Simplicial presheaves of coalgebras

SIMPLICIAL PRESHEA VES OF CO ALGEBRAS GEORG E RAPTIS Abstract. The category of simplicial R -coalgebras ov er a presheaf of commuta tive unital rings on a small Grothendiec k site is endow ed with a l eft prop er, simplicial , cofi- bran tly generated model categ ory structure where the we ak equiv a lences are the lo cal we ak equiv alences of the underlying simplicial presheav es. This mo del category is natu- rally linked to the R -l ocal homotop y theory of simplicial presheav es and the homotop y theory of si mplicial R -mo dules by Quillen adjunctions. W e study the comparison with the R -lo cal homotop y category of simpl icial preshea ve s in the sp ecial case where R is a presheaf of algebraically closed (or perfect) fields. If R is a presheaf of algebraically closed fields, we show that the R -lo cal homotop y category of simplicial preshea ves em beds fully faithfully in the homotopy category of simpl i cial R - coalgebras. 1. Introduction and st a tement of resul ts Let PSh( C ) denote the ca tegory of s et-v alued presheaves on a small Grothendieck site C . Let R be a pr esheaf of co mmutative unital rings o n C . The categor y Mod R of presheaves of R -modules is an abelia n loca lly presentable c ategory w ith a clo sed symmetric monoidal pairing that is given by the p oint wise tensor pr o duct of R -mo dules . Let Coalg R denote the catego ry of co commut ative, coas s o ciative, counital R -coalgebra s . The forgetful functor Φ : Co alg R → Mo d R is a left a djoin t where the right adjoint is the cofree R -co algebra functor. Let sP Sh( C ) denote the ca tegory of simplicial pres he aves on C . The clas s of lo c al weak equiv alences b e t ween simplicia l presheaves defines a homoto p y theory that has b een studied extensively in the literature and for ms the sub ject known as homotopical sheaf theory . Nat- urally , the sub ject b egan with the s tudy of the corr esp onding homotopy theory o f simplicial sheav es (see [6], [7 ], [19]). The shift to the mor e flexible catego ry of simplicial presheaves is due to J ardine who realize d that “. . . it is not so much t he ambient top os that is cr e ating the homotopy the ory as it is the top olo gy of the underlying site” [16 ]. On the other hand, the depe ndence on the choice of a Grothendieck site for the ambien t to po s is immater ial: a lo cal weak equiv alence of simplicia l presheves is the sa me as a lo cal weak equiv alence o f the asso ciated simplicial s he aves, and so, in particula r , the tw o homotopy theor ies are equiv a - lent (see [17]). The class of lo cal weak equiv alence s is a refinement o f the s e ctionwise weak equiv alences that takes into acco unt the to po logy on C . F or example, in the case where the top os of sheaves Sh( C ) has enough p oints, a ma p o f simplicial presheaves is a lo cal weak equiv alence if it induces a weak equiv a le nce of simplicial sets at all stalks . Note that this is the same a s a sectionwise weak equiv alence when C has the trivial top olog y . Let sCoalg R denote the catego ry o f simplicial R -coalg ebras. The pur po se of this pap er is to study the homotopy theory of simplicial R -coalgebr a s and compar e it with that of simplicial presheav es. The metho d to pursue this follows the axioma tic approach o f the theory of mo del catego ries. W e a ssume that the reade r is familiar with the theory o f mo del categorie s and what it is go o d for. F or background material, we recommend the recent monogra phs of Hirschhorn [14] and Hov ey [15]. 1 2 GEORG E RAPTIS A mo rphism f : A → B of simplicial R -coalg ebras is called a (lo c al) we ak e quivalenc e if it defines a lo cal weak eq uiv alence b et ween the under lying s implicial pr esheav es in sPSh( C ). This class o f weak e q uiv alences will be denoted here by W R . A morphism f : A → B in sCoalg R is called a Φ -monomorphism if the mo r phism betw een the underlying simplicial R -mo dules is a monomorphism. Theorem A. Ther e is a left pr op er, simplicia l, c ofibr antly gener ate d mo del c ate gory stru c- tur e on sCoalg R wher e the class of we ak e quivalenc es is W R and t he set of Φ - monomor- phisms b etwe en κ -pr esentable obje cts is a gener ating set of c ofibr ations (for any choic e of a lar ge enough r e gular c ar dinal κ ) . The choice of κ will b e clar ified in the pro of of the theore m in section 4. This result was prov ed by Go erss [12] in the case wher e C is the termina l category (i.e., sPSh( C ) = sSet) and R is given by a s ingle field. Although the structural prop erties of c o algebra s ov er a field ar e w ell-studied in the literature (e.g . see [27]), the situation in the general case of a commutativ e ring is more complicated and less well-understo o d. Among the basic reaso ns that make the case of fields sp ecial are, of course, the exa ctness o f the tensor pr o duct and the av ailable duality techniques. Both o f them play a ro le in Go erss’ pro o f. There is yet another r elev ant distinctive fea tur e o f the ca se o f fields that r elates to the (p ossibility of the) int ro duction of higher ca rdinals in the pr o of of Theorem A. According to the fundamental the or em of c o algebr as , due to Sweedler [2 7], if F is a field, every F -co a lgebra is the filtered colimit of its finite dimens io nal sub- coalgebra s. Mor eov e r, finite dimensio nal co algebras are finitely presentable ob jects in Coalg F . This prop erty may fail in the case of an arbitra ry (presheaf of ) comm utative ring(s), how ever an appropriate analogous statemen t can b e formulated in this general case if we a llow a higher r ank of presentabilit y . This is b ecause the category of R -coalg ebras is known to b e lo cally λ -pres e n table for some reg ular cardinal λ by res ults o f Barr [2] and F o x [10] (see a lso [23]). This fact tog ether with additiona l background mater ia l ab out R -coalgebra s will b e discussed in s e c tion 2. The idea to apply metho ds fro m the theo ry of lo c a lly pr esentable ca tegories in ho motopi- cal algebr a orig ina tes fro m unpublished work of J. H. Smith. In par ticular, the theor y o f combinatorial mo del categor ies, introduced by J.H. Smith, has provided the theor y o f mo del categorie s with p ow erful set-theor etical tec hniques. Our pro of of Theor em A is heavily based on such ideas. W e will assume that the reade r is familiar with the basic theo ry o f lo c ally presentable and accessible catego r ies. A detailed ac c o unt can b e found in the monog raph by Ad´ amek-Rosick´ y [1]. F or ba ckground material ab out combinatorial mo del categorie s , see the articles [3], [9], [26]. F ollo wing the work of Go erss [12], the motiv ation for studying the homotopy theor y of simplicial R -coalge br as comes fro m its co nnection with the R - lo c al homotopy theory of simplicia l presheav es, i.e., the Bo usfield lo ca lization of the mo del categor y of s implicia l presheav es at the c lass o f R -homology e quiv alences. Let us first discuss this connection in the c a se where C is the terminal category and let R b e a commutativ e ring with unit. The simplicial R -chains R { X } o f a simplicial set X form naturally a simplicial R -coalg e bra. The comultiplication is induced b y the diagona l ma p ∆ : X → X × X and the counit b y the map X → ∆ 0 . Moreov er, ther e is an adjunction (1.1) R {−} : sSet ⇄ sCo alg R : ρ where the r ight adjoint is the functor of R -p oints defined b y ρ ( A ) n = Coalg R ( R, A n ) . SIMPLICIAL PRESHEA VES OF COALG EBRAS 3 An impo rtant obs erv ation is that the ca nonical unit ma p X → ρR { X } is an is omorphism, for every simplicia l set X , when R ha s no non-triv ial idempo ten ts. Moreov er, the adjunction (1.1) is a Q uillen adjunction b etw een the standar d mo del c ategory structure on s Set and the mo del c ategory of Theor em A. Therefor e it induces a new Quillen adjunction (1.2) R {−} : L R sSet ⇆ sCo a lg R : ρ where L R sSet deno tes the Bo usfield lo calization of sSet at the R -homolog y equiv a lences, i.e., the class of ma ps f : X → Y such that H ∗ ( f , R ) is an is omorphism (see [5 ]). Reca ll that the classical Dold-K an corr esp o ndence (e.g. see [28]) shows that f is an R -homolog y equiv alence iff R { f } is a weak equiv alence of simplicial R -mo dules. Then this prompts the question whether the additional co algebraic str ucture on the sim- plicial R -chains could suffice in order to pro duce a faithful app oximation to the homotopy theory of spa ces lo ca lized at the R - ho mology equiv a le nc e s. By taking (functorial) cofibra n t and fibrant replace ments r espe ctively , one obtains a derived adjunction b etw een the resp ec- tive homotopy ca tegories, (1.3) L R {−} : Ho(L R sSet) ⇄ Ho(sCoalg R ) : R ρ and the derived unit tra ns formation gives a canonical map X → R ρ ( R { X } ) from every simplicia l set X to an R - l ocal space, i.e., a fibrant ob ject in L R sSet. Go er ss [12, Theorem C] prov ed that this map is an R - homology equiv a lence when R is an a lgebraica lly closed field. An equiv a lent sta temen t is that the functor L R {−} is fully faithful. More generally , he show ed that for any p erfect field F with a lgebraic closure F and profinite Galois gr oup G , the der ived unit map ca n b e ident ified w ith the map from X in to the fixed po in ts of the lo c alization of X at the F -homology equiv alences where X is regarde d a s a simplicial G -set endow ed with the trivial G -actio n (see [12, Theorem E ]). W e do no t k now of any appr o priate extension of this remar k able result to an a rbitrary ring . The gener a l case of an arbitra ry small site C and a pres heaf of co mm utative unita l rings R is completely ana logous. The (sectionwise) free R -mo dule functor factor s thro ugh sCo alg R and there is a Quillen adjunction, (1.4) R{−} : sP Sh( C ) inj → sCoalg R : ρ where s PSh( C ) inj denotes the mo del categ ory of simplicial presheav es by Jardine [16], [17]. Some basic facts ab out the v arious mo del category structures o n simplicial presheaves will be reviewed in s e ction 3. The functor of R -p oints ρ is defined similar ly as ab ov e, by ρ ( A )( U ) n = Coa lg R ( U ) ( R ( U ) , A ( U ) n ) . The Quillen adjunction (1.4) will b e studied in section 5. W e show tha t this adjunction can b e used to endow sCoalg R with a different mo del ca teg ory structure wher e the weak equiv alences ar e pulled back fro m s PSh( C ) inj via the functor ρ o f R -p oints. In addition, it follows ea sily , and so mewhat surpr isingly , that the adjunction (1.4) defines a Quillen equiv alence b etw een this new mo del category , denoted by sCoa lg ρ R , a nd sPSh( C ) inj . This pro duces an alterna tiv e p oint of view for the compariso n b etw ee n the homotopy theor y of simplicial pr esheav es and simplicia l R -coalgebra s as it can be als o mo delled by the identit y functor 1 : sCoalg ρ R → sCoalg R as a left Quillen functor. Similarly to the case o f a single commutativ e ring, the co mparison via the Quillen ad- junction (1.4) can only relate to the R -lo cal part of sPSh( C ) inj . A morphis m f : X → Y betw een s implicial preshe aves is called an R -homolo gy e quivalenc e if R{ f } is a weak 4 GEORG E RAPTIS equiv alence. The left Bousfield lo ca lization L R sPSh( C ) inj of sPSh( C ) inj at the class of R - homology equiv a lences ex ists and so the a djunction (1 .4) induces a new Quillen adjunction R{−} : L R sPSh( C ) inj → sCoa lg R : ρ . Based o n the structure theory of coalge bras ov er a n algebraic ally close d field and the metho ds o f Go erss [12], we prov e the fo llowing theorem in section 6. Theorem B. L et F b e a pr eshe af of algebr aic al ly close d fi elds. Then the functor L F {−} : Ho (L F sPSh( C ) inj ) → Ho(sCoa lg F ) is ful ly faithful. Theorem B is a gener alization o f Go erss’ theor e m [12, Theorem C] to the context of sim- plicial pr esheav es. W e will a ls o consider the case of a co ns tant pr esheaf at a p erfect field, following the case of s ingle p erfect field in [1 2, T heo rem E] as ment ioned ab ov e. The pre cise analogue of Theor em B in this case require s some preparato ry work and we will not attempt to summarise it here. It will b e discussed in deta il in s ection 6. Or ganization of the p ap er. In s e c tion 2, we review some categ orical prope rties of the preshea f categorie s of coalg e bras Coalg R fo cusing in particular on the pro per ty of lo ca l pr esentabilit y that is cr ucial to the pro o f of Theor em A. In section 3, we reca ll briefly (some of ) the v ario us known mo del categor ies of simplicial presheav es a nd simplicial R -mo dules . In section 4, we prov e Theo r em A. In sec tio n 5, we c ompare the model categor y of s implicial R -coalg e br as with the mo del categ o ries of simplicial pres heav es and simplicial R -mo dules. The compa r i- son with s implicia l presheaves is ba sed on the adjunction (1.4) as dis c ussed ab ov e. The com- parison with s implicia l R -mo dules is based on the forgetful functor Φ : sCoalg R → sMo d R . In sectio n 6, we prove Theorem B and discuss some generaliza tions of it to other t yp es of presheav es of fields. A cknow le dgements. I would like to thank Manfred Stelzer for the discussions a bo ut s implicia l coalgebr as and his in terest in the results o f this pa per . 2. Preliminaries on presheaf ca tegories of Coalgebras Let C b e a small category a nd PSh( C ) := F un( C op , Set) denote the catego ry of set-v alued presheav es on C . F or every ob ject U of C , there is a presheaf Y ( U ) : C op → Set, V 7→ C ( V , U ), called the r epresentable pr esheaf by U . The set of presheaves { Y ( U ) : U ∈ Ob C } defines a strong g enerator of finitely pr e sent able o b jects in PSh( C ). This is essentially a consequence of the Y o neda lemma which says that Y : C → P Sh( C ), U 7→ Y ( U ), is fully faithful. Let R be a pre s heaf of co mm utative unita l r ings on C . An R -mo dule M is a presheaf of ab elian groups on C suc h that M ( U ) is an R ( U )-module for every U ∈ Ob C and the restriction ma p M ( j ) : M ( U ) → M ( V ) is a homomo rphism of R ( U )-mo dules for every morphism j : V → U in C . A mor phism f : M → N of R -mo dules is a natural tra nsformation of the underlying preshe aves such that f | U : M ( U ) → N ( U ) is an R ( U )-mo dule ho momorphism. This defines a catego ry Mo d R of R -mo dules which is ab elian a nd lo c ally ( ℵ 0 -) prese ntable. W e r ecall that a ca tegory is ( λ -)lo cally presentable if it is co co mplete a nd has a stro ng g enerator of ( λ -)presentable SIMPLICIAL PRESHEA VES OF COALG EBRAS 5 ob jects 1 . The forg etful functor ι : Mo d R → P Sh( C ) admits a left adjoint R{−} : PSh( C ) → Mo d R that asso ciates to every pres heaf X : C op → Set, the R -module whose v alue at U ∈ Ob C is the free R ( U )-mo dule with generato rs X ( U ). A conv enient strong g e ne r ator for Mo d R is given by the set of finitely presentable ob jects {R{ Y ( U ) } : U ∈ Ob C } . The catego r y Mo d R has a close d symmetric mono idal pairing ⊗ : Mo d R × Mo d R → Mo d R that is given by the sectionwise tensor product of mo dules. More precis e ly , this takes a pair ( M , N ) of R -mo dules to the R -mo dule M ⊗ N whose v a lue at U ∈ Ob C is the R ( U )- module M ( U ) ⊗ R ( U ) N ( U ). F or every j : V → U in C , the re striction map is given by the comp os itio n o f R ( U )-mo dule ho momorphisms M ( U ) ⊗ R ( U ) N ( U ) → M ( V ) ⊗ R ( U ) N ( V ) → M ( V ) ⊗ R ( V ) N ( V ) . The unit o f this monoidal pair ing is given b y R as an R -mo dule. As for every symmetric monoidal ca tegory , there is an asso ciated c a tegory o f co commu- tative, co asso ciative, co unital comonoids with resp ect to the monoidal pairing. A pplied to (Mo d R , ⊗ , R ), this defines the categ ory Coalg R of co commutativ e, coa sso ciative, counital R -coalgebr as. More explicitly , an R -co algebra ( A, µ, ǫ ) is a n R -mo dule A together with morphisms of R -mo dules for comultiplication µ : A → A ⊗ A and counit ǫ : A → R such that the following diag r ams c o mm ute A ⊗ A tw ∼ =   A µ < < x x x x x x x x x µ " " F F F F F F F F F A ⊗ A A ⊗ A µ ⊗ 1 / / ( A ⊗ A ) ⊗ A ∼ =   A µ " " E E E E E E E E E µ < < y y y y y y y y y A ⊗ A 1 ⊗ µ / / A ⊗ ( A ⊗ A ) A ⊗ A ǫ ⊗ 1   A 1   µ / / µ o o A ⊗ A 1 ⊗ ǫ   R ⊗ A A ∼ = / / ∼ = o o A ⊗ R The isomorphisms in these diag rams ar e the cano nical o nes, given as par t of the sym- metric monoidal str ucture. A morphism of R -coalgebra s from ( A, µ, ǫ ) to ( A ′ , µ ′ , ǫ ′ ) is a 1 This is equiv alent to the definition of [1, Definition 1.17] by [1, Theorem 1.20]. 6 GEORG E RAPTIS morphism f : A → A ′ betw een the under lying R -mo dules that makes the fo llowing dia - grams commute A ⊗ A f ⊗ f / / A ′ ⊗ A ′ A µ O O f / / A ′ µ ′ O O A f / / ǫ   A ′ ǫ ′ ~ ~ } } } } } } } R The res triction maps of a n R -coalg ebra are not quite maps of coalgebra s over a r ing, since the coalgebr aic str ucture is defined s ectionwise with resp ect to different rings in gener al. How ever, note tha t given a ring ho momorphism f : R → S and an R -coalge bra ( A, µ, ǫ ), the S -mo dule A ⊗ R S is natur ally an S - c oalgebra with comultiplication defined by A ⊗ R S µ ⊗ 1 − → ( A ⊗ R A ) ⊗ R S ∼ = → ( A ⊗ R S ) ⊗ S ( A ⊗ R S ) . Given an R -coalg ebra A and an S -coalg e bra B , an R -mo dule homomorphism g : A → B is called a map of coalg ebras if the induced S -mo dule homomor phism ˜ g : A ⊗ R S → B is a map o f S -coalg ebras. The r estriction maps of an R -coa lgebra ar e maps of coalgebra s in this sense. V arious impo rtant prop erties of categor ies of coalgebr as hav e bee n s tudied by Ba r r [2] (for a sing le commutativ e ring ), F ox [10] (for an arbitrary locally presentable monoida l category ) and, mor e recently , by Porst [23], [24]. W e reca ll some of them. Prop ositio n 2.1. The for getful functor Φ : Coalg R → Mo d R has a right adjoint. Mor e over, Coalg R is c omonadic over Mo d R . Pr o of. This is proved in [2, Theorem 4.1] for the case of a sing le commutativ e r ing. The same pro of a pplies here, see [10, Corollar y 8 ] and [2 3, Theorem 1 2, Remark 15].  The main prop erty that w e need in this pap er is sta ted in the following theorem that is essentially due to Ba rr [2] a nd F ox [10]. Fir st let λ b e a regula r ca rdinal that is big ger than ℵ 0 and the ca r dinalities of the r ings R ( U ) for all U ∈ Ob C . Theorem 2 .2. (Barr [2]-F ox [10]) Coalg R is a lo c al ly λ -pr esentable c ate gory. Pr o of. This is ess ent ially contained in the la st remar ks o f [10] co mb ined with the a rguments of [2, Theo rem 3.1, Coro llary 3.2] to g et the requir ed rank of pr esentabilit y . Let us fir st prov e the theorem in the cas e where C is a discrete categor y , i.e., when there are no non-identit y mo r phisms. Consider the set A of R -coa lgebras A w ho se underlying R -mo dule Φ( A ) sa tisfies the following prop erties (i) Φ( A ) is the trivial zero mo dule everywhere except for exactly o ne ob ject U ∈ Ob C , (ii) the c ardinality of Φ( A )( U ) is ≤ max { card( R ( U )) , ℵ 0 } . By [2, Theorem 3.1], given an R -coa lgebra A , ev ery element x ∈ A ( U ), U ∈ O b C , is contained in an R -s ubco algebra tha t satisfies (i) and (ii). Similarly to [2, Cor ollary 3.2], it follows that A is a str ong generator o f Coalg R . Let A ∈ A and U ∈ Ob C as in (i) a bove. By (ii), Φ( A )( U ) is genera ted by strictly less than λ elements, so it is λ -pre s ent able a s a n R ( U )- module. B y (i), Φ( A ) is also λ -presentable (in Mo d R ). F ollowing the la st remarks of [10], it is immediate that A is als o λ -pr esentable (in Coa lg R ). Indeed it suffices to note that SIMPLICIAL PRESHEA VES OF COALG EBRAS 7 the s et of maps fr om A to an R -co a lgebra B ca n b e expressed a s the equalizer of a pair of arrows Mo d R (Φ( A ) , Φ( B )) ⇒ Mo d R (Φ( A ) , Φ( B ) ⊗ Φ( B )) × Mo d R (Φ( A ) , R ) and that equalizers co mm ute with λ -directed co limits (for a ny λ ) in the categ ory of sets. Hence it follows tha t Coalg R is lo cally λ -pres ent able. F or the gener al ca se, let C 0 denote the discrete ca tegory with s e t of o b jects Ob C and u : C 0 → C be the inclusion functor . The preshea f R clear ly restricts to a pr esheaf of commutativ e unital rings R 0 := u ∗ R on C 0 . There is an a djunction u ! : Coalg R 0 ⇄ Coalg R : u ∗ where the r ight adjoint is the o bvious for getful pullba ck functor. The left adjoint is defined by u ! ( A )( V ) = M V → U A ( U ) ⊗ R ( U ) R ( V ) where the tensor pro ducts a re endo wed with the natural R ( V )-coalgebraic str ucture as remarked ea rlier. Mor e ov er , u ∗ is faithful a nd prese r ves ( λ -)directed colimits. It follows that the set u ! ( A ) = { u ! ( A ) : A ∈ A} ( A as ab ov e) is a strong generato r o f λ -presentable ob jects and hence the result a lso follows.  Remark 2. 3. The lo cal pres ent ability of Coalg R was a lso prov ed by Porst [23] in the case o f a single commutativ e ring using somewha t differ en t metho ds based more heavily on genera l results ab out accessible categ ories from [1]. These metho ds a pply identically to the ca se of a pre s heaf of commutative ring s, but it is gener ally difficult to obtain an explicit rank o f presentabilit y by them a lo ne and most probably it will b e very big. One wa y to obtain it would b e to go through the choices of ca rdinals in the pro ofs of [1, Lemma 2.76 , T heo rem 2.72, Theo rem 2.43, The o rems 2.32 -2.34 and Theor em 2.1 9] in that o rder more or less. On the other hand, the rank s hown in Theorem 2.2 is not the le ast p os sible s ince the category o f coalgebr as ov er a field is known to b e lo cally finitely presentable. It would b e interesting to know wha t the be s t p ossible rank is exa ctly and how it is related to the divisibility prop erties of the elemen ts of R or the failure of the tenso r pro duct to be a left exa c t functor . The tensor pro duct o f R -coalg ebras ( A, µ, ǫ ) and ( B , µ ′ , ǫ ′ ) is natura lly an R -coalgebra. The underlying R -mo dule is A ⊗ B and the coa lgebraic structure is g iven by the following R -mo dule morphisms A ⊗ B µ ⊗ µ ′ − → ( A ⊗ A ) ⊗ ( B ⊗ B ) ∼ = ( A ⊗ B ) ⊗ ( A ⊗ B ) A ⊗ B ǫ ⊗ ǫ ′ − → R ⊗ R ∼ = R . There are canonical morphisms p A : A ⊗ B 1 ⊗ ǫ ′ − → A ⊗ R ∼ = A p B : A ⊗ B ǫ ⊗ 1 − → R ⊗ B ∼ = B Moreov er, the co ne ( A ⊗ B , p A , p B ) actually defines a pro duct c o ne for A and B in Coa lg R . Using this des cription of the pro ducts in Coalg R , the following theorem is an immediate consequence o f the sp ecial adjoint functor theorem a nd the fact that Φ cr e a tes co limits. Theorem 2 .4. Coalg R is a c artesian close d c ate gory. Pr o of. This is proved in [2, Theorem 5.3] for the case of a sing le commutativ e r ing. The same pro of a pplies here.  8 GEORG E RAPTIS The R -mo dule R{ X } ha s a natur a l R -coalgebra str ucture that is induced by the dia gonal map X ∆ → X × X and the unique map to the terminal ob ject X → ∗ . Hence the functor R{−} : PSh( C ) → Mo d R factors through Φ : Coalg R → Mo d R . W e deno te the functor into Coalg R again by R{ −} : PSh( C ) → Coalg R . This functor has a rig h t a djoin t ρ : Coalg R → PSh( C ) which is defined sectionwise as follows: g iven an R -coa lgebra A , then ρ ( A )( U ) = Coalg R ( U ) ( R ( U ) , A ( U )) defines a pres heaf ρ ( A ) : C op → Set. The functor ρ preserves κ -directed colimits, and so R{−} pres erves κ - presentable ob jects, for ev ery r egular cardina l κ > card( M or C ). The functor ρ is called the functor of R -p oints bec a use it picks out from every section the elements of the R -coalgebr a that co-multiply “ diagonally” . If R ( U ) ha s no no n-trivial idempo ten ts for all U ∈ Ob C , the set o f ele men ts of R{ X } ( U ) that hav e this prop erty can be identified with X ( U ). In this case , the unit transfo r mation o f the a djunction 1 → ρ R{− } is a na tural is omorphism. Note also that s ince adjoints a re es sent ially unique, the forgetful functor ι : Mo d R → PSh( C ) factors as the comp osite of the cofree R -coalg ebra functor, denoted by T : Mo d R → Coalg R , follow ed by the functor of R -p oints ρ : Coalg R → PSh( C ). 3. A quick review of the model structures on simplicial preshea v es Let C b e a sma ll Grothendieck site and sPSh( C ) denote the categor y of simplicial presheav es (of sets) on C . The ob jects are usually understo o d as diag rams F : C op → sSet and the mor phisms are natur al transfor ma tions of s uc h diagr ams. V arious mo del categor y structures o n the categ ories of simplicial pr e s heav e s and simpli- cial R -mo dules (or presheav es of chain complexes ) ar e known in the litera tur e. W e rev iew some facts ab out the four mo del categor y str uctures tha t are characterized by the following t wo sp ecifications: (i) whether the weak equiv alences are defined sectionwise or stalkwise, and (ii) whether the cofibrations ar e defined sectio n wise or they are the so-ca lled pro jective cofibrations. First we rec a ll the definition of the lo cal weak equiv alences in sPSh( C ) from [17]. The definition uses B o olean lo caliza tion in order to include the case where the asso ciated to po s of sheav es do es not have eno ugh p o int s. A different definition using sheaves o f ho motopy groups can be found also in [17]. Let L 2 : PSh( C ) → Sh( C ) deno te the s heafification functor. According to a fundamental theore m in topo s theory , due to Ba rr, there is a co mplete Bo olean algebra B and a surjective geo metr ic morphis m ℘ : Sh( B ) → Sh( C ) (se e [21, IX.9] for details). Thus we obtain a geometric mo rphism b et ween the categories o f s implicial ob jects, ℘ ∗ L 2 : sPSh( C ) → sSh( B ) . A map f : X → Y in sP Sh( C ) is a loc a l weak equiv alence if (and only if ) the map ( ℘ ∗ L 2 E x ∞ ( f ))( b ) : ( ℘ ∗ L 2 E x ∞ ( X ))( b ) → ( ℘ ∗ L 2 E x ∞ ( Y ))( b ) is a weak e quiv alence of simplicial s ets for a ll b ∈ B . Here E x ∞ denotes Kan’s fibr a nt replacement functor applied sectionwise. It can b e s hown that this definition do e s no t depe nd on the c hoice o f Bo olean lo ca lization [17]. In par ticular, a co llection of enough po in ts of Sh( C ) (if it exists) defines a Bo olean lo calization, a nd s o a lo cal weak equiv a lence is a natura l tra nsformation tha t induces a weak equiv alence of simplicia l sets at every p oint in this collection. A sectionwise weak equiv alence is always a lo cal weak equiv a lence [17, SIMPLICIAL PRESHEA VES OF COALG EBRAS 9 Lemma 9]. If C has the trivial top olog y , a lo c al w eak equiv a le nce is exactly the same as a sectionwise weak equiv alence. Note that we can dispens e w ith the fibrant replacement functor E x ∞ when the presheaves are a lready sectionwise fibrant. Thus a mo r phism f : M → N b etw ee n simplicia l R -mo dules is lo cal weak equiv alence (of the underlying simplicial presheaves) if (and only if ) ℘ ∗ L 2 ( f )( b ) is a weak equiv alence for a ll b ∈ B , s inc e a simplicial pr esheaf of ab elian gr oups is already sectionwise fibra n t. The pr oje ctive mo del c a tegory structure on sP Sh( C ) is the standar d pr o jective mo del category structure on a categor y o f diagrams in a cofibr antly generated mo de l categor y (see [14] for details). The w eak eq uiv alences (resp. fibrations ) a re the sectionwise weak equiv a - lences (resp. Kan fibratio ns) of simplicial sets. In pa rticular, the class of weak equiv alences, and so the as so ciated homotopy theory , is independent of the choice of the Grothendieck top ology on C . The cofibra tions of this mo del catego ry a re ch ara cterised by a right lifting prop erty and they will b e re ferred to as the pr oje ctive cofibra tions. F urthermor e, the pro jec- tive mo del categor y is prop er, simplicial and co fibrantly genera ted. The simplicial structure is defined by the following functorially defined ob jects, ( K ⊗ X )( U ) := K × X ( U ) X K ( U ) := Ma p sSet ( K, X ( U )) Map( X , Y ) := sPSh( C )(∆ · ⊗ X , Y ) for every simplicia l set K and simplicial preshe aves X and Y . Sets of g enerating c o fibrations and trivia l cofibr ations are defined as follows. Let C 0 denote the discrete category with set of ob jects Ob C and u : C 0 → C b e the inclusion functor. There is an adjunction u ! : sPSh( C 0 ) ⇄ sPSh( C ) : u ∗ where u ∗ is the o b vious pullback functor . Note that since u ∗ preserves κ -dir ected colimits (in fact, all co limits), the left a djoin t u ! preserves κ -pr esentable ob jects (for any κ ). The category s P Sh( C 0 ) can b e endowed with the pro duct mo del catego r y structure where all classes o f weak equiv a le nces, cofibrations and fibrations are defined point wise. The pr o jective mo del categor y str ucture on sPSh( C ) is the lifting of this pro duct mo del ca tegory (which is, incident ally , a lso an example of a pro jective mo del category ) a long the adjunction u ! ⊣ u ∗ . A detailed acco un t of the metho d of transferr ing a co fibrantly generated mo del categ ory structure along a n adjunction ca n be found in [14, Theorem 1 1.3.2]. As g enerating se ts for cofibra tions and trivial co fibrations of sPSh( C 0 ) we can choose the “ pro ducts” of s o me generating sets of the mo del ca tegory sSet. Let I = { ∂ ∆ n ֒ → ∆ n : n ≥ 0 } and J := { Λ n k ∼ ֒ → ∆ n : 0 ≤ k ≤ n, n ≥ 0 } denote the standa r d generating sets of sSet. F or every U ∈ Ob C 0 and simplicial set K , let X ( U, K ) b e the pr e sheaf on C 0 which takes the v alue K at U and ∅ els ewhere. F or every f : K → L in sSet, there is a na tural mor phism o f presheav es X ( U, f ) : X ( U, K ) → X ( U, L ). The sets I C 0 := { X ( U, f ) : U ∈ O b C 0 , f ∈ I } and J C 0 := { X ( U, f ) : U ∈ Ob C 0 , f ∈ J } are genera ting sets for cofibr ations a nd trivial cofibrations res p ectively . Consequently , the sets of mo rphisms I C := u ! ( I C 0 ) J C := u ! ( J C 0 ) are generating sets o f cofibr ations a nd triv ial cofibr ations, resp ectively , for the pro jective mo del ca tegory sPSh( C ). Moreover, they consist of finitely presentable ob jects in sPSh( C ) → . The lo c al pr oje ct ive mo del catego ry structure has the s ame co fibrations, i.e., the pro jec- tive cofibrations , and the weak equiv alences are the lo c a l weak equiv a lences. This mo del 10 GEORG E RAPTIS category was shown by Blander [4]. The no ta tion sPSh( C ) pro j will b e use d to denote it. It is clearly a left Bo us field lo caliz a tion o f the pro jective mo del categ ory , so the tr ivial fibrations are the same in both cases . The lo cal pro jective mo del categor y is again pro pe r , simplicial and cofibrantly gener ated. Note that the pro jective mo del catego ry ca n be obtained as a sp ecial case of the lo cal pro jective one by endowing the ca teg ory C with the trivia l top ology . The categor y sMo d R inherits a mo del category structure from sPSh( C ) pro j along the adjunction (cf. [18, L emma 2.2]) R{−} : sP Sh( C ) pro j ⇄ sMo d R : ι. This can b e shown easily using ag ain the standar d metho d of transferr ing a mo del c a tegory structure along a n adjoint pair (se e also Remark 5.8). A ma p f : M → N in sMo d R is a weak equiv a lence (resp. fibra tion) if ι ( f ) is so in s PSh( C ) pro j . This lo cal pr o jective mo del category will be denoted by s Mo d pro j R . Note that the trivial fibrations in sMo d pro j R are exactly the morphisms that define a trivial fibr ation of the asso ciated presheaves of simplicial sets, so they ar e the maps tha t are sectionwise a weak eq uiv alence and a Kan fibration. But a map of simplicial a belia n gro ups is a trivial fibra tion if and only if it is a weak eq uiv alence and an epimorphism. Thus we obtain the following pr o po sition that will b e needed in the pro of of Theorem A. Prop ositio n 3.1. A map f : M → N in sMo d pro j R is a trivial fi br ation iff it is a se ctionwise we ak e quivalenc e and an epimorphism. The mo del ca teg ory sMo d pro j R is prop er , simplicial a nd cofibra n tly genera ted. The simpli- cial structure o n sMo d R is induced b y the s implicial structur e of sPSh( C ). Mor e explicitly , given a simplicial s e t K a nd simplicial R -mo dules M a nd N , the simplicial s tructure is defined by the following ob jects, K ⊗ M := R{ K } ⊗ M M K ( U ) := Ma p sSet ( K, M ( U )) Map( M , N ) := sMo d R (∆ · ⊗ M , N ) where K denotes the co nstant presheaf a t the simplicia l set K . The sets o f morphisms I pro j R := R{I C } J pro j R := R{J C } are generating sets o f c o fibrations a nd tr ivial cofibra tions r e s pec tively . They a lso consis t of finitely prese n table ob jects in sMo d → R . There is also an inje ctive mo del categ ory structure on sPSh( C ) due to Heller [1 3]. The cofibrations and weak e quiv alences are the sectionwise monomorphisms and s ectionwise weak equiv alences respectively . This is a gain independent of the Gro thendiec k top ology on C . In fa c t, this mo del categor y structure is an insta nce of the more genera l injective mo del categor y structure o n a catego ry of diag rams in a combinatorial mo del catego r y [20, Prop ositio n A.2.8.2 ]. It is k nown to b e cofibrantly genera ted, s implicial and prop er. The asso ciated lo cal ho motopy theory corres p onds to the lo c al inje ct ive mo del c ategory structure on s PSh( C ) due to J ardine [1 6], [17]. The cofibrations a r e the monomor phisms and the weak eq uiv alences and the lo ca l weak equiv alences. As a consequence, it is a left Bousfield lo calization o f the injective mo del categor y a t the lo cal weak equiv a lences. It is also co fibr antly generated, simplicial a nd prop er . This mo del catego r y will b e denoted here by sPSh( C ) inj . Aga in the injective mo del catego ry is a n instance of the lo cal injective one if the catego ry C is endow ed with the trivial top ology . SIMPLICIAL PRESHEA VES OF COALG EBRAS 11 The ca tegory of simplicia l R -mo dules inher its als o a “lo cal injectiv e” mo del catego ry structure from sPSh( C ) inj using simila r metho ds as be fore in the pro jective case. T his is again cofibrantly generated, simplicial and prop er [18]. It will b e denoted her e by sMo d global R following Jardine’s terminology of glob al fi br ations . This model categ ory is different to the lo cal injective mo del category sMo d inj R that we dis cuss in s e ction 5. They hav e the same class of weak equiv alences, but the co fibrations o f sMo d inj R are exa ctly the monomorphisms . Moreov er, sMo d inj R is als o cofibrantly g enerated, s implicial and prop er (see Theorem 5.7). Let us finally note the dir ections of the v ario us left Q uillen functors, sPSh( C ) pro j 1 / / R{−}   sPSh( C ) inj R{−}   sMo d pro j R 1 / / sMo d global R 1 / / sMo d inj R 4. Proof of Theorem A W e will apply the following theorem a bo ut co mb inator ia l mo del categories . Theorem 4.1. L et C b e a lo c al ly pr esent able c ate gory, W a class of m orphisms of C and I a set of morphisms. Then the classes of morphisms W , Cof(I) and (Cof(I) ∩ W ) − inj define classes of we ak e quivalenc es, c ofibr ations and fibr ations for a c ofibr antly gener ate d mo del c ate gory struct ur e on C if and only if the fol lowing c onditions ar e satisfie d: (i) W satisfies t he 2-out -of-3 pr op erty, (ii) I − inj ⊆ W , (iii) Cof(I) ∩ W is close d u nder tr ansfi n ite c omp ositions and pushout s , (iv) the (ful l su b c ate gory sp anne d by the) class W is ac c essible and ac c essibly emb e dde d in C → . Pr o of. Every a ccessible, accessibly embedded s ubca tegory of a lo ca lly presentable catego ry is co ne - reflective by [1, Theo rem 2.53]. Hence it satisfies the so lution set co ndition at every morphism. Moreov er, it is clo sed under retracts [3 , Pr o po sition 1.19 ]. Then the sufficiency of the conditions follows from J. H. Smith’s r ecognition theor em [3, Theo rem 1.7 ]. The key part of the pro o f of [3, Theorem 1.7] is to us e the fact that W is cone-reflec tive a t I in o r der to obtain a g enerating set J for Cof(I) ∩ W , se e [3, Lemma 1 .9]. The rest of the pro o f is an easy applica tion of the more sta ndard reco g nition theor em for co fibrantly gener ated mo del categorie s, see e.g. [15, Theor em 2.1.19]. (This is a small simplification of the pr o o f given in [3] that we lear ned from G. Maltsiniotis.). The necessity of (i),(ii) a nd (iii) is obvious. Pro ofs of the necessity of (iv) can b e found in [20, Co r ollary A.2.6.6], [26, Theo rem 4.1 ] and [25].  The p ow er of this theorem, when compar e d to the standard r ecognition theorem fo r cofibrantly genera ted mo del categ ories (e.g. see [15, Theorem 2.1.19 ]), is that it do es not assume as given a s et of g e nerating trivial c o fibrations (but it do es not produce a very explicit one either), but rather its exis tence is essentially a co nsequence o f the a ccessibility prop erties of the cla ss of weak equiv a lences. Conditio n (iv) should b e no rmally the most difficult to verify in the applica tio ns of the theor em. The following prop osition will b e useful. 12 GEORG E RAPTIS Prop ositio n 4.2. Le t F : C → D b e an ac c essible functor and D ′ b e an ac c essible and ac- c essibly emb e dde d sub c ate gory of D . Then F − 1 ( D ′ ) is an ac c essible and ac c essibly emb e dde d sub c ate gory of C . Pr o of. See [1, Remark 2.5 0].  W e pro c eed to the pro o f o f Theo rem A with the v erificatio n of the conditions (i)-(iv). Condition (i) is obviously satisfied by the clas s W R . Prop ositio n 4 .3. The class of we ak e quivalenc es W R is ac c essible and ac c essibly emb e dde d in sCoalg → R . Pr o of. By Theore m 4.1, the cla s s of lo cal weak equiv alences in sPSh( C ) is accessible and accessibly embedded in sPSh( C ) → . The forge tful functor sCo alg R → sPSh( C ) is accessible being the co mpo sition of the forg etful left adjoint Φ : sCo alg R → sMo d R (whic h preserves all colimits) followed by the for getful right adjoint ι : sMo d R → sPSh( C ) (which preserves directed colimits). It follows that the class W R is accessible and accessibly embedded in sCoalg → R by Pr op osition 4.2.  Let κ b e a r egular cardinal s uch that sCoa lg R is lo ca lly κ - presentable. F o r example, this can b e the c ho ice of ca rdinal fr om Theor em 2.2. Also as s ume that κ > max { car d( M or C ) , ℵ 0 } . Let I denote the se t of Φ-monomo rphisms b etw een κ -presentable ob jects in sCoalg R . Recall that a map f : A → B in s Coalg R is called a Φ-monomor phism if the ma p b etw een the underlying simplicial R -mo dules is a monomor phis m. It is clear that every morphis m in Cof( I ) is a Φ- monomorphism. So it follows that the class Cof( I ) ∩ W R is closed under pushouts a nd transfinite comp ositions since they are created in sMo d inj R . So it remains to verify condition (ii) of Theor em 4 .1. Fir st we prove the following key lemma. Lemma 4. 4. Every map f : A → B in sCoa lg R admits a factorization f = p i in sCoa lg R such that the fol lowing ar e satisfie d: (a) i is a Φ -monomorphism, (b) the domain of p is κ -pr esentable if b oth A and B ar e κ -pr esentable, (c) Φ( p ) is a trivial fibr ation in sMo d pro j R . Pr o of. Let R denote the cons tant simplicial R -coalg ebra at R viewed a s an R -coa lg ebra. This is the same as R{ ∆ 0 } . (Recall that ∆ n denotes the constant simplicial presheaf whose v alue is the standar d n -simplex everywhere.). The fold map R ⊕ R → R in sCoalg R admits a factoriza tion as required, induced b y the facto rization in sP Sh( C ), ∆ 0 ⊔ ∆ 0 i 0 ⊔ i 1 − → ∆ 1 p → ∆ 0 and applying the functor R{− } : s P Sh( C ) → sCoalg R (This functor will b e discusse d in detail in sec tio n 5.1). In the general case of a map f : A → B , define the mapping cylinder M ( f ) in the standard way by a pusho ut dia gram in sCo alg R , A ∼ = A ⊗ R 1 ⊗R{ i 0 } / / f   A ⊗ R{ ∆ 1 }   B j / / M ( f ) The mapping c y linder construction y ields a factoriz ation of f : A → B as A i → M ( f ) p → B SIMPLICIAL PRESHEA VES OF COALG EBRAS 13 in the us ual wa y . The ma p Φ( i ) is clearly a monomorphism o f s implicial pres heav es , so (a ) is satisfied. If A is κ - pr esentable then so is A ⊗ R{ ∆ 1 } , and therefore (b) is also satisfied. Note that p is a split epimorphism since pj = 1 B . The map Φ(1 ⊗ R{ i 0 } ) : Φ( A ) → Φ( A ⊗ R{ ∆ 1 } ) ∼ = Φ( A ) ⊗ R{ ∆ 1 } is a sectionwise w eak equiv alence and a mono morphism. Since the pushout squa r e above also defines a pushout of s implicia l R -mo dules, it follows that Φ( j ) is a sectionwise weak eq uiv - alence. Hence Φ( p ) is a sectionwise epimorphism and weak equiv alence, so by P r op osition 3.1, it is a trivia l fibration in sMo d pro j R as requir ed by (c).  Prop ositio n 4.5. I − inj ⊆ W R . Pr o of. Let q : X → Y b e a map in I − inj. It suffices to show that there is lift to every diagram in s Mo d R · t   / / Φ( X ) Φ( q )   · / / Φ( Y ) where t is chosen from the gener ating s e t for co fibrations of the mo del categor y sMo d pro j R as discussed in s ection 3. By the choice of the ca rdinal κ , the map q is a κ -directed c olimit of κ -presentable o b jects in sCoa lg → R . Since the morphism t is finitely presentable in sMo d → R and Φ pr e serves colimits, there is a factor ization · / / t   Φ( A ) Φ( f )   Φ( α ) / / Φ( X ) Φ( q )   · / / Φ( B ) Φ( β ) / / Φ( Y ) where A and B are κ - presentable and the right-hand side square is the image o f a comm u- tative squar e in sCoa lg R under Φ. If w e factor iz e the map f as in Lemma 4 .4, we obtain a commutativ e diagram Φ( A ) Φ( i )   Φ( α ) / / Φ( X ) Φ( q )   · > > } } } } } } } } } / / t   Φ( C ) Φ( h 2 ) ; ; Φ( p )   · h 1 > > / / Φ( B ) Φ( β ) / / Φ( Y ) By the assumption that Φ( p ) is a tr ivial fibr a tion in sMo d pro j R , it follows that there exists a lift h 1 as indica ted in the diagram. By the co nstruction o f the facto r ization in Lemma 4.4, C is κ -presentable, so the map i is in I . Therefor e there is a morphism h 2 : C → X such that the following dia gram in sCo alg R commutes A i   α / / X q   C h 2 > > ~ ~ ~ ~ ~ ~ ~ β p / / Y 14 GEORG E RAPTIS Then comp ositio n h = Φ( h 2 ) h 1 provides a lift to the original diagr am and hence the result follows.  By Theo r em 4.1, it follows that the cla sses of w eak equiv alences W R and cofibr ations Cof( I ) define a co fibrantly ge ner ated mo del categor y s tr ucture on sCoalg R . It is left pro per bec ause sMo d inj R is left prop er (see Theorem 5 .7) and the forg etful functor Φ : sCo a lg R → sMo d inj R preserves pushouts. The simplicia l structure o n sCoalg R is defined as follows. F or every simplicia l set K a nd simplicial R -coalg ebra A , the tensor str ucture is induced by sMo d R , i.e., (4.1) K ⊗ A := R{ K } ⊗ A. More explicitly , the n -simplices of ( K ⊗ A )( U ), U ∈ Ob C , is the tensor pro duct R ( U )- coalgebr a R ( U ) { K n } ⊗ R ( U ) A ( U ) n where the coalgebraic struc tur e o n the free R ( U )-mo dule R ( U ) { K n } is induced by the canonical maps ∆ : K n → K n × K n and K n → ∗ . This defines a functor sSet × sCoalg R → sCoalg R which preserves colimits in b oth v ariables. Hence it extends to a n adjunction of t wo v a riables in the sense of [15, Definition 4.1 .1 2] by the sp e cial adjoint functor theo rem. Then it suffices to show that given a monomorphism i : K ֒ → L b etw een finitely pres ent able simplicial s e ts and a Φ-monomo rphism f : A → B in I , then the morphism of the pusho ut pro duct i  f : K ⊗ B ∪ K ⊗ A L ⊗ A → L ⊗ B is a Φ-monomor phism and it is trivia l if either i or f is trivial. Both the domain and co domain of i  f ar e aga in κ -presentable. Since Φ preserves pushouts, the morphism Φ( i  f ) is isomorphic to i  Φ( f ) : K ⊗ Φ( B ) ∪ K ⊗ Φ( A ) L ⊗ Φ( A ) → L ⊗ Φ( B ) . This is a mono morphism and it is trivia l if either i or Φ( f ) is trivia l b ecause sMo d inj R is a simplicial mo del categ ory by Theo rem 5.7. Hence the simplicia l structure of (4.1) ma kes the mo del c ategory sCoalg R int o a s implicial mo del categ ory . This conc ludes the pro of of Theo rem A. Remark 4.6. It is not clear whether ther e is a mo del catego r y structure such that the cofibrations a re all the Φ-monomo rphisms. The standard argument to show tha t this class is cofibrantly g e nerated (e.g. see [3, P rop osition 1.12]) do es not apply her e s ince sCoalg R is not closed under the intersection of subob jects in sMo d R bec ause ⊗ : Mo d R × Mo d R → Mo d R is not left exact in g eneral. F or the same reaso n, the tensor pro duct of simplicial R -coalg ebras, which by Theo rem 2.4 g ives the pro duct functor in sCoalg R , do es not define a monoidal mo del categ o ry in general. 5. Comp arison with simplicial preshea ves and simpl icial R -modules 5.1. The homotopy theory of simplicia l pr esheav es a nd simplicial R -coa lg ebras are linked by the functor of simplicial R -chains R{−} : sP Sh( C ) → sCo alg R . This ta kes a simplicial presheaf X : C op → sSet to the s implicial R -coalg ebra R{ X } whos e underly ing R -mo dule is the fr ee R -module on X (denoted also by R{ X } ) a nd the coa lg ebraic structur e is induced by the cano nical maps X ∆ → X × X SIMPLICIAL PRESHEA VES OF COALG EBRAS 15 X → ∆ 0 . More explicitly , R{ X } : ∆ op → Coalg R is defined po int wise, by R{ X } n ( U ) = R ( U ) { X ( U ) n } with the coa lg ebraic structure induced similarly p oint wise. This functor ha s a right adjoint ρ : sCo alg R → sPSh( C ) inj which is defined p oint wis e as follows: given A : ∆ op → Coa lg R , then ρ ( A ) n ( U ) = Co alg R ( U ) ( R ( U ) , A n ( U )) defines a simplicial presheaf ρ ( A ) : ∆ op → PSh( C ). This adjunction is induced by the analogo us adjunction R{− } : PSh( C ) ⇄ Coalg R : ρ fro m section 2. Prop ositio n 5. 1. The adjunction R{− } : s P Sh( C ) inj ⇄ sCoa lg R : ρ is a Quil len adjunc- tion. Pr o of. W e check that R{−} pr e serves cofibratio ns and trivial cofibrations. A generating set I inj for the class of mono morphisms in sPSh( C ) is given by a ll monomo rphisms betw een κ -presentable ob jects. This is a c onsequence of the gener al statement of [3, Pr op o sition 1.12] combined with some basic pr op erties of the rank o f presentabilit y of presheaves, see e.g. [1, Example 1.31]. The image of a κ -presentable ob ject under R{−} is again κ -pre sent able. Thu s the mono morphisms in I inj maps to (generating) c o fibrations in s Coalg R . It follows that R{−} preser ves c o fibrations. It also preserves tr ivial cofibra tions b ecause it pres e r ves all weak eq uiv alences (cf. [18, L emma 2.1]).  There is a refinement of the Quillen adjunction ab ov e that offers a more pr e cise com- parison. This is obta ine d b y lo ca lizing the category of simplicial presheaves at the c la ss of R -ho mo logy equiv alences, i.e., the morphisms f : X → Y such that R{ f } is a weak equiv alence. Note that every lo cal weak equiv alence is an R -homology equiv alence (e.g. s ee [18, Lemma 2 .1]). The class of R -ho mo logy equiv alence s is the cla ss o f weak equiv alences for a ne w mo del categor y structure on sPSh( C ) which c a n b e obta ined as a left Bousfield lo- calization o f s PSh( C ) inj . F o r background material ab out the Bousfield lo calization of mo del categorie s, see Hir schhorn [14]. Theorem 5.2. The left Bousfield lo c alization L R sPSh( C ) inj of t he mo del c ate gory sP Sh( C ) inj at the class of R -homolo gy e quivalenc es exist s , and R{−} : L R sPSh( C ) inj ⇄ sCoalg R : ρ is a Quil len adjunction. Pr o of. The cla ss of R -ho mology equiv alences is the inv e rse imag e o f W R , which is accessible and a ccessibly embedded in sCo alg → R by Pro po sition 4.3, by the accessible functor R{−} → : sPSh( C ) → → sCoalg → R . There fo re it is is acces sible and accessibly embedded in sP Sh( C ) → by Prop os ition 4.2. The ex istence of the Bo usfield lo caliza tion follows fr o m Theorem 4.1: conditions (i), (ii) and (iv) are sa tis fie d and (iii) is an easy co nsequence of the co r resp onding condition fo r sCoalg R and P rop osition 5.1. Then it is clea r that R{−} : L R sPSh( C ) inj → sCoalg R is a le ft Quillen functor .  As a co nsequence, there is a derived adjunction L R{−} : Ho(L R sPSh( C ) inj ) ⇄ Ho(sCoa lg R ) : R ρ betw een the R -lo cal homotopy categ o ry of simplicial pres heav es a nd the homotopy ca tegory of simplicial R -coalgebras . 16 GEORG E RAPTIS 5.2. Assume that R ( U ) has no non-trivial idempotents for all U ∈ O b C . In this ca s e, the unit transfo r mation 1 → ρ R{−} is a natural isomor phism. The a djunction ( R{−} , ρ ) can be used to pro duce a new mo del categor y structure o n sCoalg R . W e will need the following elementary lemma. Lemma 5.3. L et i : X ֒ → Y b e a monomorphism in sP Sh( C ) . If (5.1) R{ X } / / R{ i }   A j   R{ Y } / / C is a pushout squar e in sCoalg R , then the adjoint squar e X / / i   ρ ( A ) ρ ( j )   Y / / ρ ( C ) is a pushout in sPSh( C ) . Pr o of. Since pushouts ar e computed p oint wise, it suffices to c heck this in the case of a pushout diagram (5.1) in Co a lg R where R is a sing le comm utative ring and R {−} : Se t → Coalg R . In this cas e, the R - coalgebr a C is iso morphic to the direct s um o f the R - c oalgebra A with ⊕ α ∈ Y − X R α where each R α is iso morphic to R r e garded as an R - coalgebr a . Then it is easy to chec k that the R -p oints of C is the disjoint union of the R -p oints o f A with the set Y − X .  A morphism f : A → B in sCo alg R is called a ρ -w eak equiv alence (resp. ρ -fibra tion) if the map ρ ( f ) is a lo cal weak equiv a lence (r e sp. global fibra tion, i.e., a fibration in the mo del category sP Sh( C ) inj ). Let W ρ R and Fib ρ R denote the cla sses of ρ -weak equiv alences and ρ - fibrations r esp e c tiv ely , and let Co f ρ R denote the class of ρ -c ofibrations, tha t is, morphisms that have the left lifting pro pe r t y with res pect to all maps that are b oth ρ -weak equiv alences and ρ -fibrations . Theorem 5.4. Ther e is a pr op er, simplicial, c ofibr antly gener ate d mo del c ate gory sCo alg ρ R whose underlying c ate gory is sCoalg R and the we ak e quivalenc es, fibr ations and c ofibr ations ar e define d by the classes W ρ R , Fib ρ R and Cof ρ R r esp e ctively. Mor e over, the adjunction R{−} : sP Sh( C ) inj ⇄ sCoalg ρ R : ρ is a Quil len e quivalenc e. Pr o of. Using the sta ndard metho d of tr ansferring a mo del categor y struc tur e along an ad- junction [14, Theor em 11.3.2 ], it suffices to chec k that for every pushout diagra m in sCoa lg R R{ X } / / R{ i }   A j   R{ Y } / / C where i : X ∼ ֒ → Y is a trivia l cofibration, then the morphism j is ρ -weak equiv alence. But this follows directly b y Lemma 5.3. F o r genera ting sets of cofibratio ns and trivial cofibr ations, SIMPLICIAL PRESHEA VES OF COALG EBRAS 17 we ca n choose R{I inj } and R{J inj } resp ectively , wher e I inj and J inj denote g e nerating sets of sP Sh( C ) inj . W e s how that sCoa lg ρ R is a simplicia l mo del catego ry . Let I = { ∂ ∆ n ֒ → ∆ n | n ≥ 0 } and J = { Λ n k ֒ → ∆ n | 0 ≤ k ≤ n } be the standard generating sets of co fibrations and tr ivial cofibrations of sSet. The simplicia l structure is the sa me as tha t o f Theo rem A, i.e., it is defined by ⊗ : sSet × sCo a lg ρ R → sCoalg ρ R K ⊗ A = R{ K } ⊗ A. By [15, Corolla r y 4.2.5], it suffices to show tha t the pusho ut pro ducts in I ✷ R{I inj } are ρ - cofibrations and thos e in J ✷ R{I inj } a nd I  R{J inj } a re ρ -weak equiv alences . The pushout pro duct of i : K → L with f : A → B is the canonic a l morphism i  f : K ⊗ B ∪ K ⊗ A L ⊗ A → L ⊗ B . The pushout pr o duct of i : K ֒ → L in s Set with R{ j } : R{ X } ֒ → R{ Y } is R{ i  j } , where i  j denotes the pusho ut pro duct o f i a nd j with resp ect to the simplicial s tructure of sPSh( C ). T hus the required r esult follows from the fact that sPSh( C ) inj is a s implicial mo del categ o ry . The mo del categ ory sCo alg ρ R right prop er b ecause s PSh( C ) inj is right pro per a nd ρ pre- serves pullbac ks. Left prop erness follows easily from Lemma 5.3 and the fact that sPSh( C ) inj is left prop er. Lastly we show that R{−} is a left Q uillen equiv alence. It suffices to check that the derived unit tr ansformation is a natural iso morphism. This holds b ecause R{− } ⊣ ρ is a coreflection and ρ pr eserves the weak equiv a le nces by definition. More ex plicitly , the der ived unit tra nsformation of the Quillen adjunction is defined as follows: given an ob ject X o f sPSh( C ) inj , let g : R{ X } ∼ → X f be a functor ial fibrant repla c emen t in sCoalg ρ R obtained by a n a pplication of the small-ob ject argument to the se t o f triv ial cofibratio ns R{J inj } . The n the derived unit trasformation a t X can b e repr esented by the ma p X ∼ = ρ R{ X } ρ ( g ) − → ρ ( X f ) . By Lemma 5.3, this is in J inj -cell, so in particular it is a lo cal weak equiv a lence.  Prop ositio n 5.5. The identity functor 1 : sCoa lg ρ R → s Coalg R is a left Quil len functor b etwe en the mo del c ate gories of The or em 5.4 and The or em A. Pr o of. Let I inj be the g enerating se t o f mo nomorphisms in sP Sh( C ) that co nsists o f the monomorphisms b e t ween κ -pres ent able ob jects and J inj a generating set of trivial cofibra- tions. The n R{ I inj } a nd R{J inj } a re genera ting se ts of cofibratio ns and tr ivial co fibrations for sCoa lg ρ R . Every morphism in R{I inj } is a Φ-monomo rphism b etw een κ -pr esentable ob- jects and therefore the identit y functor preserves cofibra tio ns. Every mo rphism in R{J inj } is a lo cal weak equiv a lence. Hence it follows that 1 : sCoalg ρ R → sCoalg R is a left Quillen functor.  Remark 5.6 . By Lemma 5.3, it is eas y to see that the cofibrant ob jects in sCoalg ρ R are exactly the ob jects of the form R{ X } for some simplicial presheaf X . Thus a co fibrant replacement functor in sCoalg ρ R is given b y the co unit transformation of the adjunction ( R{−} , ρ ), i.e., the natural mor phism R{ ρ ( A ) } → A is a cofibr ant replace men t of the simplicial R -coalg ebra A . 18 GEORG E RAPTIS 5.3. By P rop osition 2.1, the forgetful functor Φ : sCoalg R → sMod R is a left a djoint . It is a left Quillen functor as long as there is a mo del c a tegory s tructure on sMo d R where the weak equiv a lences are the lo cal weak equiv a lences of the inderlying s implicial presheaves and there are eno ugh cofibr ations. The following theorem is undoubtly well-known to the exp erts but we were not able to find an exact refer ence for it in the litera tur e. Theorem 5. 7. Ther e is a pr op er, simplici al, c ofibr antly gener ate d mo del c ate gory st ructur e on sMo d R wher e the c ofibr ations ar e the monomorphisms and the we ak e quivalenc es ar e the lo c al we ak e qu ivalenc es of the underlying simplicial pr eshe aves. Pr o of. The pro of will follow the metho d of Theorem 4.1. Let us denote again by W R the cla ss of loca l w eak equiv alences in s Mo d R . The cla ss of loca l weak equiv alences in sPSh( C ) is access ible and access ibly embedded in s PSh( C ) → inj by Theorem 4.1. The forgetful functor ι : sMo d R → sPSh( C ) is accessible, therefor e condition (iv) of Theor em 4.1 ho lds by Prop ositio n 4.2. The clas s o f monomorphisms Mo no in sMo d → R is cofibra n tly genera ted by a set of mo nomorphisms. This is more g enerally true in every Grothendieck a belia n categor y , see [3, P rop osition 1.12, Remar k 1.13]. Next we show that the class Mono ∩ W R is closed under pushouts and transfinite comp ositions. The closur e under transfinite comp ositions is obvious (since those ca n be computed in sP Sh( C ) inj ), so it suffices to show that for every pushout squa re A j   / / X f   B / / Y where j ∈ Mono ∩ W R , then f ∈ W R . Let ℘ : Sh( B ) → Sh( C ) b e a Bo ole an lo caliza tion of Sh( C ). Recall the definition of the lo cal w eak equiv alences from section 3 a nd the terminology used there. There is a pushout diagram ℘ ∗ L 2 ( A )( b ) j ( b )   / / ℘L 2 ( X )( b ) f ( b )   ℘ ∗ L 2 ( B )( b ) / / ℘ ∗ L 2 ( Y )( b ) for a ll b ∈ B . Since ℘ ∗ and L 2 are geometric mor phis ms, they preserve mo no morphisms, so j ( b ) is a mo nomorphism. It is also a weak equiv ale nce o f s implicial sets by assumption. Then it follows that f ( b ) is also a weak equiv a lence a nd s o co nditio n (iii) of Theore m 4.1 follows. It rema ins to verify co ndition (ii) of Theo r em 4.1. Let I pro j R = R{ I C } b e the g ener- ating set of cofibrations for s Mo d pro j R as defined in s ection 3 . Clea rly I pro j R ⊆ Mono , so Mono − inj ⊆ I pro j R -inj. But if f ∈ I pro j R -inj then f is a lo cal weak equiv alence, so (ii) fol- lows. Hence by Theorem 4.1, there is a co fibrantly g enerated mo del categ ory , de no ted by sMo d inj R , as r equired. This mo del categor y is right prop er b e cause sPSh( C ) inj is rig ht prop er and the for getful functor ι : s Mo d inj R → sPSh( C ) inj is a right Quillen functor . The pr o of that it also left prop er is similar with the a rguments ab ov e based on Bo olea n lo caliza tion. It remains to show that the mo del catego ry is also simplicia l. Le t i : K ֒ → L be an inclusion o f simplicial s e ts and p : M → N a fibration in sMo d inj R . Since p is also a fibration SIMPLICIAL PRESHEA VES OF COALG EBRAS 19 in sMo d global R , which is a simplicial mo del categor y by [1 8, Lemma 2.2], the canonical map M L → N L × N K M K is a Ka n fibr ation a nd it is triv ial if either i or p is trivia l. This concludes the pro of of the theorem.  Remark 5.8. It is clear from the pro of that for e very set of monomo r phisms I in sMo d R such that I − inj ⊆ W R , there is a left prop er , simplicial, cofibrantly gene r ated mo del category structure on sMo d R with class of cofibrations Cof(I) and weak equiv alences W R . It is also r ight prop er if R{ i } : R{ X } → R{ Y } is in Cof(I) for every pro jective co fibration i : X → Y of simplicial presheav es. The following pr o po sition is now o b vious. Prop ositio n 5.9. The for getful functor Φ : sCoalg R → sMo d inj R is a left Quil len fu n ctor. 6. Proof of Theorem B 6.1. W e r emind the rea der o f certain facts a bo ut the structure of coalge bras over a p erfect field. F or mor e details, see [12], [27]. Let F b e a per fect field. An F -co algebra is called simple if it has no no n-trivial sub- coalgebr as. Every simple F - coalgebr a is finite dimensiona l and the dua l F -algebr a is a finite field extension of F . T he ´ etale part ´ E t ( A ) of an F -coalgebr a A is the sum of all the simple sub c oalgebra s of A . This sum is known to b e direct, see [27, p. 166 ]. Accor ding to the de c omp osition t he or em (see [8, p.42], [12]), the inclusion ´ E t ( A ) ⊆ A is a na tural s plit monomor phism o f c o algebra s . If F is algebra ic a lly clo sed, then F is the unique s imple F -co algebra up to isomor phism. The ´ etale pa rt of an F -coa lgebra A in this case can b e identified with the canonical counit map F { ρ ( A ) } → A , that is , there is a natural isomor phism (6.1) F { ρ ( A ) } ∼ = → ´ E t ( A ) . Note of co urse that an a rbitrary change o f fields F ⊆ K may give rise to K -p oints of A ⊗ F K that are not induced by F -p oints o f A . But any K -p oint of A ⊗ F K is already an F -p oint of A ⊗ F F where F ⊆ F deno tes the a lgebraic clo sure (see [22, Sec tio n 3]). Therefor e the isomorphism (6.1) is natural with res p ect to field extensions o f a lgebraically closed fields, that is, if F ⊆ K are alg ebraically closed fields and A is an F -coa lgebra, then there is a natural bijection ρ ( A ) ∼ = → ρ ( A ⊗ F K ). The isomorphism (6.1) can b e extended to a descr iption o f the ´ etale part of the F - coalgebr a A in the ge neral ca se where F is a p erfect field. This is essentially a c o nsequence o f Galois theory . Let F b e the algebra ic closur e o f F and G the Galois gr o up. The Ga lo is group G is r e garded as a profinite gro up, so a G -a ction is always understo o d to b e contin uous. Recall that a G -action is contin uous if and o nly if every element ha s a finite orbit. Let A := A ⊗ F F denote the a sso ciated F -c oalgebra . The set of F -p oints of A generate the ´ etale part o f A by (6.1) ab ov e. More over, it is naturally a G -set. More explicitly , the G -actio n is defined as fo llows: given an F -p oint f : F → A a nd g ∈ G , then ( g · f )( x ) = (1 ⊗ g ) f g − 1 ( x ) . 20 GEORG E RAPTIS In other w ords, there is a commutativ e diagra m F f / / g   A 1 ⊗ g   F g · f / / A The a s so ciated F -coalg ebra F { ρ ( A ) } is also natur ally endow ed with a G -action. The G - action is defined by the fo rmula g ( X x i f i ) = X g ( x i )( g · f i ) . This makes the canonical ev alua tio n ma p F { ρ ( A ) } → A X x i f i 7→ X f i ( x i ) inv aria n t under the G -a ction. The ´ etale part ´ E t ( A ) o f A is naturally is omorphic to the G -inv aria n ts of F { ρ ( A ) } , i.e., ther e is a natural isomor phism (see [12, P rop osition 2.8]) (6.2) F { ρ ( A ) } G ∼ = ´ E t ( A ) . More g enerally , if X is a G -set, the G -inv ar iants of the F -coalgebra F { X } form naturally an F - coalgebr a . Let Set( G ) denote the categ ory of sets with a co n tinuous G -action. This is a Grothendieck top os, e.g. s ee [21, p. 5 96]. Ther e is a well-defined functor F {−} G : Set( G ) → Coa lg F . This has a r ight adjoint that is defined o n o b jects by the formula ρ G ( A ) = Coalg F ( F , A ⊗ F F ) . 6.2. W e ca n now pr ov e Theor em B . Let F b e a preshea f o f algebr aically closed fields on C . The main argument of the pro of is in the following pro p os ition. Prop ositio n 6.1 . The functor ρ : sCoalg F → sPSh( C ) inj sends we ak e quivalenc es to F - homolo gy e quivalenc es. Pr o of. Let f : A → B b e a weak equiv alence of simplicia l F -co algebra s . The natur a lit y of the splitting of the ´ etale par t of coalge br as is resp ected along ex tensions of alg e br aically closed fields, so it follows that the map F { ρ ( f ) } : F { ρ ( A ) } → F { ρ ( B ) } is a retract o f f in the catego ry of mor phis ms b et ween simplicial F - coalgebra s. Since weak equiv alences a re closed under retracts, the r e quired result follows.  T o finish the pro of of Theorem B, it suffices to s how that the natura l der ived unit map X → R ρ ( F { X } ) is a natura l is omorphism in the F -lo ca l homotopy categ ory . This follows directly from the fact that both functors of the Quillen adjunction (6.3) F { −} : L F sPSh( C ) inj ⇄ sCoalg F : ρ SIMPLICIAL PRESHEA VES OF COALG EBRAS 21 preserve the weak equiv alences , so the derived unit transfor mation is induced b y the unit transformatio n of the co r eflection (6 .3). More explicitly , let F { X } ∼ → F { X } f be a func- torial fibrant replacement in sCoalg F . The derived unit map of the Quillen adjunction can be represented by the natura l map X ∼ = ρ ( F { X } ) → ρ ( F { X } f ) . By P rop osition 6.1, this is a n F -homolo gy equiv alence, s o an is omorphism in the homotopy category of L F sPSh( C ) inj . This completes the pro of of Theo r em B. The following is an immediate c orollar y . Corollary 6.2. L et X and Y b e simplicial pr eshe aves in sPSh( C ) inj and F b e a pr eshe af of algebr aic al ly close d fields. Then X ∼ = Y in Ho(L F sPSh( C ) inj ) if and only if F { X } ∼ = F { Y } in Ho(sCoalg F ) . 6.3. Let F b e the constant presheaf a t a per fect field F . Theorem B together with the isomorphism (6 .2 ) can b e used to give a nice description of the derived unit transfor ma tion of (6.3) in this case. The main idea is aga in based on the na tur al splitting of the ´ etale part of an F -coalg ebra, but now this can b e r elated to the G -inv ariants o f the F -points ra ther than with the F -p oints directly . This brings the actio n of the Galois gro up int o the picture. The arg umen ts are completely a nalogous to [1 2, pp.5 41-54 3 ], so we only sketc h the neces sary details. Let F denote the a lgebraic clos ure of F and G the profinite Galo is gro up. Let sPSh( C , G) denote the catego ry o f simplicia l presheaves of G -sets. Say that a morphism b etw e e n sim- plicial pr esheav es of G -sets is a lo c al we ak e quivalenc e (resp. c ofibr ation ) if the mor phism of the underlying simplicia l pr esheav es (of sets), b y forgetting the G -ac tion, is a lo cal weak equiv alence (r e sp. monomorphism). Theorem 6.3. The c ate gory sPSh( C , G) to gether with t he classes of lo c al we ak e quivalenc es and c ofibr ations define a c ombinatorial mo del c ate gory. Pr o of. The categ ory sPSh( C , G) can b e equiv ale ntly viewed a s the catego ry of presheaves of simplicial G -sets. The categor y of simplicial G -sets, denoted by sSet( G ), has a combinatorial mo del categor y structure where the cofibr a tions are the monomo rphisms and the weak equiv alences are the weak eq uiv alences of the underlying simplicial sets [11]. Then ther e is an injective mo del categor y struc tur e on sP Sh( C , G) where the cofibrations and the weak equiv alences are defined sectionwise. T his is again a combinatorial mo del category , see [20, Prop os itio n A.2 .8.2]. The required model categ ory will b e obtained as a left Bousfield lo calization of this injective mo del category by an applicatio n of Theo rem 4.1. W e check that the conditions ar e satisfied: (i) and (ii) are obvious. Let i : ∗ → G denote the o b vious inclusion and i ∗ : sPSh( C , G) → sP Sh( C ) be the for getful functor. Condition (iii) holds bec ause i ∗ preserves mono morphisms and pushouts. F or (iv), note that the class of lo c al weak equiv a lences in sPSh( C , G) → is the inv er se image o f the c la ss of lo cal weak equiv alences in sPSh( C ) under the access ible functor i ∗ : sPSh( C , G) → sP Sh( C ). Since the class o f lo cal weak e quiv alences in sPSh( C ) is acces sible and accessibly embedded (by Theorem 4.1), so is a ls o the class of lo cal weak equiv alences in sPSh( C , G) → by Prop osition 4 .2. Hence the conditions of Theorem 4 .1 are s atisfied and so the result follows.  This mo del categor y will b e denoted by sPSh( C , G) inj . Let F denote the c o nstant presheaf at F . The next pro po sition s hows that the comparis o n Q uillen adjunction b etw een L F sPSh( C ) inj and sCo alg F factors thro ugh the mo del category sPSh( C , G) inj . 22 GEORG E RAPTIS Prop ositio n 6.4. Ther e ar e Quil len adjunctions p ∗ : sPSh( C ) inj ⇄ sPSh( C , G) inj : ( − ) G F { −} G : sPSh( C , G) inj ⇄ sCoalg F : ρ G . Pr o of. The left adjoint p ∗ is the pullback functor induced by the unique functor G → ∗ , i.e., p ∗ ( X ) is the s implicia l presheaf X endow ed with the triv ial G -action. It is clea r that p ∗ preserves cofibratio ns and trivial cofibra tions, so it is a left Quillen functor. The right adjoint is the limit functor whic h, in this case, is just the functor of G -fixed p oints. F or a simplicial preshea f of G -sets X , the s implicial F -coa lgebra F { X } G is defined sectionwise by the formula F { X } G ( U ) n = F { X ( U ) n } G . The rig h t adjoint ρ G is defined b y the fo rmula ρ G ( A )( U ) n = Coalg F ( F , A ( U ) n ⊗ F F ) . F {−} G clearly pre serves cofibra tions. Moreover, there is a natura l isomorphism (see the pro of of [12, Lemma 4.3]), (6.4) F ⊗ F F { X } G ∼ = → F { X } from which it follows that F { −} G sends F -homo logy equiv alences to w eak eq uiv alences. In particula r, it preserves trivia l cofibrations and so it is a left Quillen functor.  Let L F sPSh( C , G) inj denote the left Bous field lo calization o f sPSh( C , G) inj at the cla s s of F -homolog y equiv alences of the underlying s implicial presheav es. The pr o of that this Bousfield lo calization exists is simila r to the pr o of of Theo rem 5 .2. Moreover, there is an induced Quillen adjunction (cf. [12, Le mma 4.3]), (6.5) F {−} G : L F sPSh( C , G) inj ⇄ sCoalg F : ρ G and an a sso ciated derived adjunction (cf. [12, Pro po sition 4.4 ]), L F {−} G : Ho(L F sPSh( C , G) inj ) ⇄ Ho(sCoa lg F ) : R ρ G . Note tha t the natural isomor phism (6.4) s hows that the a djunction (6.5) is a co r eflection, i.e., the unit transfor mation of the adjunction is a natura l isomor phism. The following theorem is the analog ue of Theo rem B for the presheaf F . Theorem 6.5. The functor L F {−} G : Ho(L F sPSh( C , G) inj ) → Ho(sCoa lg F ) is ful ly faith- ful. Pr o of. Similarly to the pro of of Theor em B, it suffices to show that ρ G preserves the weak equiv alences, i.e., it sends weak equiv alences to F -homolog y equiv alences. This is a conse- quence of the natural splitting of the ´ etale part of an F -coalg e bra s imilarly to the pro of of Prop ositio n 6 .1. The res ult follows from the iden tification of the ´ etale par t by isomo phism (6.2) and the isomorphis m (6.4).  The la st theorem ca n b e used to give a nice description of the unit transfor mation of the derived adjunction L F {− } : L F sPSh( C ) inj ⇄ sCo alg F : R ρ. Let X be a simplicia l presheaf and p ∗ ( X ) be the s implicia l presheaf X endo wed with the trivial G - a ction. Let p ∗ ( X ) ∼ → p ∗ ( X ) f be a functorial fibran t replacement o f p ∗ ( X ) in L F sPSh( C , G) inj . W e hav e the following co rollary . SIMPLICIAL PRESHEA VES OF COALG EBRAS 23 Corollary 6.6. The c anonic al derive d unit map X → ( R ρ )( F { X } ) c an b e identifie d, up t o a natur al isomorphism in t he homotopy c ate gory, with the map X → ( p ∗ ( X ) f ) G . Pr o of. Note that the Q uillen a djunction ( F {− } , ρ ) is the comp osition of the Quillen ad- junctions of P rop osition 6.4. By Theorem 6.5, the derived unit transfor mation of the Q uillen adjunction ( F {−} G , ρ G ) is a na tural isomor phism. Hence the result follows.  6.4. W e end with a rema rk ab out the general c ase o f an arbitra ry preshea f F o f p er fect fields. The no n-functoriality of algebraic clo sures becomes the main iss ue in treating this ca se using similar ar guments. On the other ha nd, note that the class o f F -homo lo gy equiv alence s depe nds only on the characteristics o f the fields inv olved. W e only comment o n the followin g sp ecial case. Supp ose that the Gro thendieck site C ha s a ter mina l ob ject denoted by 1 . E xamples include the site o f o pen subsets of a top olog ical space. Let F b e an arbitra ry presheaf of p erfect fields on C a nd let F 1 denote the co nstant presheaf at F (1) = F . Th us there is a morphism of presheav es F 1 → F . Let F be the algebraic clos ure o f F , G the profinite Galois g r oup and F 1 the consta nt pr esheaf a t F . The Quillen a djunction (6.6) F 1 {−} : L F sPSh( C ) inj ⇄ sCoalg F 1 : ρ can written a s the co mp os itio n o f the following three adjunctions: the Quillen equiv alence 1 : L F sPSh( C ) inj → L F 1 sPSh( C ) inj : 1 and the Quillen adjunctions p ∗ : L F 1 sPSh( C ) inj ⇄ L F 1 sPSh( C , G) inj : ( − ) G F 1 {−} G : L F 1 sPSh( C , G) inj ⇄ sCoalg F 1 : ρ G . Therefore the der ived unit transformation o f (6.6) ca n b e express ed in terms of the derived unit tr ansformation of the Quillen adjunction ( F 1 {−} , ρ ) a s descr ibed in Cor ollary 6.6. References [1] Ad ´ amek J.; Rosic k ´ y J. , L o c al ly pr esentable and ac c e ssible c atego ries , London Math. So c. Lect. Note Series, No. 189, Cambridge Universit y Press, 1994. [2] Barr M. , Co algebr as over a c ommutative ring , J. Algebra 32 (1974), no. 3, 600-610. 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