Locally constant functors

LOCALL Y CONS T ANT FUNCTORS DENIS-CHARLES CISINSKI T o Mi chael Batanin, for his nic e questions Abstract. W e study lo cally constan t co efficien ts. W e first study the the- ory of homotop y Kan ext ensions with lo cally consta nt coefficient s in model categories, and explain how it characterizes the homotopy theory of small cat- egories. W e explain ho w to interpret this in terms of left Bousfield localization of categories of diagrams with v alues in a combinato rial mo del category . At last, w e explain ho w the theory of homot op y Kan extensions in deriv ators ca n be used to understand l ocall y constan t functo rs. Contents 1. Homology with lo cally consta nt coefficients 1 2. Mo del structures for lo cally cons tant functors 6 3. Lo cally constant co efficients in Gr othendieck deriv ators 8 4. Galois corr esp ondence and homotopy distributors 12 References 16 1. Homo l ogy with locall y const ant coefficients 1.1. Given a mo del categor y 1 V with sma ll colimits, and a small categor y A , w e will write [ A, V ] for the category of functors from A to V . W eak equiv alences in [ A, V ] are the termwise w ea k e quiv alences. W e denote by Ho ([ A, V ]) the lo caliza tion of [ A, V ] by the c la ss of weak equiv alences. 1.2. W e denote by LC ( A, V ) the full sub ca teg ory o f the categ ory Ho ([ A, V ]) whos e ob jects are the lo cally constant functors, i.e. the functors F : A / / V such that fo r a ny ma p a / / a ′ in A , the map F a / / F a ′ is a weak equiv a lence in V , or equiv a lently , an iso morphism in Ho ( V ) (wher e F a is the ev aluation of F at the ob ject a ). Note that for any functor u : A / / B , the in verse image functor (1.2.1) u ∗ : [ B , V ] / / [ A, V ] Date : March 2008. 1 W e mean a Quillen mo del category . Ho w ev er, we could tak e any kind of mo del category giving rise to a go o d theo ry of homotop y colimits (i.e. to a Grothend iec k deriv ator); see the w ork of Andrei R˘ adulescu-Ban u [RB06] for more general examples. 1 2 DENIS-CHARLES CISINSKI preserves weak equiv alences, so that it induces a functor (1.2.2) u ∗ : Ho ([ B , V ]) / / Ho ([ A, V ]) The functor u ∗ obviously pr eserves locally consta nt functors, so that it induces a functor (1.2.3) u ∗ : LC ( B , V ) / / LC ( A, V ) . Theorem 1 .3. L et u : A / / B b e a functor who se nerve is a simplicial we ak e quivalenc e. The n the functor (1.2.3) is an e quivalenc e of c ate gories. 1.4. The pro of of Theorem 1.3 will need a little pre pa ration. Define a functor betw een small categor ies u : A / / B to b e a we ak e quivalenc e if the functor (1.2.3) is an equiv a lence of categor ies. A s mall category A is aspheric al if the map A / / e is a weak equiv alenc e (where e denotes the terminal categ ory). A functor u : A / / B will b e said to b e aspheric al if for any ob ject b o f B , the functor A/b / / B / b is a weak equiv alence (where A/b = ( u ↓ b ) is the comma ca tegory of ob jects of A ov er b , and B /b = (1 B ↓ b )). Theorem 1.3 asser ts that any functor w ho se ner ve is a simplicial weak equiv a lence is a weak equiv alence in the sense define a bove. This will fo llow fro m the following result. Theorem 1 .5. L et W b e a class of functors b etwe en smal l c ate gories. We assume that W is a we ak b asic lo c alizer in the sense of Gr othendie ck [Mal05a] , which me ans that the fol lowing pr op erties ar e satifie d. La Any identity is in W . The class W satisfies the t wo out of thr e e pr op erty. If a map i : A / / B has a r etr action r : B / / A such that ir is in W , then i is in W . Lb If a smal l c ate gory A has a terminal obje ct, then the m ap fr om A to the terminal c ate gory is in W . Lc Given a functor u : A / / B , if for any obje ct b of B , the induc e d functor A/b / / B / b is in W , then u is in W . Then any functor b etwe en sm al l c ate gories whose nerve is a simplicia l we ak e quiv- alenc e is in W . Pr o of. See [Cis06, Theorem 6.1.18 ].  1.6. T o pr ov e Theor em 1.3, we will pro ve that the class of w eak equiv alences sa tis- fies the prop er ties listed in the previous theorem. Pro p erty La is easy to chec k. It th us re ma ins to prov e properties Lb and Lc. In other w or ds, we hav e to prove that any category with a terminal ob ject is aspherical, a nd that a ny aspherical functor is a w eak equiv alence. W e will use the theory of homoto py Kan extensions in V . Given a functor u : A / / B , the functor (1.2.1) has a left adjoint (1.6.1) u ! : [ A, V ] / / [ B , V ] which admits a to tal left derived functor (1.6.2) L u ! : Ho ([ A, V ]) / / Ho ([ B , V ]) The functor L u ! is a lso a left adjoin t o f the functor (1.2.2); see [Cis03, RB06]. When B is the termina l ca teg ory , w e will write L lim − → A = L u ! for the corresp onding homotopy co limit functor . LOCALL Y CONST ANT FUNCTORS 3 1.7. F o r each ob ject b o f B , we have the following pullback sq ua re of categor ie s A/b u/b   v / / A u   B / b w / / B (1.7.1) (where w is the o bvious for getful functor). Given a functor F from A (resp. B ) to V , we will write (1.7.2) F /b = v ∗ ( F ) (resp. F /b = w ∗ ( F ) ) . This gives the follo wing form ula for a functor F : B / / V (1.7.3) u ∗ ( F ) /b = ( u /b ) ∗ ( F / b ) . It is a fact that left homotop y Kan extensions can be computed p o int wise (like in ordinary category theory), which can be for mu lated like this: Prop ositi on 1.8 . F or any functor F : A / / V , and any obje ct b of B , t he b ase change map L lim − → A/b F /b / / L u ! ( F ) b is an isomorphism in Ho ( V ) . Pr o of. See [Cis03] or, in a more gener al con text, [RB06, Theorem 9.6.5].  Prop ositi on 1 .9. L et I b e a smal l c ate gory. A morphism F / / G in Ho ([ I , V ]) is an isomorph ism if and only for any obje ct i of I , the map F i / / G i is an isomorphi sm in Ho ( V ) . Pr o of. See [Cis03] or [RB06, Theorem 9.7.1].  Prop ositi on 1.10 . L et u : A / / B b e a we ak e quivalenc e of smal l c ate gories. Then, for any lo c al ly c onstant functor F : B / / V , the map L lim − → A u ∗ ( F ) / / L lim − → B F is an isomorphism in Ho ( V ) . Pr o of. Giv en a small catego ry I and an ob ject X , deno te by X I the constant functor from I to V with v alue X . Let F : B / / V b e a lo cally constant fu nctor. Using the fact that (1.2.3) is fully faithful, we see that Hom Ho ([ B , V ]) ( F, X B ) / / Hom Ho ([ A, V ]) ( u ∗ ( F ) , u ∗ ( X B )) is bijectiv e. As u ∗ ( X B ) = X A , the iden tifications Hom Ho ([ B , V ]) ( F, X B ) ≃ Hom Ho ( V ) ( L lim − → B F, X ) Hom Ho ([ A, V ]) ( u ∗ ( F ) , X A ) ≃ Hom Ho ( V ) ( L lim − → A u ∗ ( F ) , X ) and the Y one da Lemma applied to Ho ( V ) ends the pro of.  4 DENIS-CHARLES CISINSKI Corollary 1.11 . Le t I b e an asph eric al c ate gory, and F : I / / V b e a lo c al ly c onstant functor. Then for any obje ct i of I , the map F i / / L lim − → I F is an isomorphism in Ho ( V ) . Pr o of. Apply Prop osition 1.10 to the functor from the terminal category to I de- fined by i .  Prop ositi on 1.12. A ny smal l c ate gory which has a terminal obje ct is aspheric al. Pr o of. Let I be a s ma ll categ o ry with a terminal ob ject ω . This means that the functor p from I to the terminal c a tegory has a r ight a djoint (which is has to b e fully faithful). But A  / / Ho ([ A, V ]) obviously defines a 2-functor, whic h implies that the functor p ∗ : Ho ( V ) / / Ho ([ I , V ]) is fully faithful and that the ev aluation functor at ω is a left adjoint to p ∗ . In particular, for any funct or F from I to V , w e have L lim − → I F = F ω . If mor eov e r F is lo cally cons tant, then it follows from Pr op osition 1.9 that the co-unit map F / / p ∗ F ω = p ∗ L lim − → I F is an isomorphism. This shows that p ∗ is also essentially surjective and ends the pro of.  Corollary 1.13. A functor u : A / / B is asp heric al if and only if for any obje ct b of B , the c ate gory A/b is aspheri c al. Pr o of. As t he class of weak eq uiv alences satifies the tw o out o f three pro p erty , this follows from the fact that the catego ry B /b has a terminal ob ject (namely ( b, 1 b )).  1.14. A functor u : A / / B is lo c al ly c onstant if for any map b / / b ′ in B , the functor A/b / / A/b ′ is a weak equiv alence. F or exa mple, a ny aspheric al functor is lo cally constant. Prop ositi on 1.15 (F ormal Serre sp ectral sequence) . If u : A / / B is lo c al ly c onstant, then the functor (1.6.2) pr eserves lo c al ly c onstant functors. In p articular, it induc es a functor L u ! : LC ( A, V ) / / LC ( B , V ) which is a left adjoint to t he fun ctor (1.2.3) . Pr o of. Let F b e a lo cally constant functor, and β : b / / b ′ be a map in B , W e hav e to show tha t the induced map L u ! ( F ) b / / L u ! ( F ) b ′ LOCALL Y CONST ANT FUNCTORS 5 is an iso morphism in Ho ( V ). Denote b y j β : A/b / / A/b ′ the functor induced b y β (which is a w eak equiv alence b y assumption on u ). W ith the notations (1.7.2), we have j ∗ β ( F / b ′ ) = F /b . This coro llary thus follows immediately from P rop osition 1.8 and from Propo sition 1.10.  Prop ositi on 1.16 (F o rmal Quille n Theorem A) . A ny aspheric al functor is a we ak e quivalenc e. Pr o of. Let u : A / / B be an aspher ical functor. Let F : B / / V be a lo ca lly constant functor. W e will prov e that the co-unit map L u ! u ∗ ( F ) / / F is an isomorphism. Accor ding to Prop osition 1.9, it is sufficient to prov e that for any ob ject b o f B , the map ( L u ! u ∗ ( F )) b / / F b is an is omorphism in Ho ( V ). This follo ws immediately from the computations below. ( L u ! u ∗ ( F )) b ≃ L lim − → A/b u ∗ ( F ) /b (Prop osition 1.8) = L lim − → A/b ( u/b ) ∗ ( F / b ) (F or mula (1 .7 .3)) ≃ L lim − → B /b F /b (Prop os ition 1.10 applied to u/b ) ≃ F b (beca use ( b, 1 b ) is a terminal ob ject of B /b ). Consider now a lo cally constan t functor F : A / / V . W e will show that the unit map F / / u ∗ L u ! ( F ) is an isomor phism. B y virtue of Pr op osition 1.9, we are reduced to prov e that, for any ob ject a of A , the map F a / / ( u ∗ L u ! ( F )) a is an isomorphism. W e compute again F a ≃ L lim − → A/u ( a ) F /u ( a ) (Corollar y 1 .11 for I = A/ u ( a ) and i = ( a, 1 u ( a ) )) ≃ L u ! ( F ) u ( a ) (Prop ositio n 1.8) = ( u ∗ L u ! ( F )) a and this ends the pro of.  Pr o of of The or em 1.3. It is sufficient to c hec k that the class of weak eq uiv alences satisfy the prop e r ties listed in Theo rem 1.5. T he cla ss of w ea k equiv alences obvi- ously satifies pro pe rty La . Prop er ty Lb follows from Prop osition 1.12, and prop er ty Lc from Prop osition 1.16.  6 DENIS-CHARLES CISINSKI Corollary 1.17 . L et u : A / / B a functor b etwe en smal l c ate gories. Th e nerve of u is a simplicial we ak e quivalenc e if and only if for any mo del c ate gory V , the functor u ∗ : LC ( B , V ) / / LC ( A, V ) is an e quivalenc e of c ate gories. Pr o of. Theorem 1.3 asserts this is a necessar y condition. It is very eas y to chec k that this is also sufficient : we can either use P rop ositio n 1.1 0 and [Cis06, 6.5.11], or we can use the fact that, for a given s ma ll categor y A , the homo topy colimit of the constan t functor indexed b y A whose v alue is the terminal simplicial set is precisely the nerve of A (which is completely o bvious if we co nsider for example the Bousfield-Ka n construction of homotop y colimits).  2. Model structures f or local l y const ant functors 2.1. W e consider now a left prop er combinatorial model category V a nd a small category A (see [Bek00] for the definition of a com binatorial mo del categ o ry). The category of functors [ A, V ] has t wo ca no nical model structures. The pr oje ctive mo del structur e o n [ A, V ] is defined as follows: the weak equiv a lences ar e the termwise weak equiv alences , a nd the fibrations are the termwise fibrations. The inje ctive mo del st ructur e on [ A, V ] is defined dually: the w eak equiv ale nces are the termwise w eak equiv alence s , and the cofibrations are the termwise cofibrations . One can chec k that the identit y functor is a left Quillen eq uiv alence fro m the pro- jective mo del str ucture to the injective mo del structure (this is just a n abstract wa y to say that all the cofibra tions of the pro jective model s tr ucture are termwise cofibrations, whic h is easy to chec k; see for example [Cis0 6, Lemma 3.1.12]). Thes e t wo mo del str uctures are left prop er . 2.2. W e fix a (r egular) cardina l α with the following pr op erties (see [Dug01]). (a) Any ob ject of V is a α -filtered colimit of α -small ob jects. (b) The cla ss of w eak equiv alences of V is s table b y α -filtered colimits. (c) There exists a cofibrant resolution functor Q which preserves α -filtered colimits. Given an ob ject a of A , we deno te by a ! : V / / [ A, V ] the le ft adjoint to the ev alua tio n funct or at a . W e define S as the (essentially small) set of maps of shap e (2.2.1) a ! ( QX ) / / a ′ ! ( QX ) asso cia ted to each map a / / a ′ in A and ea ch α - accessible o b ject X (a nd Q is some fixed cofibrant resolution funct or sa tisfying the condition (c) ab ov e). W e define the pr oje ctive lo c al m o del structur e on [ A, V ] as the left Bousfield lo calization o f the pro jective model structure on [ A, V ] b y S . The inje ctive lo c al mo del st ructur e on [ A, V ] is the left Bousfield lo calizatio n of the injectiv e mo del structure on [ A, V ] by S . It is clear that the identit y functor is still a left Quillen equiv alence from the pro jective local mo del str ucture to the injective lo ca l model structure. The w eak equiv a le nces of these t wo mo del structures will b e called the lo c al we ak e quivalenc es . LOCALL Y CONST ANT FUNCTORS 7 Prop ositi on 2 .3. A functor F : A / / V i s fi br ant in t he pr oje ctive (r esp. in- je ctive) lo c al mo del structure if and o nly if it is fi br ant for the p r oje ctive ( re sp. inje ctive) mo del structur e and if it is lo c al ly c onstant. Pr o of. It is sufficient to prov e this for the pro jective local mo del structure. Note first that, thanks to condition (b), for a ny α - filtered category I a nd any functor F from I to [ A, V ], the natural map L lim − → I F / / lim − → I F is an isomo rphism in Ho ([ A, V ]). Hence it remains a n isomorphism in the homotopy category of the pr o jective lo cal mo del s tr ucture. This implies that lo cal weak equiv alences are stable b y α -filtered colimits. Conditions (a) and (b) thus imply that for any ob ject X of V , and an y arrow a / / a ′ in A , the map a ! ( QX ) / / a ′ ! ( QX ) is a loca l weak equiv alence. If a is an ob ject of A , X is an ob ject of V , a nd F a functor from A to V , then Hom Ho ( V ) ( X, F a ) ≃ Hom Ho ([ A, V ]) ( L a ! ( X ) , F ) ≃ Hom Ho ([ A, V ]) ( a ! ( QX ) , F ) . It is now easy to see that, if mo reov er F is fibra nt for the pro jectiv e mo del structure, then it is fibr ant for the pro jective lo cal mo del str ucture if and only if it is locally constant.  Corollary 2.4. The lo c alization of [ A, V ] by the class of lo c al we ak e quivalenc es is LC ( A, V ) . Corollary 2 .5. The i nclusion functor LC ( A, V ) / / Ho ([ A, V ]) has a left ad- joint. Prop ositi on 2.6 . Le t u : A / / B b e a functor b etwe en sm al l c ate gories. Then the functor u ∗ : [ B , V ] / / [ A, V ] is a right Quil len functor for the pr oje ctive lo c al mo del structur es. If mor e over the nerve of u is a simplici al we ak e quivalenc e, then the functor u ∗ is a right Quil len e quivalenc e. Pr o of. The left adjoint u ! of u ∗ preserves cofibratio ns: this is o bviously a left Quillen functor for the pro jective mo del structures. It is thus sufficient to c heck that u ∗ preserves fibrations be tw een fibrant ob jects; see [JT07, Prop os ition 7.1 5]. It follows from P rop ositio n 2.3 that fibrations b etw een fibrant ob jects a re just fibrations of the pro jective mo del structur e betw een fibr a nt ob jects of the pro jective mode l structure which are lo cally constant. It is clear that u ∗ preserves this prop erty . This prov es that u ∗ is a rig ht Quillen functor. The la st asser tion follows fr o m Theorem 1 .3.  R emark 2.7 . According to the preceding prop osition, the functor u ! has a tota l left derived functor L u ! : LC ( A, V ) / / LC ( B , V ) . It also has a total left derived functor L u ! : Ho ([ A, V ]) / / Ho ([ A, V ]) . 8 DENIS-CHARLES CISINSKI but, in genera l, the diagr a m (in whic h i A and i B denote the inclusion functors) LC ( A, V ) L u ! / / i A   LC ( B , V ) i B   Ho ([ A, V ]) L u ! / / Ho ([ B , V ]) do es not (even essen tially) comm ute. Ther e is only a natural map L u ! i A ( F ) / / i B L u ! ( F ) . How ever, Pro p o sition 1.15 asser ts that this natural map is an isomo rphism whenever u is lo c a lly constant. Prop ositi on 2 .8. L et u : A / / B b e a functor b et we en smal l c ate gories. A s sume that the functor u op : A op / / B op is lo c al ly c onstant. Then the fun ctor u ∗ : [ B , V ] / / [ A, V ] is a left Quil len functor for the inje ctive lo c al mo del st ructur es. If mor e over the nerve of u is a simplici al we ak e quivalenc e, then the functor u ∗ is a left Quil len e quivalenc e. Pr o of. W e know that u ∗ is a left Quillen functor for the injective mo del structure. Hence, by virtue of Prop o sition 2 .3, it is sufficient to pr ove that the total right derived functor R u ∗ : Ho ([ A, V ]) / / Ho ([ B , V ]) preserves lo cally constant functor s. But this latter prope rty is just Prop osition 1.15 applied to V op . The last ass ertion follows aga in from Theor em 1.3.  3. Locall y const ant coefficients in Grothendieck deriv a tors 3.1. W e start this section b y fixing some notations. Given a co co mplete mo del category V , we denote b y Ho ( V ) the deriv ato r asso- ciated to V ; see [Cis03, Theorem 6.11]. Let A b e a small categor y . W e will consider the category [ A op , s S et ] of simplicia l presheav es o n A endo wed with the pr o jective mo del structure. Given a s ubc ategory S of A , we denote b y L S [ A op , s S et ] the left Bousfield loca lization of the pro jective mo del str ucture o n [ A op , s S et ] b y S (where S is seen as a set of maps in [ A op , s S et ] via the Y oneda em bedding). The fibrant ob jects of L S [ A op , s S et ] are the simplicial presheav es F on A which are term wise Kan complexes and which sends the maps of S to simplicial homotopy equiv a lences. In particular, in case A = S , L A [ A op , s S et ] is the pro jective lo cal model structure on [ A op , s S et ] studied in the previous section. 3.2. Given tw o prederiv ator s D and D ′ , we deno te b y Hom ( D , D ′ ) the category of mor phisms of deriv a tors; see [Cis03]. If D and D ′ are der iv ators, w e denote by Hom ! ( D , D ′ ) the full sub categor y of Hom ( D , D ′ ) w ho se o b jects are the morphisms of prederiv a to rs which preser ves left homo to py Ka n extensions (whic h are c alled co contin uo us morphisms in [Cis03]). Given a (sma ll) catego ry , w e deno te b y A the prederiv ator whic h asso ciates to each sma ll c a tegory I the catego ry [ I op , A ] of pre s heav e s on I with v alues in A . This defines a 2-functor from the 2-categor y of small ca teg ories to the 2-categor y LOCALL Y CONST ANT FUNCTORS 9 of pr ederiv ato r s. Note that we hav e a Y o ne da Lemma for prederiv a tors: g iven a small categor y A and a prederiv a tor D , the functor (3.2.1) Hom ( A , D ) / / D ( A op ) , F  / / F (1 A ) is an equiv alence of catego ries. Theorem 3.3. F or any derivator D , the c omp osition by the Y one da emb e dding h : A / / Ho ([ A op , s S et ]) induc es an e quivalenc e of c ate gories Hom ! ( Ho ([ A op , s S et ]) , D ) ≃ Hom ( A, D ) . Pr o of. This is a translation of [Cis02, Cor ollary 3.26] using (3.2.1).  3.4. W e denote b y Hom S ( A , D ) the full s ub ca tegory of morphisms A / / D such that the induced functor A / / D ( e ) sends the maps of S to isomorphims (where e denotes the terminal categ ory). A formal consequence of Theor e m 3.3 is: Theorem 3 .5. F or any d erivator D , the c omp osition by the Y one da morphism h : A / / Ho ( L S [ A op , s S et ]) induc es an e quivalenc e of c ate gories Hom ! ( Ho ( L S [ A op , s S et ]) , D ) ≃ Hom S ( A, D ) . Pr o of. This fo llows immediately fro m Theorem 3 .3 and fro m the univ ersal prop erty of left Bousfield lo caliza tion for deriv ators; see [T a b07, Theorem 5.4 ].  3.6. Given a small ca teg ory A and a deriv a to r D , we define (3.6.1) LC ( A, D ) = Hom A ( A, D ) . It is clear that for a mo del catego ry V , we ha v e b y definition (3.6.2) LC ( A, V ) = LC ( A, Ho ( V )) . Corollary 3.7. L et u : A / / B b e a functor b etwe en smal l c ate gories. Then t he nerve of u is a simplicial we ak e quivalenc e if and only if for any derivator D , the functor u ∗ : LC ( B , D ) / / LC ( A, D ) is an e quivalenc e of c ate gories. Pr o of. As a ny mo del category giv es r ise to a deriv ato r, this is certa inly a sufficien t condition, b y vir tue of Corollar y 1.17. It thus remains to prove that this is a necessary condition. The nerv e of the functor u is a simplicial weak equiv alence if and only if the nerve of u op : A op / / B op is so. This r esult is thus a co nsequence of Prop os ition 2.6, of Theor e m 3.5, and of the fac t that a ny Quillen equiv alence induces an equiv alence of deriv a tors.  Lemma 3.8. L et A b e a smal l c ate gory. Th e inclusion morphism i : Ho ( L A [ A op , s S et ]) / / Ho ([ A op , s S et ]) (define d as the right adjoi nt of the lo c alization morphism) pr eserves left homotopy Kan ext ensions. Pr o of. It is sufficient to chec k that it pr eserves homotopy colimits; see [Cis02, Prop os ition 2 .6]. This reduces to chec k that lo cally cons tant functors a re stable by homotopy colimits in the model categor y of simplicial presheaves o n a small category , which is obvious.  10 DENIS-CHARLES CISINSKI Prop ositi on 3.9 . F or any derivator D and any smal l c ate gory A , the inclusion functor LC ( A, D ) / / Hom ( A, D ) has a left ad joint and a right adj oint. Pr o of. W e have a lo caliza tion morphism γ : Ho ([ A op , s S et ]) / / Ho ( L A [ A op , s S et ]) which has a rig ht adjoint in the 2-categor y o f prederiv a tors i : Ho ( L A [ A op , s S et ]) / / Ho ([ A op , s S et ]) . W e know that γ is co contin uous (as it c omes from a left Quillen functor; see [Cis03, Prop os ition 6.2]). The pre vious lemma asserts that i is co contin uous as well. It th us follows from the fact Hom ! ( − , D ) is 2-functor and from Theorem 3.5 that the inclusion functor LC ( A, D ) / / Hom ( A, D ) (whic h is induced by γ ) has a left adjoint (which is induced b y i ). Applying this to the opp os ite deriv ator D op (and repla cing A b y A op ) also giv es a righ t adjoint.  Corollary 3. 10. L et u : A / / B b e a functor b et we en smal l c ate gories. F or any derivator D , t he inverse image functor u ∗ : LC ( B , D ) / / LC ( A, D ) has a left ad joint and a right adj oint. 3.11. It is p o ssible to constr uct a prederiv ator LC ( A, D ) such that (3.11.1) LC ( A, D )( e ) = LC ( A, D ) (where e still denotes the terminal categor y). If D is a deriv ato r, and A is a small category , then we define a deriv a to r D A by the for mula (3.11.2) D A ( I ) = D ( A op × I ) . It is easy to see that D A is again a deriv ator . Mor eov e r, the homotopy colimits in D A can be computed termwise; see [Cis 0 2, Prop osition 2.8]. In the case where D = Ho ( V ) for a model ca tegory V , we get the formula (3.11.3) Ho ( V ) A ( I ) = Ho ([ A × I op , V ]) . The preder iv ator LC ( A, D ) is the full subprederiv ator of D A defined by the for mula (3.11.4) LC ( A, D )( I ) = LC ( A, D I op ) . In other words, LC ( A, D )( I ) is the full s ubc a tegory of D ( A op × I ) whose ob jects are the ob jects F of D ( A op × I ) such that the induced functor dia ( F ) : A / / [ I op , D ( e )] sends any mo rphism of A to isomor phis ms. Theorem 3.12. F or any smal l c ate gory A , and any derivator D , t he pr e derivator LC ( A, D ) is a deriva tor, and the ful l inclusion LC ( A, D ) / / D A has a left ad joint and a right adj oint. LOCALL Y CONST ANT FUNCTORS 11 Pr o of. The proof will follow essentially the same lines a s the pro o f of Prop osition 3.9. Recall that there is an in ternal Hom for prederiv ator s: if D and D ′ are pred- eriv ator s, w e define a pr ederiv ator Hom ( D , D ′ ) by the for mula Hom ( D , D ′ )( I ) = Hom ( D , D ′ I op ) for any small catego r y I ; see [Cis02, Cor ollary 5.3 ]. If mor eov e r D and D ′ are deriv a- tors, we define a pre deriv ator Hom ! ( D , D ′ ) as a full subprederiv a tor of Hom ( D , D ′ ) as follows: for each small category I , we put Hom ! ( D , D ′ )( I ) = Hom ! ( D , D ′ I op ) . Then Hom ! ( D , D ′ ) is aga in a deriv ator; see [Cis02, P rop ositio n 5.8]. The o rem 3.3 gives the following r esult. If A is a small ca tegory , then for any deriv ator D , the Y one da ma p h : A / / Ho ([ A op , s S et ]) induces an equiv alence of deriv ators Hom ! ( Ho ([ A op , s S et ]) , D ) ≃ Hom ( A, D ) = D A . Similarly , Theorem 3.5 implies that the Y oneda map h : A / / Ho ( L A [ A op , s S et ]) induces an equiv alence of deriv a tors Hom ! ( Ho ( L A [ A op , s S et ]) , D ) ≃ LC ( A, D ) . Thanks to Lemma 3.8 and to the fact that Hom ! ( − , D ) is a 2-functor, the adjunc- tion γ : Ho ([ A op , s S et ]) ⇄ Ho ( L A [ A op , s S et ]) : i th us induces an adjunction i ∗ : Hom ! ( Ho ([ A op , s S et ]) , D ) ⇄ Hom ! ( Ho ( L A [ A op , s S et ]) , D ) : γ ∗ . In par ticula r, we see that LC ( A, D ) is a deriv a tor (as it is equiv alent to the deriv a tor Hom ! ( Ho ( L A [ A op , s S et ]) , D )), and we get an adjunction of deriv ator s D A ⇄ LC ( A, D ) . Applying this to the opp osite deriv ator D op gives the o ther adjoint.  R emark 3.13 . The preceding result can b e in terpreted a s follows in terms of model categorie s. Consider a s mall catego ry A and a complete and cocomplete mo del category V . Then Ho ( V ) is a der iv ator, so that LC ( A, Ho ( V )) is a deriv ator a s well. Denote by LC ( A, V ) the full sub categ o ry of [ A, V ] whose o b jects a r e the lo cally constant functors. One can then verify that the pr ederiv ator asso ciated to the catego ry LC ( A, V ) (by inv er ting the ter mwise weak equiv alence s) is ca nonically equiv alent to LC ( A, Ho ( V )); this ca n b e expres sed b y the for mula Ho ( LC ( A, V )) ≃ LC ( A, Ho ( V )) . This means that the left Bousfield lo calizations discussed in 2 .1 for co mbinatorial mo del categor ies alwa ys exist in the setting of deriv ator s. Theorem 3.12 implies that such Bousfield lo calizations actually exist in the setting of ABC c o fibration categorie s developp ed in [RB06]. 12 DENIS-CHARLES CISINSKI 4. Galo is cor respond ence and homotopy distributors 4.1. Let A and B be small categ ories. W e get fro m Theorem 3 .5 the f ollowing canonical equiv alence of categ ories Hom ! ( Ho ( L B [ B op , s S et ]) , Ho ( L A [ A op , s S et ])) ≃ Hom B ( B , Ho ( L A [ A op , s S et ])) ≃ Ho ( L A × B op [ A op × B , sSet ]) . (4.1.1) Moreov er, w e hav e an e q uiv alence o f categ ories (4.1.2) Ho ( L A × B op [ A op × B , sSet ]) ≃ Ho ( Cat / A × B ) where Ho ( Cat / A × B ) denotes the lo caliza tion of the category of sma ll categories ov er A × B by the class of functors (o ver A × B ) whose nerve are simplicial weak equiv alences; this follows for example from [Cis06, Corollaries 4.4.20 and 6.4 .27] and from the fact B and B op hav e the same homotopy t y p e . The induced equiv ale nce o f categories (4.1.3) S : Ho ( Cat / A × B ) / / Hom ! ( Ho ( L A [ A op , s S et ]) , Ho ( L B [ B op , s S et ])) can b e describ ed very explicitely: its compo sition with the lo calization functor from Cat / A × B to Ho ( Cat / A × B ) is the functor (4.1.4) s : Cat / A × B / / Hom ! ( Ho ( L A [ A op , s S et ]) , Ho ( L B [ B op , s S et ])) which can b e described as follows. Consider a functor C / / A × B . It is determined by a pair of functors p : C / / A a nd q : C / / B . The functor q induces an in v erse image morphism (4.1.5) q ∗ : L B [ B op , s S et ] / / L C [ C op , s S et ] which happ ens to b e a right Quillen functor for the pro jective loca l mo del struc- tures; s ee P rop osition 2.6 . It thus defines a cont inuous morphism of deriv ators (see [Cis03, Prop ositio n 6.12]) (4.1.6) R q ∗ : Ho ( L B [ B op , s S et ]) / / Ho ( L C [ C op , s S et ]) . Using the equiv a lences of type Ho ( L B [ B op , s S et ]) ≃ LC ( B op , Ho ( sSet )), we see that R q ∗ corres p o nds to the restrictio n to the deriv ator s LC ( B op , Ho ( sSet )) and LC ( C op , Ho ( sSet )) o f the inverse imag e map q ∗ : D B op / / D C op for D = Ho ( sSet ) (whic h is coc o ntin uous, by virtue of [Cis02, P rop ositio n 2.8]). W e th us conclude from Lemma 3.8 that (4.1.6) is also co contin uous . The functor p induces a left Quillen functor for the pro jective local mo del structure s (b y P rop ositio n 2.6 aga in) (4.1.7) p ! : L C [ C op , s S et ] / / L A [ A op , s S et ] . This defines a co contin uous morphism of deriv ator s (b y the dual of [Cis03, Prop o- sition 6.12]) (4.1.8) L p ! : Ho ( L C [ C op , s S et ]) / / Ho ( L A [ A op , s S et ]) . The functor (4.1 .4) is simply defined by sending the pa ir ( p, q ) to the comp osition of (4.1.6) and (4.1.8). (4.1.9) s ( p, q ) = L p ! R q ∗ : Ho ( L B [ B op , s S et ]) / / Ho ( L A [ A op , s S et ]) . Prop ositi on 4.2. Given a fu n ctor ( p, q ) : C / / A × B , the fol lowing c onditions ar e e quivalent. (a) The morphism (4.1.9) is c ontinu ous ( i.e. pr eserves homotop y limits). LOCALL Y CONST ANT FUNCTORS 13 (b) The morphism (4.1.8) is c ontinuous. (c) The funct or L p ! R q ∗ : Ho ( L B [ B op , s S et ]) / / Ho ( L A [ A op , s S et ]) pr eserves terminal obj e cts. (d) The functor L p ! : Ho ( L C [ C op , s S et ]) / / Ho ( L A [ A op , s S et ]) pr eserves ter- minal obje cts. (e) The morphism (4.1.8) is an e quivalenc e of deri vators. (f ) The functor L p ! : Ho ( L C [ C op , s S et ]) / / Ho ( L A [ A op , s S et ]) is an e quiva- lenc e of c ate gories. (g) The nerve of p is a simplic ial we ak e quivalenc e. Pr o of. The f unctor (4.1.7) is a left Quillen equiv alence (for t he pro jective lo cal mo del structures) if and only if for any small category I , the induced functor p ! : [ I , L C [ C op , s S et ]] / / [ I , L A [ A op , s S et ]] is a left Quillen e q uiv alence. This prov es that the conditions (e) and (f ) a re equiv a- lent . It is obvious that condition (e) implies condition (b). The fact that condition (g) implies condition (f ) can b e o btained, for ex ample, using Theorem 1.3. It is clear that condition (b) implies conditions (a) and (d), and that conditions (a) or (d) implies co ndition (c). T o finish the pro of, w e will sho w that the condition (c) implies (g). Under the eq uiv alences of t ype Ho ( L X [ X op , s S et ]) ≃ Ho ( Cat /X ), the functor L p ! corres p o nds to the functor fro m H o ( Cat / A ) to Ho ( Cat / A ) which is induced by co mpo sition with p . Similarly , the functor R q ∗ corres p o nds to the functor from Ho ( Cat /B ) to Ho ( Cat / C ) which sends a functor X / / B to the pro jection X × h B C / / C (where X × h B C denotes the homotopy fib er pro duct of X and C ov er B ). Thes e descriptions show immediately that the condition (c) implies (g). This ends the pro of.  4.3. W e refer to [Mal05a, Mal05b, Cis06] fo r the notion o f smo o th functor and of prop er functor (with r esp ect to the minimal basic loca lizer). The first r eason we are interested b y this notion is that these functors have very go o d pro p erties with resp ect to ho motopy Ka n extensions; s ee [Mal05a, Sec tion 3.2]. The second r eason of our in terest for this class of functors is the follo wing statemen t. Prop ositi on 4.4. The c ate gory of smal l c ate gories is endo we d with a structur e of c ate gory of fibr ant obje cts in the sense of Br own [B ro73] , for whic h the we ak e quivalenc es ar e the functors whose nerve is a simplicial we ak e quivalenc e, and the fibr ations ar e the smo oth and pr op er fun ctors. Mor e over, the factorizations int o a we ak e quivalenc e fol lowe d by a fibr ation c an b e made functorial ly. Pr o of. An y functor to the t erminal catego ry is smo oth and prop er (s o t hat any small category will b e fibr a nt). F unctors which are smooth and prop er are stable under ba se change and compositio n (see [Mal0 5a, Co rollar y 3.2.4 a nd Prop ositio n 3.2.10]). It follows from [Cis06, Corollaries 6.4.8 and 6 .4.18] and from [Mal0 5a, Prop os ition 3.2 .6] that t he cla ss of trivial fibr ations (i.e. of s mo oth and pro p er functors whic h are w eak e q uiv alences) is s ta ble b y pullbacks. By virtue of [Cis 0 6, Prop os ition 6 .4.14], the pullback of a weak equiv alence by a smo oth and prop er functor is a w ea k equiv alence. T o finish the pro of, it is s ufficient to prove that any functor can factor (functoria lly ) through a weak equiv alence follow ed b y a smo oth and prop er functor, which is a consequence of [Cis06, Theorem 5.3.14 ].  14 DENIS-CHARLES CISINSKI Corollary 4.5. The lo c alization of the ful l sub c ate gory of Cat / A × B whose obje cts ar e the fun ctors ( p, q ) : C / / A × B such that p and q ar e smo oth and pr op er by the cla ss of we ak e quivalenc es is c anonic aly e quivalent t o Ho ( Cat / A × B ) . 4.6. The simplicial localiza tion L ( Cat ) of Cat b y the class of w e a k equiv alences can be describ ed using the structure of catego r y of fibrant ob jects given b y Prop o sition 4.4. In particular, the simplicial set Hom L ( Cat ) ( A, B ) can b e describ ed a s the nerv e of the category Map ( A, B ), whic h is defined as the full sub categ ory of Cat / A × B whose ob jects ar e the functors ( p, q ) : C / / A × B such that p is a trivial fibration (i.e. a functor whic h is smo o th, pro p er, and a weak equiv a lence). It is easy to see from Pro po sition 4.4 that the fundamen tal gro up o id of Map ( A, B ) is equiv alent to the full sub categ ory of Ho ( Cat / A × B ) whose ob jects are the functors ( p, q ) : C / / A × B such that p is a weak equiv alence. In other words, Prop osition 4.2 can now b e reform ulated as follo ws. Corollary 4.7 (Galo is reco nstruction theo rem) . The gr oup oid π 1 ( Map ( A, B )) is c anonic al ly e quivalent to the c ate gory of c o c ontinu ous morphisms of de rivators fr om Ho ( L B [ B op , s S et ]) to Ho ( L A [ A op , s S et ]) which pr eserve finite homotopy limi ts. 4.8. Le t us explain wh y th e preceding coro llary can b e in terpreted as a Galois reconstructio n theor e m. Given a small ca tegory A , if we think of Ho ( L A [ A op , s S et ]) as the “top os of representations of the ∞ -gr oup oid as s o ciated to A ”, it is na tural to define the functor of p oints of Ho ( L A [ A op , s S et ]) by B  / / Hom ex ! ( Ho ( L B [ B op , s S et ]) , Ho ( L A [ A op , s S et ])) (where Hom ex ! ( Ho ( L B [ B op , s S et ]) , Ho ( L A [ A op , s S et ])) denotes the categor y o f co- contin uous morphis ms of deriv a tors which preserve finite ho motopy limits). This is a 2 -functor fro m τ 6 2 L ( Cat ) to the category of group oids which is corepr esentable precisely by A . This can b e re formulated b y saying that we can reco ns truct the homotopy type of A from the “ top os” Ho ( L A [ A op , s S et ]). This is the deriv ator version o f T o¨ en’s homotop y Galois theory [T o¨ e02]. 4.9. Corollary 4.5 can also b e used to understand the compatibilities of the equiv- alences of categorie s of type (4.1.3) with comp o s ition of morphisms of deriv a tors. More precisely , we hav e a bica tegory Ho ( Dist ), whose ob jects a re the s mall ca te- gories, and w ho se ca tegory o f morphisms from A to B is the ho mo topy category Ho ( Cat / A × B ) (comp osition is defined b y homotopy fib er pro ducts). W e will fin- ish this s ection b y ex plaining how Corolla ry 4.5 implies that the functors (4.1.3) define a bifunctor from Ho ( Dist ) to the 2-categor y o f deriv ators . Define a bicat- egory SP as follows. The o b e cts of SP are the s mall categories. Given tw o small categorie s A a nd B , the ca tegory o f morphisms SP ( A, B ) is the full sub categ o ry of Cat / A × B whose ob jects are the functors ( p, q ) : C / / A × B such that p and q are smo o th and proper. The compo sition law of SP is defined by fiber pro ducts (whic h is meaningful, as the smo oth and prop er functors a re stable by pullbacks and comp o sitions). W e denote b y Der ! the 2-categor y whose ob jects ar e the deriv ators, and whose morphisms are the coco nt inuous morphims (2-cells are just 2-cells in the 2-category of prederiv ator s). Lemma 4.10. The functors (4.1 .4) de fine a bifunctor s : SP op / / Der ! LOCALL Y CONST ANT FUNCTORS 15 Pr o of. Consider a commutativ e diagra m G t   ~ ~ ~ ~ ~ ~ ~ u   @ @ @ @ @ @ @ E p   ~ ~ ~ ~ ~ ~ ~ q @ @ @ @ @ @ @ F r   ~ ~ ~ ~ ~ ~ ~ s   @ @ @ @ @ @ @ A B C in whic h the square is a pullback, and all the maps are smo oth and prop er . Note that for an y smoo th and prop er map ϕ , b oth ϕ and ϕ op are lo cally constant; see [Cis06, Corollar y 6.4 .8 ]. By vir tue of Prop ositions 1.15 and 2.8, we can apply [Mal05a, Prop o sition 3.2.28] to get that the base ch ange map L u ! R t ∗ / / R r ∗ L q ! is a n iso morphism in Hom ! ( Ho ( L E [ E op , s S et ]) , Ho ( L F [ F op , s S et ])). In particular, we ge t a cano nic a l isomorphism L s ! L u ! R t ∗ R p ∗ ≃ L s ! R r ∗ L q ! R p ∗ . These isomor phisms, together with the functors (4.1.4) define a bifunctor: to chec k the coherence s, w e are r educe to c heck that commut ative squares with the Be ck- Chev alley prop er ty are stable b y co mpo sitions, which is w ell kno wn to hold.  4.11. W e define now a bicategor y Ho ( SP ) as follows. The ob jects ar e the small categorie s, a nd g iven tw o ob jects A and B , the c a tegory o f mor phisms from A to B is Ho ( SP ( A, B )), that is the lo calisation of SP ( A, B ) by the clas s of weak equiv alences. W e hav e a lo calization bifunctor (4.11.1) γ : SP / / Ho ( SP ) . Corollar y 4 .5 can now b e reformulated: the canonical bifunctor (4.11.2) j : Ho ( SP ) / / Ho ( Dist ) is a biequiv alence. Prop ositi on 4.12. The e qu ivalenc es of c ate gories (4.1 .3) define a bifunctor S : Ho ( Dist ) op / / Der ! . Pr o of. F or a ny sma ll ca teg ories A , B and C , we have Ho ( SP ( A, B ) × SP ( A, B )) = Ho ( SP ( A, B )) × Ho ( SP ( B , C )) . The universal pro p erty of lo caliza tions and Lemma 4.10 imply tha t we g et a bi- functor S ′ : Ho ( SP ) op / / Der ! . W e deduce fro m this and f rom the biequiv alence (4.11 .2) that there is a unique w ay to define a bifunctor S from Ho ( Dist ) op to Der ! from the eq uiv alences of categories (4.1.3) such tha t S ′ j = S .  Scholium 4 .13 . Theorem 3 .5 asserts that for a ny deriv ator D , w e hav e an equiv a- lence of deriv ator s Hom ! ( Ho ( L A [ A op , s S et ]) , D ) ≃ LC ( A, D ) . 16 DENIS-CHARLES CISINSKI As Hom ! ( − , D ) is a 2-functor , w e deduce from the preceding propos ition that w e get a bifunctor LC ( − , D ) : Ho ( Dist ) / / Der ! . which sends a small category A to LC ( A, D ). This defines a bifunctor LC ( − , − ) : Ho ( Dist ) × Der ! / / Der ! , which defines a n enrichmen t of Der ! in homotopy distr ibutors. References [Bek00] T. Bek e, She afifiable homotopy mo del c ate gories , Math. Pr o c. Camb. Phil. So c. 129 (2000), 447–475. [Bro73] K. S. Brown, Abstr act homotop y and gener alize d she af c ohomol o gy , T rans. A mer. Math. Soc. 1 86 (1973), 419–458. [Cis02] D.-C. Cisinski, Pr opri ´ et´ es unive rsel le s et exte nsions de Kan d´ eriv ´ ees , preprin t, 2002. [Cis03] , Images dir e ctes c ohomo lo giques dans les c at ´ egories de mo d ` eles , Annales Math ´ ematiques Blaise P ascal 10 (2003) , 195–244. [Cis06] , L es pr ´ ef aisc e aux c omme mo d ` eles des t y p es d’ho motopie , Ast´ eris que, vo l. 308, Soc. M ath. F rance, 2006. [Dug01] D. Dugger, Combinatorial mo del c ate gories ha ve pr esentations , Adv. Math. 164 (2001), no. 1, 177–201 . [JT07] A. Jo yal and M. Tierney , Quasi-c ategories vs Se gal sp ac e s , Categories in Algebra, Ge- ometry and Physics, C ontemp. Math., vo l. 431, Amer. Math. Soc., 2007, pp. 277–326. [Mal05a] G . Maltsiniotis, L a th´ eorie de l’homotopie de Gr othendie ck , As t´ eri sque, vol. 30 1, So c. Math. F rance, 2005. [Mal05b] , Structur es d’asph ´ e ri cit´ e , Annales Math´ ematiques Blaise Pascal 12 (2005), no. 1, 1–39. [RB06] A. R˘ adulescu-Ban u, Cofibr ations in ho motopy the ory , arXiv:math/0610009, 2006. [T ab07] G. T abuada, Higher K-the ory via universal invariants , ar Xiv:0706.2420, 2007. [T o ¨ e02] B. T o¨ en, V ers une interpr ´ etation galoisienne de la th´ eorie de l’ho motopie , Cahiers de topologie et g ´ eom´ etrie diff´ erentielle cat ´ egoriques XLIII-3 (2002), 257–312 . LA GA, CNRS (UMR 7539), Institut Galil ´ ee, Universit ´ e P aris 13, A venue Jean-Baptiste Cl ´ ement, 93 430 Villet a neuse, France E-mail addr ess : cisins ki@math.u niv-paris13.fr URL : http://ww w.math.un iv-paris13.fr/~cisinski/