). So by Lem.3.3.7 and the un iv ersal prop ert y of ( K sc h ) S 1 γ , w e get a natural bijection: [ ψ ∗ M red (( K sc h ) S 1 γ ) , B ] ′ dg Alg ( H ) ∼ = H 2 . On the other hand, (*) implies [ ψ ∗ Λ γ M , B ] ′ dg Alg ( H ) ∼ = H 3 . No w, elemen ts in [ s M , ∧ ξ ] which is ident iﬁed with u in [ s M , ∧ ξ ] ′ are { u ∗ g | g ∈ G ( Q ) } bu t b y assu mption th ese are equal to u . So H 2 = H 3 , w hic h implies Λ γ M ≃ M red (( K sc h ) S 1 γ ). F or the case π 2 ( K ) ﬁnitely generated ab elia n, f is n ot a w eak equiv alence. Let r : K → K Q b e a ﬁb erw ise r ationalization so that π 1 ( r ) : π 1 ( K ) ∼ = π 1 ( K Q ) and π i ( r ) ⊗ Q : π i ( K ) ⊗ Q ∼ = π i ( K Q ). Consider the induced map r ∗ : Λ γ K → Λ γ K Q . By Lem.4.2.7, the pull-bac k by π 1 ( r ∗ ) preserv es s emi-simp le Q - represent ations so maps π 1 ( r ∗ ) red : π 1 (Λ γ K ) red → π 1 (Λ γ K Q ) red and A red ( r ∗ ) : A red ( K Q ) → A red ( K ) are induced. By 3.3.7, ( α ∗ ) ∗ T PL ( K Q ) → T PL ( K ) is equiv alent to T A red ( α ∗ ) : T A red ( K Q ) → T A red ( K ). So by Thm.4.1.3, Thm .4.2.8 and naturalit y of the constru ction of the iterated Hirs ch extensions, w e see M red (Λ γ K Q ) ∼ = M red (Λ γ K ) as und erlying dga’s. Remark 4.2.10. If we deﬁne a ”c ate gory of dg-algb er as over pr o-r e ductive gr oup oids” appr opriately, it is e quivalent to the su b-c ate gory of Ho ( SHT ) whose morphisms ar e those which pr eserve se misimple lo c al systems. We c an pr ove Pr op.4.2.9, u si ng this c ate gory similarly. T o de al with mor e gener al b ase lo ops than the c enter,esp e cial ly lo ops which do not pr eserve semisimple lo c al systems, it wil l b e inevitable to c onsider T annakian dgc’s as the u niversality of the c omp onent is not c ontaine d in the se misimple sub c ate gory. Example 4.2.11. L et K b e a c onne cte d simplicial set such that π 1 ( K ) = Z and M (= M red ( K )) = ∧ ( t, s 1 , s 2 , s 3 , s 4 , u 1 , u 2 , u 3 ) deg t = 1 , deg s i = 2 , deg u j = 3 , wher e 1. the gener ator g = 1 ∈ Z act on M by g · t = t , g · ( s i , s i +1 ) = ( s i , s i +1 )  0 1 − 1 0  ( i = 1 , 3 ), g · u j = − u j ( j = 1 , 2 ), g · u 3 = u 3 , and 2. the diﬀer e ntial is given by ds 1 = ds 2 = 0 , ds 3 = ts 1 , ds 4 = ts 2 , du 1 = s 1 s 2 , du 2 = 2 tu 1 − s 1 s 3 − s 2 s 4 , du 3 = s 1 s 4 − s 2 s 3 . F or example, the action of π 1 on π 2 ⊗ Q ( π i = π i ( K ) ) is as fol lows (se e the pr o of of Thm.3.4.5). g · ( s ∗ 4 , s ∗ 3 , s ∗ 2 , s ∗ 1 ) = ( s ∗ 4 , s ∗ 3 , s ∗ 2 , s ∗ 1 )     0 1 0 − 1 − 1 0 1 0 0 0 0 1 0 0 − 1 0     Her e, ( s ∗ i ) is the dual b asis of ( s i ) . (i) γ = e (the unity). Λ e M = ∧ ( t, ¯ s i , s i , ¯ u j , u j ) i =1 ,..., 4 ,j =1 , 2 , 3 , which is alr e ady minimal. F or example, d ¯ u 2 = 2 t ¯ u 1 + ¯ s 1 s 3 + s 1 ¯ s 3 + ¯ s 2 s 3 + s 2 ¯ s 3 . In this c ase, π 1 (Λ e K ) ∼ = π 2 ⋊ π 1 and b y the description of the 31 minimal mo del, we c an c ompute the action of π 1 (Λ e K ) on the homotopy gr oup. F or example, the action of ¯ s 1 (c onsider e d as an element of π 2 ⊂ π 2 ⋊ π 1 ) on π 2 (Λ e K ) ⊗ Q ∼ = Q h ¯ u ∗ 3 , ¯ u ∗ 2 , ¯ u ∗ 1 , s ∗ 4 , s ∗ 3 , s ∗ 2 , s ∗ 1 i is g iven by           1 1 1 − 1 1 1 1 1 1 1           (ii) γ = g . Λ g M = ∧ ( t, ¯ u 3 , u 3 ) with d = 0 , which is minimal (iii) γ = g 2 . Λ g 2 M i s not minimal. A minimal mo del of Λ g 2 M i s N 2 := ∧ ( t, ¯ u 1 , ¯ u 3 , u ′ 2 , u 3 ) , deg t = 1 , deg ¯ u 1 = deg ¯ u 3 = 2 , deg u ′ 2 = deg u 3 = 3 , d = 0 . N 2 is identiﬁe d with a sub algbr a of Λ g 2 M whose inclusion is a qu asi-isomorphism, by t 7→ t , ¯ u j 7→ ¯ u j , u ′ 2 7→ u 2 − t ¯ u 2 / 2 , u 3 7→ u 3 . (iv) γ = g 4 . A minimal mo del of Λ g 4 M is N 4 := ∧ ( t, ¯ s 1 , ¯ s 2 , s ′ 3 , s ′ 4 , ¯ u ′ 1 , ¯ u ′ 3 , u ′ 2 , u ′ 3 ) with deg s ′ i = deg ¯ u ′ j = 2 , deg u ′ j = 3 , and ds ′ 3 = ds ′ 4 = d ¯ u ′ 1 = 0 , d ¯ u ′ 3 = − ¯ s 1 s ′ 4 + ¯ s 2 s ′ 3 , du ′ 2 = du ′ 3 = 0 . N 4 is identiﬁe d with a sub algbr a of Λ g 4 M who se inclusion is a quasi-isomorphism, by x 7→ x ( x = t, ¯ s 1 , ¯ s 2 ), s ′ i 7→ s i − t ¯ s i / 4 ( i = 3 , 4 ), ¯ u ′ 1 7→ ¯ u 1 + ( ¯ s 1 ¯ s 4 − ¯ s 3 ¯ s 2 ) / 4 , ¯ u ′ 3 7→ ¯ u 3 − ¯ s 3 ¯ s 4 / 4 , u ′ 2 7→ u 2 − ( ¯ s 3 s ′ 3 + ¯ s 4 s ′ 4 + t ¯ u 2 ) / 4 . u ′ 3 7→ u 3 − t ¯ u 3 / 4 − 13 t ¯ s 3 ¯ s 4 / 16 + ( ¯ s 3 s 4 − ¯ s 4 s 3 ) / 4 . Remark 4.2.12. L et [ γ ] ∈ H 2 ( π 1 ; ( π 2 ) γ ) b e the class asso ciate d to the extension 0 → ( π 2 ) γ → π 1 (Λ γ K ) → π 1 → 1 . When γ is the unity, cle arly [ γ ] = 0 . In the ab ove example, for al l γ , the class [ γ ] ⊗ Q ∈ H 2 ( π 1 ; ( π 2 ) γ ⊗ Q ) is zer o. But this is not true in gener al, and a c ounter example exists even in the nilp otent c ase. F or example, let M = ∧ ( t 1 , t 2 , t 3 , s ) with deg t i = 1 , deg s = 2 , dt i = 0 , and ds = t 1 t 2 t 2 , and let γ = ( t 1 = 1 , t 2 = 0 , t 2 = 0) . Then Λ γ M = ∧ ( t 1 , t 2 , t 3 , ¯ s, s ) with d ¯ s = − t 2 t 3 , which imply [ γ ] ⊗ Q = [ − t 2 t 3 ] 6 = 0 (se e Thm.4.2.8). 4.3 Equiv alence with algebraically go o d spaces In th is su b section, w e will sho w a homotop y categ ory of T annakian d gc’s w ith sub sidiary data is equiv alent to a homotop y catego ry of some sp aces. W e r estrict our d iscussion to the p oin ted case. L et s S et c ∗ denote th e catego ry of p ointed connected simplicial sets. W e s ay a conn ected p ointe d simplicial set K is algebr aic al ly go o d if π 1 ( K ) is algebraically go o d and π i ( K ) is a ﬁnite dimensional Q -v ector s pace for eac h i ≥ 2. we denote by s Set g d ∗ the full sub cate gory of s Set c ∗ consisting of algebraically go o d s paces. Deﬁnition 4.3.1. (1) Deﬁne a c ate gory T an + ∗ as fol low s. 1. An obje ct is a triple ( T , Γ , φ ) c onsisting of a T annakian dgc T , a discr e te gr oup Γ and an e quivalenc e of close d k -c ate gories φ : Z 0 T ∼ → Rep(Γ) such that the fol lowing diagr am is c ommutative. Z 0 T φ / / Z 0 ω T $ $ H H H H H H H H H Rep(Γ) ω Γ   V ect , wher e ω Γ is the for getful functor. 32 2. A morphism ( T , Γ , φ ) − → ( T ′ , Γ ′ , φ ′ ) is a p air F = ( F , F g r ) of a morphism of T annakian dgc’s F : T − → T ′ and a gr oup homom orphism F g r : Γ ′ − → Γ suc h that the f ol low ing diagr am is c ommutative. Z 0 T Z 0 F / / φ   Z 0 T ′ φ ′   Rep(Γ) ( F gr ) ∗ / / Rep(Γ ′ ) We say a morphism F : ( T , Γ , φ ) − → ( T ′ , Γ ′ , φ ′ ) ∈ T an + ∗ is a we ak e qu ivalenc e i f F : T → T ′ is a qu asi- e quivalenc e and F g r is an isomorphism. Ho ( T a n + ∗ ) denotes the lo c alization of T an + ∗ obtaine d by inverting al l we ak e quivalenc es. (2) T an + g d ∗ denotes the ful l sub c ate gory c onsisting of ( T , Γ , φ ) ’s such that π i ( T ) := [ T , T PL S i ] dg Cat cl ∗ is a ﬁnite dimensional Q -ve ctor sp ac e for i ≥ 2 and Γ is algebr aic al ly go o d (By L em.3.4.6, π i ( T ) is isomorphic to the dual of the i -th inde c omp osable mo dule of the minimal mo del of the c orr esp onding dga). The c orr esp onding ful l su b c ate gory of Ho ( T an + ∗ ) is denote d by Ho ( T an + g d ∗ ) . (3) L et F 0 , F 1 : ( T , Γ , φ ) − → ( T ′ , Γ ′ , φ ′ ) ∈ T an + ∗ b e two morphisms. We say F 1 and F 2 ar e r igh t homotopic (r esp. left h omotopic ), written ∼ r (r esp. ∼ l ) if F gr 1 = F gr 2 and F 0 and F 1 : T → T ′ ar e right homotopic (r esp. left homotopic) as morphisms of dg Cat cl ∗ . W e shall sho w Ho (s Set g d ∗ ) and Ho ( T an + g d ∗ ) are equiv alen t. W e do n ot claim T an + ∗ has a mo del category structure bu t we ha ve the f ollo wing lemma. Lemma 4.3.2. L et ( T , Γ , φ ) , ( T ′ , Γ ′ , φ ′ ) ∈ T an + ∗ . Supp ose T is c oﬁbr ant and T ′ is ﬁbr ant in dg Cat cl ∗ . Then, for two morphism F 0 , F 1 : ( T , Γ , φ ) → ( T ′ , Γ ′ , φ ′ ) ∈ T an + ∗ , F 0 ∼ r F 1 if and only if F 0 ∼ l F 1 . ∼ r (so ∼ l ) is an e quivalenc e r elation on Hom T an + ∗ (( T , Γ , φ ) , ( T ′ , Γ ′ , φ ′ )) and ther e exists a natur al bije ction: Hom Ho ( T an + ∗ ) (( T , Γ , φ ) , ( T ′ , Γ ′ , φ ′ )) ∼ = Hom T an + ∗ (( T , Γ , φ ) , ( T ′ , Γ ′ , φ ′ )) / ∼ r . Pr o of. Note that if F 0 , F 1 : T → T ′ ∈ T an ∗ are right or left homotopic in dg Cat cl ∗ , the in duced m orp hisms (Z 0 F 0 ) ∗ , (Z 0 F 1 ) ∗ : Aut ⊗ ( ω Z 0 T ′ ) → Aut ⊗ ( ω Z 0 T ) are equal. By using this fact, we see right or left homotopic morphisms r epresen t the same morp hism in Ho ( T an + ∗ ). So the lemma is standard, see [1] or [14, Thm.1.2.10]. Remark 4.3.3. The author do e s not know whether the statement c orr esp onding to L em.4.3.2 in the un- p ointe d c ase is true. In this c ase, we wil l c onsider gr oup oids inste ad of gr oups. As morphisms of gr oup oids have non-trivial homotopic r elation, if we deﬁne a right or left homoto pic r elation similarly, the pr o of do es not g o on similary. Deﬁne tw o f unctors T + PL : s Set c ∗ − → ( T an + ∗ ) op , R h− , − , −i : ( T an + ∗ ) op − → s Set c ∗ as follo ws. F or K ∈ s Set c ∗ , T + PL ( K ) = (T PL ( K ) , π 1 ( K ) , φ K ), where φ K is a ﬁ xed functorial equiv alence Z 0 (T PL ( K )) ∼ → Lo c( K ) ∼ → Rep( π 1 ( K )). F or ( T , Γ , φ ) ∈ T an + ∗ , R h T , Γ , φ i is the follo wing pullback. R h T , Γ , φ i / /   h QT i p   B Γ φ ∗ / / B Π 1 h QT i 33 Here, QT is a ﬁxed fun ctorial coﬁbrant replacement of T , Π 1 h QT i is the fu ndamenta l group oid, p is the canonical map whic h giv es an isomorphism of fun damen tal group oid, and φ ∗ is the follo wing comp osition. B Γ → B Au t ⊗ ( ω Γ ) φ ∗ ∼ = B Aut ⊗ (Z 0 ω QT ) ∼ = B π 1 h QT i → B Π 1 h T i , see App endix A.2 , where the ﬁrst map is indu ced by the natural ”ev aluation” Γ → Aut ⊗ ( ω Γ ) and the in verse of the isomorphism Aut ⊗ (Z 0 ω T ) ∼ = π 1 h QT i is th e follo wing comp osition: π 1 h QT i ∼ = [ QT , T PL S 1 ] dg Cat cl ∗ Z 0 → [Z 0 ( QT ) , Rep ( Z )] Cat cl ∗ ∼ = Aut ⊗ (Z 0 ω QT ) , where the last isomorph ism sends an action of 1 ∈ Z to an automorphism. Th is is in fact, an isomorphism by Thm. ?? and 5.2.1. Note that T + PL preserve s weak equiv alences so it induces a fu n ctor T + PL : Ho (s Set c ∗ ) − → Ho ( T an + ∗ ) op . As th e m ap p : h QT i − → B Π 1 h QT i is a ﬁbration, the ab o ve p ullbac k is a homotop y pullbac k. So R h− , − , −i also pr eserv es wea k equ iv alences and it d eﬁnes a fun ctor b et w een homotop y categorie s. W e ha ve an obvio us natur al map Φ : Hom s Set c ∗ ( K, R h T , Γ , φ i ) − → Hom T an + ∗ (( QT , Γ , φ ) , T + PL ( K )) . W e can easily see if K is reduced, i.e., K 0 ∼ = ∗ , this map is a b ij ection and pr eserv es and reﬂects homotop y relation so Φ induces a bijection Φ : Hom Ho (s Set c ∗ ) ( K, R h T , Γ , φ i ) − → Hom Ho ( T an + ∗ ) (( T , Γ , φ ) , T + PL ( K )) b y Lem.4.3.2. Th us the pair (T + PL , R h− , − , −i ) is an ad j oin t pair. Theorem 4.3.4. (1) The ab ove adjoint p air induc es an e quivalenc e b etwe en ful l sub c ate gories : T + PL : Ho (s Set g d ∗ ) ∼ / / Ho ( T an + g d ∗ ) op : R h− , − , −i . o o (2) L et K b e a c onne cte d p ointe d simplicial se t whose fu ndamental g r oup is algebr aic al ly go o d and whose higher homotopy gr oups ar e of ﬁnite r ank as Ab elian gr oups. Then, the unit K → R h T PL ( K ) , π 1 ( K ) , φ K i is the ﬁb erwise r ationalization. Pr o of. (1) F or K ∈ s S et g d ∗ , we shall sho w the unit map u K : K − → R h T + PL ( K ) i is a w eak equiv alence. w e use the notation of Thm.4.1.3. By deﬁnition, the map π 1 ( u K ) : π 1 ( K ) → π 1 ( R h T + PL ( K ) i ) is an isomorphism . F or i -th homotop y group ( i ≥ 2), w e ma y assume K is w eak equ iv alent to the i -th leve l of the P ostniko v tow er of K . C onsider the ﬁb er sequence F f → K p i − 1 → K ( i − 1) , where F is the ﬁ b er of typ e K ( π i ( K ) , i ). By the pro of of Lem.4.1.5, th e corresp onding m orphism f ∗ : T PL K → T PL F is equiv alen t to the morphism T ( π red 1 , M ) → T ( e, V ( π ∨ i , i )) ∈ T an ∗ induced b y a morph ism ( π red 1 , M ) → ( e, V ( π ∨ i , i )) ∈ d g Alg red 0 , ∗ whic h maps V j to 0 ( j < i ) an d V i to π ∨ i isomorphically . So by Lem.3.4.6, π i ( R h f ∗ i ) : π i ( R h T PL F i ) → π i ( R h T PL K i ) is an isomorphism. As the unit F → R h T PL F i is a wea k equiv alence by the classical Sulliv an’s theory , we see the map π i ( K ) → π i ( R h T PL K i ) is an isomorphism . T his imp lies π i ( u K ) is an isomorphism. Thus, u K is a w eak equ iv alence and T + PL is f ully faithful. By a s imilar argument , we see a map F : ( T , Γ , φ ) − → ( T ′ , Γ ′ , φ ′ ) ∈ T an + g d ∗ is a w eak equ iv alence if the induced map R F : R h T , Γ , φ i − → R h T ′ , Γ ′ , φ ′ i is a w eak equiv alence so T + PL is essentia lly surj ectiv e. One can p ro ve (2) b y a similar argument. 34 5 Corresp ond ence with schema tic homotop y t yp es 5.1 Deﬁnitions W e recall the notion of schematic homotopy t yp es fr om [22, 25]. Let R − Alg denote th e cat egory of (discrete) comm utativ e unital R -algebras for a commutati v e u nital k -algebra R . Deﬁnition 5.1.1. (1) s Pr ( k ) denotes the catego ry of simplicial p reshea ves on Aﬀ k := ( k − Alg ) op . In other wor ds, s Pr ( k ) is the c ate gory of functors k − Al g − → s Set and natur al tr ansformatio ns. 1. F or a simplicial pr eshe af X ∈ Ob(s Pr ( k )) and a R - p oint s ∈ X ( R ) 0 , ( R ∈ k − Alg ) the i -th homotop y presheaf π pr e i ( X, s ) with b ase p oi n t s is a pr eshe af on Aﬀ R = ( R − A lg ) op deﬁne d by π pr e i ( X, s )( S ) := π i ( X ( S ) , u ( s )) for S ∈ R − A lg . H e r e , u ( s ) denot es th e image of s by th e unit u : R → S , and if i = 0 , we ignor e s and su pp ose R = k . The i -th homotop y sheaf π i ( X, s ) with base p oint s is the she aﬁﬁc ation of π pr e i ( X, s ) with r e sp e ct to the f aithful ly ﬂat quasi-c omp act top olo gy on A ﬀ R . We say X ∈ s Pr ( k ) is c onne cte d if the 0 -th homotopy she af π 0 ( X ) is one p oint she af. 2. W e r e gar d s Pr ( k ) as a mo del c ate gory via the ob ject w ise p ro jectiv e m o del str u cture , deno te d by s Pr ( k ) ob j , wher e a morphism f : X → Y ∈ s Pr ( k ) is a we ak e quivalenc e (r esp. a ﬁbr ation) if and only if f R : X ( R ) → Y ( R ) ∈ s Set is a we ak e quiv alenc e (r esp. ﬁbr ation) of simpl icial sets for any R ∈ k − Alg . We c al l the ab ove we ak e quivalenc e an ob jec t wise equiv alence . 3. W e also c onsider the local pr o jectiv e mo del stru ctur e [18, 19] on s Pr ( k ) , denote d by s Pr ( k ) lo c , wher e we ak e quivalenc es and c oﬁbr ations ar e deﬁne d as fol lows. (a) A morphism f : X → Y is a we ak e quivalenc e, c al le d a lo cal equiv alence , i f and only if it induc es isomorpisms of 0 -th homotopy she aves π 0 ( X ) ∼ = π 0 ( Y ) and of i -th homotopy she aves π i ( X, s ) ∼ = π i ( Y , f ( s )) for any i ≥ 1 , any R ∈ k − Alg , and any b ase p oint s ∈ X ( R ) 0 . (b) A morphism is a c oﬁbr ation if and only if i t is a c oﬁbr ation in the obje ctwise pr oje ctive mo del structur e. We say an obje ct X ∈ s Pr ( k ) is a lo c al obje ct if a ﬁbr ant r eplac ement X → RX in s Pr ( k ) lo c is an obje ctwise e quivalenc e. 4. W e set s Pr ( k ) ∗ := ∗ / s Pr ( k ) , s Pr ( k ) ob j ∗ := ∗ / s Pr ( k ) ob j , and s Pr ( k ) lo c ∗ := ∗ / s Pr ( k ) lo c . (2) ([22, 25]) We say a p ointe d simplicial pr eshe af X ∈ s Pr ( k ) ∗ is a p oi n ted s c hematic homotop y typ e i f the fol low ing c onditions ar e satisﬁe d. 1. X is a c onne c te d lo c al obje ct. 2. The homotopy she af π i ( X, ∗ ) is r epr esente d by an aﬃne g r oup scheme over k for any i ≥ 1 and it is unip otent for i ≥ 2 . We say a p ointe d sc hematic homotopy typ e X ∈ s Pr ( k ) ∗ is a p oint ed aﬃne stac k if π 1 ( X, ∗ ) is r epr esente d by a u ni p otent aﬃne g r oup scheme over k . We denote the ful l sub c ate gory of s Pr ( k ) ∗ c onsisting of p ointe d schematic homotopy typ e s by SHT ∗ . Ho ( SHT ∗ ) denotes the c orr e sp onding ful l su b c ate gory of Ho (s Pr ( k ) ob j ∗ ) (or Ho (s Pr ( k ) lo c ∗ ) ). We say an obje c t U ∈ s Pr ( k ) is a sc hematic homotop y type if ther e exists a p oint ∗ ∈ U ( k ) 0 such that the p ointe d obje ct ( U, ∗ ) ∈ s Pr ( k ) ∗ is a p ointe d schematic homotopy typ e. We denote by S HT the fu l l sub c ate gory of s Pr ( k ) c onsisting of schematic homotopy typ es. (3) ([22]) L et K ∈ s S et (r esp. s S et ∗ ) b e a c onne cte d simplicial set. We denote by K b e a c onstant simplicial pr eshe af g iven by K ( R ) = K for R ∈ k − Alg . The sc h ematization of K is a morphism K − → ( K ⊗ k ) sc h ∈ Ho (s Pr ( k ) ob j ) 35 (r esp. Ho (s Pr ( k ) ob j ∗ ) ) which is initial in the c ate gory K / Ho ( SHT ) (r esp. K / Ho ( SHT ∗ ) ). Her e, K / Ho ( SHT ) is the fu l l sub c ate gory of the over c ate gory K / Ho (s Pr ( k )) c onsisting of obje cts K → X su ch that X is a schematic homotopy typ e. K / Ho ( SHT ∗ ) is si milar. The ab o v e d eﬁnition of p oin ted sc hematic homotop y t yp es is d iﬀeren t f rom th e original one bu t they are sho w n to b e equ iv alen t, see [25, Cor.3.16]. T he existence of the sc h ematizatio n in the p oin ted categ ory was pro v ed in [22] (w e w ill repr ov e this later, based on the ab o v e deﬁ n ition). The existence of the u np oi n ted sc hematization follo ws from Cor.5.2.7. W e sh all deﬁne an adj oint pair (T PL , h−i ) : s Pr ( k ) → (dg Cat cl ) op (or s Pr ( k ) ∗ → (dg Cat cl ∗ ) op ). This is an analogue of the adjoin t pair (T PL , h−i ) : s S et → (dg Cat cl ) op deﬁned in sub section 3.4. W e ﬁr s t deﬁne a notion of lo ca l sys tems on a simplicial presheaf. In order to ensure some functorialit y , w e take care ab o ut the deﬁnition of the category of p ro jectiv e m o dules. The follo wing deﬁnition of (1) can b e found in [23, 1.3.7], for example. Let R − Mo d denote the category of R -mo dules and R -linear map s . Deﬁnition 5.1.2. (1) F or R ∈ k − Alg we deﬁne a c ate gory R − Proj as fol lows. 1. An obje ct is a p air ( M − , φ − ) c onsisting of a f u nction M − which assigns e ach R -algebr a S a ﬁnitely gener ate d pr oje ctive S -mo dule M S and a function φ − which assigns e ach morphism f : S → S ′ ∈ R − Alg an isomorphism φ f : M S ⊗ S S ′ → M S ′ ∈ S ′ − Mo d such that for e ach se quenc e S f → S ′ f ′ → S ′′ ∈ R − A lg the fol lowing diagr am is c ommutative. ( M S ⊗ S S ′ ) ⊗ S ′ S ′′ φ f ⊗ S ′′ / /   M ′ S ⊗ S ′ S ′′ φ f ′   M S ⊗ S S ′′ φ f ′ ◦ f / / M ′′ S Her e, the left vertic al arr ow is induc e d fr om the asso ciativity isomorp hism and unit isomorph ism. Sometimes we omit φ . 2. A morphism F : ( M , φ ) → ( N , ϕ ) is a function assigning e ach R -algebr a S a morphism F S : M S → N S ∈ S − Mo d such that for any f : S → S ′ ∈ R − A lg the fol lowing diagr am is c ommutative M S ⊗ S S ′ F S ⊗ S ′ / / φ f   N S ⊗ S S ′ ϕ f   M S ′ F S ′ / / N S ′ We r e gar d R − Proj as a close d k -c ate gory in the obvious way. F or example the i nternal hom Hom ( M , N ) is given by Hom ( M , N ) S = Hom ( M S , N S ) , the right hand side is the internal hom in S − Mo d . F or f : R → R ′ ∈ k − Alg we have a morphism f ∗ : R − Proj → R ′ − Proj ∈ R − Cat cl deﬁne d by f ∗ ( M ) ′ S = M f ∗ S ′ , wher e S ′ ∈ R ′ − Alg . Note that f ′ ∗ ◦ f ∗ = ( f ′ ◦ f ) ∗ (2) L et X b e an obje ct of s Pr ( k ) . L et R − Proj iso b e the ful l sub c ate gory of R − Proj c onsisting of al l obje cts and al l isomorphisms. A lo cal sy s tem L on X is a c ol le ction { L R } R ∈ k − Alg of fu nctors L R : [∆( X ( R ))] op → R − Proj iso such that for e ach morphism f : R → R ′ ∈ k − Alg the f ol low ing diagr am i s c ommutative. [∆( X ( R ))] op L R / / (∆ X f ) op   R − Proj f ∗   [∆( X ( R ′ ))] op L R ′ / / R ′ − Proj 36 A morphism α : L → L ′ of lo cal systems is a c ol le ction { α R } of natur al tr ansformations α R : L R ⇒ L ′ R : [∆( X ( R ))] op → R − Proj c omp atible with ring homomorphisms. We denote by Lo c( X ) the c ate gory of lo c al systems on X . W e r e gar d L o c( X ) as a close d k -c ate gory in the obvi ous way. Let X ∈ s Pr ( k ) ∗ b e a conn ected ob ject such that π 1 ( X, ∗ ) is represente d b y an aﬃne group sc heme G o ve r k . I t is easy to see Lo c( X ) is equiv alen t to the category of ﬁ nite d imensional represent ations of G . Deﬁnition 5.1.3. L et X b e a simplicial pr eshe af and L b e a lo c al system on X . The de Rham complex of L -v alued p olynomial forms C PL ( X ; L ) ∈ C ≥ 0 ( k ) is deﬁne d as fol lows. F or e ach q ≥ 0 , the de g r e e q p art is the sub mo dule of Y R ∈ k − Alg Y p ≥ 0 Y σ ∈ X ( R ) p ∇ ( p, q ) ⊗ k L R ( σ ) R c onsisting of { ω σ } R ∈ k − Alg ,σ ∈ ∆ X ( R ) ’s such that 1. for e ach algebr a R ∈ k − Alg and morphism a : τ → σ ∈ ∆ X ( R ) , a ∗ ω σ = ω τ , and 2. for e ach map f : R → R ′ ∈ k − Alg and simplex σ ∈ X ( R ) , (id ⊗ k f ∗ )( ω σ ) = ω X f σ . Her e, a ∗ denotes the tensor of a ∗ : ∇ ( | σ | , q ) → ∇ ( | τ | , q ) and L R ( a ) : L R ( σ ) → L R ( τ ) ( | · | denote the dimension of a simplex), and (id ⊗ k f ∗ )( ω σ ) denotes the i mage of ω σ by the tensor of the identity of ∇ ( | σ | , q ) and the fol lowing c omp osition. L R ( σ ) R → L R ( σ ) R ⊗ R R ′ φ f → L R ( σ ) R ′ = L R ′ ( X f σ ) . F or q ≤ − 1 , we set C q PL ( X, L ) = 0 . The diﬀe r ential d : C q PL ( X, L ) → C q +1 PL ( X, L ) is deﬁne d fr om that of ∇ ( ∗ , q ) . W e deﬁne a fun ctor T PL : s Pr ( k ) − → (dg Ca t cl ) op b y setting Ob(T PL ( X )) = O b(Lo c ( X )) , Hom T PL ( X ) ( L , L ′ ) = C PL ( X ; Hom ( L , L ′ )) for X ∈ s Pr ( k ). Th e comp ositio n and the closed tensor structure are deﬁn ed in the ob vious wa y . A r igh t adjoin t h−i : (dg Cat cl ) op − → s Pr ( k ) of T PL is deﬁned by h C i ( R ) n := Hom dg Cat cl ( C, T PL ( h R × ∆ n )) for C ∈ dg Cat cl , n ≥ 0 and R ∈ k − Al g . Here h R denotes the Y oned a embedd ing of S pecR , and h R × ∆ n is a simplicial presheaf giv en b y h R × ∆ n ( S ) := h R ( S ) × ∆ n ∈ s Set . F or a p ointe d simplicial pr esheaf X ∈ s Pr ( k ) ∗ , w e d eﬁne a closed d gc with a ﬁb er fu nctor T PL ( X ) ∈ dg Cat cl ∗ b y the follo wing pullbac k square: T PL ( X ) / /   T PL ( X u ) T PL ( pt )   V ect / / T PL ( ∗ ) , where X u ∈ s Pr ( k ) denotes the underlyin g unp oi n ted sim p licial presheaf. On the other h and, for an ob ject C ∈ dg Cat cl ∗ , h C i is n aturally p oi n ted as h V ect i ∼ = ∗ so w e hav e an adj oin t pair (T PL , h−i ) : s Pr ( k ) ∗ → (dg Cat cl ∗ ) op . F or connected ob ject X ∈ s Pr ( k ) ∗ , T PL ( X ) is equiv alent to T PL ( X u ) as closed dgc’s. 37 Lemma 5.1.4. The adjoint p airs (T PL , h−i ) : s Pr ( k ) ob j → (dg Cat cl ) op and (T PL , h−i ) : s Pr ( k ) ob j ∗ → (dg Cat cl ∗ ) op ar e Quil len p airs. Pr o of. When we use Prop.A.1.2, th e pro of is similar to that of Lem.3.4.3. Note that for a lo cal system L on h R × ∆ n , C PL ( h R × ∆ n , L ′ ) ∼ = C PL (∆ n ; L R | id R × ∆ n ). 5.2 Equiv alence b etw een schem atic homotop y types and T annakian dg-categories In this subsection, we p ro ve the f ollo wing theorem. Note that for connected K ∈ s Set ∗ L T PL ( K ) is naturally (quasi-)equiv alent to T PL ( K ) which is d eﬁned in 3.4. Theorem 5.2.1 (cf. Cor.3.57 of [27]) . (1) The derive d adjunction ( L T PL , R h−i ) : Ho (s Pr ( k ) ob j ∗ ) − → Ho (dg Cat cl ∗ ) induc es an e quivalenc e b etwe en ful l sub c ate gories: L T PL : Ho ( SHT ∗ ) ∼ / / Ho ( T an ∗ ) op : R h−i . o o (2) F or c onne cte d p ointe d simplicial set K , the unit of the adjunction K − → R h T PL ( K ) i ∈ Ho (s Pr ( k ) ob j ∗ ) is the schematization over k . W e pro ve th is using the follo wing theorem of [22]. Theorem 5.2.2 (Thm .0.0.2 and Cor.2.2.3 of [22]) . L et c Alg /k b e the c ate gory of c ommutative and unital c osimplicial k -algebr as with augmentations. L et O : s Pr ( k ) lo c ∗ → (c Alg /k ) op b e the functor such that for a simplicial pr e she af X ∈ s Pr ( k ) lo c ∗ , O ( X ) n = Hom( X n , O 0 ) , wher e O 0 is the tautolo gic al pr eshe af k − Alg ∋ R 7→ R ∈ Set and Hom is the set of morphisms b etwe en pr eshe aves. Then O is a left Qu il len f u nctor and its derive d f unctor L O : Ho (s Pr ( k ) lo c ∗ ) → Ho (c Alg /k ) op induc es an e quivalenc e b etwe en the ful l sub c ate gory of aﬃne stacks and one of c onne cte d (i.e. H 0 ∼ = k ) c osimplicial algebr as. This theorem can b e formula ted usin g d g-algebras instead of cosimplicial algebras as follo ws. Let A P L : s Pr ( k ) lo c ∗ → (dg Al g /k ) op b e the functor deﬁ ned by A P L ( X ) = C PL ( X, 1 ). This is a left Quillen functor whose right adjoin t | − | is giv en by | A | ( R ) n := Hom dg Alg ( A, A P L ( h R × ∆ n )) It is w ell-kno wn that the T hom-Sulliv an co chain functor T h : c Alg /k → dg Al g /k induces equiv alence of homotop y categories, and it is easy to see the follo wing diagram commuta tiv e up to natural quasi-isomorphism. s Pr ( k ) lo c ∗ O / / A P L ' ' N N N N N N N N N N N (c Alg /k ) op T h   (dg A lg /k ) op . So we ma y r ep hrase Th m .5.2.2 that the f unctor L A P L : Ho (s Pr ( k ) lo c ∗ ) → Ho (dg A lg /k ) op induces an equiv- alence b et we en aﬃne stac k s and 0-connected d ga’s. W e shall state some fund amen tal r esu lts. Deﬁnition 5.2.3. F or a p ointe d schematic homotopy typ e X , we put A red ( X ) := A red ( L T PL ( X )) Prop osition 5.2.4. (1) L et X ∈ s Pr ( k ) ∗ . L T PL X is c omplete. If X is c onne c te d, L T PL X is a T annakian dg-c ate gory with a ﬁb er functor. L T PL X and T A red ( X ) ar e e quivalent. (2) F or T ∈ T an ∗ , let T ss denote the ful l sub dgc of T c onsisting of semisimple obje cts of Z 0 T . The inclusion 38 T ss − → T induc es an isomorphism R h T i − → R h T ss i ∈ Ho (s Pr ( k ) ob j ∗ ) . (3) F or any obje ct T ∈ T an ∗ , R h T i ∈ Ho (s Pr ( k ) ob j ∗ ) is c onne cte d. (4) L et X , Y ∈ s Pr ( k ) ∗ b e c onne cte d obje cts which is c oﬁbr ant in s Pr ( k ) ob j ∗ . F or a lo c al e qu ivalenc e f : X → Y , the c orr esp onding morphism f ∗ : T PL Y → T PL X is a q u asi-e quivalenc e. (5) L et F ∈ s Pr ( k ) ∗ b e a p ointe d aﬃne stack. The unit of the adjoint F − → R h L T PL ( F ) i ∈ Ho (s Pr ( k ) ob j ∗ ) is an isomorphism. Pr o of. The pr o of of (1) is similar to that of Thm.3.4.5. F or (2), b y (1) and Lem.2.2.4, for an y L ∈ s Pr ( k ), [ L, R h T ss i ] s Pr ( k ) ∼ = [ T ss , L T PL ( L )] dg Cat cl ∼ = [ T , L T PL ( L )] dg Cat cl ∼ = [ L, R h T i ] s Pr ( k ) . (3) follo ws from [11, T h m.3.2]. (4) follo ws from Pr op.A.1.2 as inv ariance of C spl ( − ; L ) u n der lo cal equiv- alence b et ween connected ob jects are we ll-kno wn (see e.g.[22 ]). (5) follo ws from th e reformulatio n of Thm.5.2.2 and (2) as R h L T PL F i ∼ = R h T ss A P L ( QF ) i ∼ = R | A P L ( QF ) | ∼ = F . F or an aﬃne group sc heme G , d eﬁne a simplicial presheaf K ( G, 1) ∈ s Pr ( k ) ∗ b y K ( G, 1)( R ) := K ( G ( R ) , 1). Lemma 5.2.5. (1) L e t G b e a r e ductive aﬃne gr oup scheme. L et K ∈ s Pr ( k ) ∗ b e a ﬁbr ant mo del of K ( G, 1) in s Pr ( k ) lo c ∗ . The unit of the adjoint K − → R h L T PL ( K ) i ∈ Ho (s Pr ( k ) ob j ∗ ) is a lo c al e quivalenc e. (N ote that lo c al e quivalenc es ar e stable under obje ctwise e quivalenc es so we apply the notion to the morphism of Ho (s Pr ( k ) ob j ∗ ) ) (2) L et X b e a p ointe d c onne c te d simplicial pr eshe af suc h that π 0 ( X ) ∼ = ∗ and π 1 ( X ) is r epr esente d by an aﬃne gr oup scheme G . Supp ose X is ﬁbr ant in s Pr ( k ) lo c ∗ . L et K ∈ s Pr ( Q ) ∗ b e a ﬁbr ant mo del of K ( G red , 1) in s Pr ( k ) lo c ∗ and p : X − → K ∈ s Pr ( k ) ∗ b e a ﬁbr ation such that π 1 ( p ) : π 1 ( X ) − → π 1 ( K ) is isomorph ic to the c anonic al map G − → G red . L et F ∈ s Pr ( k ) ∗ b e the ﬁb er of p at the b ase p oint. Then the se quenc e R h L T PL ( F ) i − → R h L T PL ( X ) i − → R h L T PL ( K ) i is a homotopy ﬁb er se quenc e in s Pr ( k ) ob j ∗ . Pr o of. Let Q F b e a coﬁbrant r eplacemen t of F . W e ﬁrst sho w A red ( QF ) ( ∼ = C PL ( QF ; 1 )) is quasi- isomorphic to A red ( X ) with tr ivial group action as a dga. As an explicit mo del of F w e use e X deﬁned as follo ws. W e den ote by F ib (Loc( X ) ss ) the p resheaf of group oid of ﬁb er fu nctors of Lo c( X ) ss , the category of semi-simple lo ca l systems on X . Precisely , f or an algebra R ∈ k − Alg , An ob ject of F ib (Lo c ( X ) ss )( R ) is a morp hism ω : Lo c( X ) ss → R − Proj ∈ Cat cl . and a m orp hism is a natural transformation p r eserv- ing tensors. By the deﬁn ition of R − Proj the corresp onden ce R 7→ F ib (Loc( X ) ss )( R ) is functorial. Let N ( F ib (Loc( X ) ss )) b e the corresp onding simplicial p resheaf obtained b y taking nerv e in the ob ject wise manner. This is a ﬁbrant mo del of K ( G red , 1) in s Pr ( k ) lo c ∗ ( G = π 1 ( X )). W e hav e a natural morp hism p ′ : X − → N ( F ib (Lo c ( X ) ss )) suc h that π 1 ( p ′ ) : π 1 ( X ) − → π 1 N ( F ib (Loc( X ) ss )) is isomorphic to G − → G red . W e set e X ( R ) n = { ( σ, l ) | σ ∈ X ( R ) n , l ∈ F ib (Loc ( X ) ss )( ω σ (0) , ω ∗ ) } 39 where ω x : Lo c ( X ) ss → R − Proj denotes the R -ﬁb er functor comes fr om x ∈ X ( R ) 0 . Note that e X has a G red -action given b y g · ( σ, l ) = ( σ , g ◦ l ). W e shall sho w A red ( e X ) is isomorphic to A red ( X ). O ( G red ) is considered as a lo cal system on X by X ( R ) n ∋ σ 7− → O (Hom F ib (Lo c( X ) ss ) ( ω σ (0) , ω ∗ )) . Here, Hom F ib (Lo c( X ) ss ) ( − , − ) denotes the sheaf of natur al tran s formations whic h is represented by an aﬃne sc heme o ver R . Note that if X is coﬁbr an t, e X is also coﬁbrant . A map φ : A red ( X ) − → A red ( e X ) is deﬁn ed as follo w s. An element of A red ( X ) is presented as P i α i ⊗ f i , α i ∈ ∇ ( n, ∗ ), f i ∈ O (Hom F ib (Lo c( X ) ss ) ( ω ∗ , ω σ (0) )) lo cally on σ ∈ X ( R ) n . W e deﬁne an elemen t φ ( P i α i ⊗ f i ) to b e P i α i ⊗ l ( f i ) ∈ ∇ ( n, ∗ ) ⊗ R on ( σ, l ). Here, l is considered as a map O (Hom F ib (Lo c( X ) ss ) ( ω σ (0) , ω ∗ )) − → R . A map ϕ : A red ( e X ) − → A red ( X ) is deﬁned as follo ws. A red ( e X ) has an G red -action induced fr om the action on e X . The corresp ond ing comod uled map in duce an isomorpism ρ : A red ( e X ) ∼ = ( O ( G red ) ⊗ A red ( e X )) G red . Via this isomorphism, an elemen t of A red ( e X ) are presented as P j v j ⊗ a j , v j ∈ O ( G red ), a j ∈ A red ( e X ). W e deﬁne an elemen t ϕ ( P j v j ⊗ a j ) to b e P j a j ( σ , l ) ⊗ R [( l ◦ ι ⊗ id) ◦ µ ∗ ( v j )] lo cally on σ ( l ∈ F ib (Loc( X ) ss )( ω σ (0) , ω ∗ ) is arb itrarily ﬁxed. ) Here, µ ∗ : O ( G red ) ∼ = O (Hom ( ω ∗ , ω ∗ )) − → O (Hom( ω ∗ , ω σ (0) )) ⊗ R O (Hom( ω σ (0) , ω ∗ )) is th e co comp osition and ι : O (Hom ( ω ∗ , ω σ (0) )) − → O (Hom( ω σ (0) , ω ∗ )) is the coinv erse map. φ and ϕ are morphisms of G red -equiv ariant dga and in verse to eac h ot her. Thus, A red ( e X ) ∼ = A red ( X ). Consider the case that G is reductiv e and X is a ﬁbran t mo d el of K ( G, 1) in s Pr ( k ) lo c ∗ . Clearly F is con tractible so A red ( X ) ≃ ( G, k ). and L T PL ( K ) ∼ = T ss ( G red , k ) ( ∼ = Rep( G red )). Then, π pr e n ( R h T PL ( K ) i )( R ) ∼ = [ h R ∧ S n , R h T PL ( K ) i ] s Pr ( k ) ob j ∗ ∼ = [ T ss ( G, k ) , T PL ( h R ∧ S n )] dg Cat cl ∗ Here, h R ∧ S n ∈ s Pr ( k ) ∗ b e an ob ject deﬁned b y th e follo wing push out diagram h R / / id × v   ∗   h R × S n / / h R ∧ S n , where v is the base p oin t of S n . If n ≥ 2, th is is clearly ∗ . If n = 1, Th e last term is bijectiv e to the set Hom Cat cl /R − Proj ( T ss ( G, k ) c , Rep R ( Z )) / ∼ Here Rep R ( Z ) is the closed k -category deﬁned as follo ws. 1. An ob ject of Rep R ( Z ) is a p air ( V , r ) of V ∈ V ect and R -linear representa tion r : Z → GL R ( V ⊗ k R ) of Z , and 2. a morph ism f : ( V 0 , r 0 ) → ( V 1 , r 1 ) is a m orp hism f : V 0 → V 1 of V ect su c h that f ⊗ k R is compatible with the actions. The au gmentati on ω Rep R ( Z ) : Rep R ( Z ) → R − Proj is given by the forgetful fu n ctor. ∼ is an equ iv alence relation su c h that F 1 ∼ F 2 if and only if there is a natural isomorphism i : F 1 ⇒ F 2 suc h th at i pr eserv es tensors and ( ω Rep R ( Z ) ) ∗ i is the identit y . This set is b ij ective to Aut ⊗ ( ω T ss ( G,k ) c : T ss ( G, k ) c → R − Proj ) in 40 the wa y that an action of 1 ∈ Z corresp on d s a natur al automorphism. Thus we ha v e π pr e 1 ( R h T PL ( K ) i ) ∼ = Aut ⊗ ( ω ) ∼ = G and obtain (1) (see Ap p endix A.2). By (1), Thm.3.3.4 and Lem.3.3.5, there is a commutativ e diagram in d g Cat cl ∗ : T ss ( G, k ) / /   T ss ( G, A red X ) / /   T ss ( e, A red X )   T PL ( K ) ss / / T PL ( X ) ss / / T PL ( QF ) ss suc h that the left and mid dle v ertical arrows are quasi-equiv alences. By the ab o v e assertion, the righ t v ertical arro w is also a quasi-equiv alence so the b ottom h orizon tal sequence is a homotopy coﬁb er sequence and this imply the claim of (2). Pr o of of Thm.5.2.1 . Prop.5.2.4 (5), Lem.5.2.5 and the long exact sequence of homotop y sheav es sho w the unit u X : X − → R h L T PL ( X ) i is a lo cal equiv alence for X ∈ SHT ∗ Consider the comm u tativ e diagram in Ho (s Pr ( k ) ob j ∗ ): R h L T PL ( X ) i i   u / / R h L T PL ( R h L T PL ( X ) i ) i R h L T PL ( i ) i   R loc R h L T PL ( X ) i u ′ / / R h L T PL ( R loc R h L T PL ( X ) i ) i , where u = u R h L T PL ( X ) i , u ′ = u R loc R h L T PL ( X ) i , and i is a ﬁ brant replacemen t in s Pr ( k ) lo c ∗ . By Prop.5.2.4,(3), R h L T PL ( i ) i ◦ u is an isomorphism so u ′ ◦ i is and we ma y regard R h L T PL ( X ) i as a retract of R loc R h L T PL ( X ) i in Ho (s Pr ( k ) ob j ∗ ). By a c haracterization of lo cal ob jects, this imply R h L T PL ( X ) i is also a lo cal ob ject. As a lo cal equiv alence b et w een lo cal ob jects is an ob ject wise equ iv alence, we see u X is an isomorp hism. Thus L T PL : Ho ( SHT ∗ ) − → Ho ( T an ) is fully faithful. By Lem.3.3.5 and 5.2.5, w e s ee a morph ism f : T → T ′ ∈ T an is a qu asi-equiv alence if and only if R h f i : R h T ′ i → R h T i is a we ak equiv alence so L T PL is essential ly surjectiv e. The follo wing is a corollary of Thm.5.2.1. Corollary 5.2.6 (cf. Rem.4.43 of [27]) . L et X ∈ SHT ∗ b e a schematic homoto py typ e and M b e a minimal mo del of A red ( X ) . L et V i denote the i -th inde c omp osable mo dule of M . F or i ≥ 2 , ther e exists an isomorphism of aﬃne gr oup sc hemes : π i ( X ) ∼ = ( V i ) ∨ . F or i = 1 , ther e is an isomorphism of aﬃne schemes (without gr oup structur e) R u ( π 1 ( X )) ∼ = ( V 1 ) ∨ Her e, we r e gar d ( V i ) ∨ as an aﬃne gr oup scheme whose c o or dinate ring is the p olynomial ring on V i . Pr o of. Let i ≥ 2. Th en, b y T hm.5.2.1 π pr e i ( X )( R ) ∼ = π pr e i ( R h L T PL ( X ) i )( R ) ∼ = [ h R ∧ S i , R h L T PL ( X ) i ] s Pr ( k ) ob j ∗ ∼ = [ L T PL ( X ) , L T PL ( h R ∧ S i )] dg Cat cl ∗ , see the pro of of Lem.5.2.5 f or the notation. If we deﬁne A R i ∈ dg Al g /k b y ( A R i ) 0 = k , ( A R i ) i = R, and ( A R i ) j = 0 for j 6 = 0 , i , w e easily see T ( e, A R i ) ≃ L T PL ( h R ∧ S i ) so b y Lem.3.3.7, π pr e i ( X )( R ) ∼ = [ M , A R i ] dg Alg /k ∼ = Hom k − Mo d ( V i , R ) and claim for i ≥ 2 follo ws. The case of i = 1 follo ws from a s im ilar argumen t using the homotop y ﬁb er F of the map X → K ( π 1 ( X ) red , 1). Note that π 1 ( F ) ∼ = R u ( π 1 ( X )) and A red ( F ) ≃ A red ( X ) (see th e pro of of Lem.5.2.5). 41 W e obtain the un p oin ted v ersion of Thm.5.2.1 Corollary 5.2.7. (1) The derive d adjunction ( L T PL , R h−i ) : Ho (s Pr ( k ) ob j ) − → Ho (dg Cat cl ) op induc es an e quivalenc e b etwe en sub c ate gories: L T PL : Ho ( SHT ) ∼ / / Ho ( T an ) op : R h−i . o o (2) Any c onne cte d simplicial set K ∈ s Set adm its the schematization K → ( K ⊗ k ) sc h in the u np ointe d c ate gory. It is r e alize d as the unit of the adjunction K − → R h T PL ( K ) i ∈ Ho (s Pr ( k ) ob j ) . Pr o of. This is clear f rom Thm.5.2.1. A App endix A.1 V arian t s of simplicial de Rham t heorem In this subsection, we state some v arian ts of de Rham theorem for simplicial sets, whic h is used in s ections 4 and 5. Th eir pro ofs are muc h similar to the pro of in [6, section 2, section 3], so w e only indicate h o w to mo dify . A.1.1 Twisted de Rham theorem Let K b e a s im p licial set and L b e a lo cal system on K . W e denote b y C spl ( K ; L ) the n ormalized co chain complex of K with L coeﬃcien ts. An element of the degree q -part C q spl ( K ; L ) is a f unction u which assigns eac h q -simplex σ ∈ K q an elemen t u ( σ ) ∈ L ( σ ) su c h that f or an y degenerate simplex σ , u ( σ ) = 0. W e ha ve the S tok es map as usual: ρ K, L : C PL ( K ; L ) − → C spl ( K ; L ) ∈ C ≥ 0 ( k ) . The twisted de Rham theorem is the follo wing Prop osition A.1.1. F or a simplicial set K and a lo c al system L on K , ρ K, L is a qu asi-isomorphisms . Pr o of. W e hav e t w o functors: C PL ( − ; L ) , C spl ( − ; L ) : s Set /K − → C ≥ 0 ( k ) deﬁned b y ( φ : L → K ) 7− → C PL ( L ; φ ∗ L ) , C spl ( L ; φ ∗ L ) . resp ectiv ely . W e tak e { ∆ n → X | n ≥ 0 } as mo dels instead of { ∆ n } and apply the argum ent in [6, section 2, section 3]. It is clear C PL ( − ; L ) and C spl ( − ; L ) is corepresentable and acyclic on mo dels (in a suitably mo diﬁed sense). A.1.2 Twisted de Rham theorem for simplicial presheav es W e use notations deﬁned in subsection 5.1. Let X ∈ s Pr ( k ) b e a s implicial presheaf and L b e a lo ca l system on X . W e deﬁne a non-negativ ely graded co c hain complex C spl ( X ; L ) as follo ws. An elemen t of C q spl ( X ; L ) is a collectio n { u R } R ∈ k − Alg suc h that u R ∈ C q spl ( X ( R ); L R ) and f or f : R → R ′ ∈ k − A lg an d σ ∈ X ( R ), f ∗ ( u R ( σ )) = u R ′ ( X f ( σ )). the diﬀerentia l is deﬁned in the comp onent -wise mann er. W e ha ve a Stok es map ρ X, L : C PL ( X ; L ) − → C spl ( X ; L ) ∈ C ≥ 0 ( k ) . Let I = { h R × ∂ ∆ n → h R × ∆ n | R ∈ k − Alg , n ≥ 0 } , where h R denotes the Y oneda em b eddin g of S pecR . 42 Prop osition A.1.2. F or an I -c el l obje ct X ∈ s Pr ( k ) and a lo c al system L on X , the Stokes map ρ X, L is a quasi-isomorphism. Pr o of. The pro of is similar to that of Pr op.A.1.1. In this case, we tak e { h R × ∆ n → X } as mo dels. Note that for a lo cal system L ′ on h R × ∆ n , C spl ( h R × ∆ n , L ′ ) ∼ = C spl (∆ n ; L R | id R × ∆ n ) so C spl ( − ; L ) is acyclic on mo dels. A.1.3 de Rham theorem for cubical sets Let Set  b e the category of cubical sets. An ob ject of Set  consists of a collect ion of sets { K n } n ≥ 0 , face maps ∂ ǫ i : K n → K n − 1 (0 ≤ i ≤ n , ǫ = 0 , 1), and d egeneracy maps s i : K n → K n +1 (0 ≤ i ≤ n ) w hic h satisfy the stand ard cubical identiti es. W e denote by  n ∈ Set  the standard n -cub e. W e regard S et  as a mo del catego ry w ith th e mo d el structure giv en in [20, 21], where trivial ﬁbrations are precisely those whic h ha ve right lifting prop ert y with r esp ect to the maps ∂  n →  n ∈ Set  ( n ≥ 0, of course, ∂  n is the b ound ary of  n ). W e den ote by  ( n, ∗ ) the dg-algebra of k -p olynomial f orms on  n . T his is th e commutativ e graded algebra o ve r k freely generated by t 1 , . . . , t n and dt 1 , . . . , dt n with deg t i = 0, d eg dt i = 1. (It is isomorphic to ∇ ( n, ∗ ).) F or eac h q ≥ 0, W e regard  ( ∗ , q ) as a cubical set (or cubical ab elian group ). F ace maps and d egeneracy m aps are deﬁn ed by the p ullbac k of corr esp onding maps b etw een the stand ard cub es. F or example, ∂ ǫ i :  ( n, q ) →  ( n − 1 , q ) is deﬁned as follo ws. ∂ ǫ i ( t j ) =    t j j < i ǫ j = i t j − 1 j > i W e need the follo wing. Lemma A.1.3. F or e ach q ≥ 0 , the u nique morphism  ( ∗ , q ) → ∗ ∈ Set  is a trivial ﬁbr ation. Pr o of. W e imitate the p ro of of ﬁ brancy of sim p licial ab eli an group s (see for example, [15, Lem.3.4]). W e shall sho w the morph ism of the claim has r igh t lifting prop ert y with resp ect to the m aps ∂  n →  n ( n ≥ 0). Supp ose 2 n elements x ǫ i ∈  ( n − 1 , q ), i = 1 , . . . , n , ǫ = 0 , 1 such that ∂ ǫ 1 i x ǫ 2 j = ∂ j − 1 x ǫ 1 i for i < j , are giv en. (This is equiv alen t to giving a map ∂  n →  ( ∗ , q ) ∈ Set  .) W e use indu ction. Let 1 < l ≤ n . Supp ose we ha ve y ∈  ( n, q ) s u c h th at for l ≤ ∀ i ≤ n , ∀ ǫ , ∂ ǫ i y = x ǫ i . (F or l = n , put y = (1 − t n ) s n ( x 0 n ) + t n s n ( x 1 n ).) Consider an elemen t z ǫ := x ǫ l − 1 − ∂ ǫ l − 1 y . Cubical iden tities imply ∀ i ≥ l − 1, ∀ ǫ ′ , ∂ ǫ ′ i z ǫ = 0. Set y ′ := (1 − t l − 1 ) s l − 1 z 0 + t l − 1 s l − 1 z 1 + y . Agai n cubical iden tities imply ∂ ǫ i y ′ = x ǫ i for l − 1 ≤ ∀ i ≤ n , ∀ ǫ . Th us w e can constru ct a lifting indu ctiv ely . Let K ∈ S et  . T he n otion of a lo cal s ystem on K is deﬁned similarly to the case of simplicial sets. F or a lo cal system L on K , we deﬁne the de Rham complex C PL ( K ; L ) ∈ C ≥ 0 ( k ) using  ( ∗ , ∗ ). Let C  ( K ; L ) b e the normalized co chain complex of K with L -co eﬃcient s. Explicitly , an el emen t of C q  ( K ; L ) is a function u whic h assigns eac h q -c ub e σ ∈ K q an elemen t u ( σ ) ∈ L ( σ ) suc h that u ( σ ) = 0 for a d egenerate cub e σ . The diﬀerential d : C q  ( K ; L ) → C q +1  ( K ; L ) is as usu al, give n b y du = q X i =1 ( − 1) i [( ∂ 0 i ) ∗ u − ( ∂ 1 i ) ∗ u ] . Here, ( ∂ ǫ i ) ∗ u = L ( ∂ ǫ i ) − 1 ◦ u ◦ ∂ ǫ i . W e ha ve a S tok es map ρ K, L : C PL ( K ; L ) − → C  ( K ; L ) ∈ C ≥ 0 ( k ) . 43 Prop osition A.1.4. F or a cubic al set K and a lo c al system L on K , ρ K, L is a qu asi-isomorphism. Pr o of. The pro o f is similar to that of Prop.A.1.1. W e use Lem.A.1.3 to prov e C PL ( − ; L ) is corepresentable. W e can deﬁne an adjoint pair T PL : Set  / / (dg Ca t cl ) op : h−i . o o W e can see this is a Quillen pair, using Lem.A.1.3 and Prop.A.1.4. This fact is implicitly used in th e p ro of of Lem.4.1.5 . A.2 T annakian theory In this su bsection, we su mmarize T annakian theory of [11]. It s tates a du ality b etw een aﬃne group schemes and certain closed k -ca tegories. Th e catego ry corresp onding to a group scheme is the category of ﬁ nite dimensional rep resen tations of it. W e giv e a charact erization of s u c h categorie s and describ e h o w the group sc heme is reco v ered. Let V ect ′ b e the category of all ﬁnite d imensional k -v ector s paces and linear maps. Deﬁnition A.2.1. A close d k - c ate gory T ∈ Cat cl (se e Def.2.1.1) is said to b e a neutr al T annakian c ate g ory if it satisﬁes the fol low ing c onditions. 1. T is an Ab elian c ate gory. 2. Hom T ( 1 , 1 ) ∼ = k . 3. Ther e exists a morphism ω : T → V ect ′ ∈ Cat cl which is faithful and exact. In the ab ov e d eﬁnition, w e use V ect ′ instead of V ect in order to mak e neutral T ann akian cat egories stable under equiv alences of Cat cl . (Recall that V ect is a small full su b category of V ect ′ suc h that the natural inclusion V ect → V ect ′ is an equ iv alence, see the paragraph ab o v e Def.2.1.3 .) Example A.2.2. L et Γ (r esp. G ) b e a discr ete g r oup (r esp. an aﬃne gr oup scheme over k ). The close d k -c ate gory of ﬁnite dimensional k -r epr esentations of Γ (r esp. G ) Rep(Γ) (r e sp. Rep( G ) ) is a neutr al T annakian c ate gory. Let k − Al g d en ote the category of commutat iv e and un ital k -algebras and Grp denote the category of groups. Example A.2.3. L et G : k − Alg → Grp b e a functor. A ﬁnite dimensional r epr esentation of G is a p air of a ve ctor sp ac e V ∈ V ect and a natur al tr ansformation r − : G = ⇒ GL( V ) : k − Alg → Grp , wher e GL( V ) is the func tor given by k − Al g ∋ R 7→ GL R ( V ⊗ k R ) ∈ G rp . Ther e is an obvious notion of morphisms b etwe en ﬁnite dimensional r epr esentations of G . We deno te the c ate gory of ﬁnite dimensiona l r epr esentations of G by Rep( G ) and the for getful functor Rep( G ) → V ect by ω G . Rep ( G ) has an close d tensor structur e such that ω G is a morphism of close d k -c ate gories. Then R ep ( G ) is a neutr al T annakian c ate gory. If G is r epr esente d by an aﬃne gr oup scheme G , Rep( G ) is e quivalent to the c ate gory Rep( G ) . F or a neutral T ann akian categ ory T and a morphism ω : T − → V ect ∈ Cat cl whic h is exact and faithful, w e deﬁ n e a functor Aut ⊗ ( ω ) : k − A lg → G rp b y k − Alg ∋ R 7− → Au t ⊗ ( ω ⊗ k R ) ∈ G rp . 44 Here Aut ⊗ ( ω ⊗ k R ) is th e group of tensor pr eserving natural isomorphisms α : ω ⊗ k R ⇒ ω ⊗ k R : T − → R − Mo d ( ω ⊗ k R : T → R − Mo d is the morphism from T to the closed k -category of R -mo dules deﬁned b y T ∋ t 7→ ω ( t ) ⊗ k R ∈ R − Mod). W e deﬁn e a morph ism of closed k -categories e ω : T → Rep (Aut ⊗ ( ω )) by T ∋ t 7− → ( ω ( t ) , ev ( t ) : Au t ⊗ ( ω ) = ⇒ GL ( ω ( t ))) ∈ Rep(Aut ⊗ ( ω )) , where ev ( t ) is th e ev aluation of elemen ts of Aut ⊗ ( ω ) at t . Theorem A.2.4 (Thm.2.11 of [11]) . We use the ab ove notations. (1) The functor Aut ⊗ ( ω ) is r epr esente d by an aﬃne gr oup scheme, which i s c al le d the T annakian dual of ( T , ω ) . (2) The ab ove morphism e ω : T − → Rep(Aut ⊗ ( ω )) is an e quiv alenc e of c ate gories. (3) Supp ose ( T , ω ) = (Rep( G ) , ω G ) for some aﬃne gr oup scheme G . L et G b e the functor c orr esp onding to G . Ther e is a natur al isomorphism G ∼ = Aut ⊗ ( ω ) deﬁne d by G ( R ) ∋ g 7→ g · ∈ Au t ⊗ ( ω G ⊗ k R ) . Corollary A.2.5. The pr o-algebr aic c ompletion Γ alg of a discr ete gr oup Γ is isomorphic to the T annakian dual of Rep(Γ) which i s e quipp e d with the natur al ”evaluation” Γ → Aut ⊗ ( ω ) = Aut ⊗ ( ω )( k ) given by Γ ∋ γ 7− → [Rep(Γ) ∋ ( V , r ) 7→ ( r ( γ ) : V → V )] . The maximal r e ductive quotient G red of an aﬃne gr oup scheme G is isomorphic to the T annakian dual of the ful l sub c ate gory of Rep( G ) c onsisting of semi-simple r epr e se ntations. (This is close d under tensors and internal homs.) Ac kno wledgemen ts The author is grateful to Masana Harada f or many v aluable discussions and commen ts to improv e p re- sen tations of the p ap er. He also th ank Daisuke Kishimoto f or letting him kn o w the b o ok [10]. He thank Akira Kono f or constan t encouragement . He thank T akuro Mo chizuki and th e members of Homotopical Algebraic Geometry seminar, esp ecially , Isam u Iw anari and Hiro yuki Minamoto, for atten tion to this wo rk. References [1] D. 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