Diagrams and torsors J.F. Jardine December 21, 2004 Introduction Suppose that I is a small category, and that X : I ! S is a functor taking values in simplicial sets. Such functors are also commonly called I-diagrams of simplicial sets. Suppose further that X is a diagram of equivalences in the sense that any morphism ff : i ! j of I is mapped to a weak equivalence ff* : Xi ! Xj of simplicial sets. Then it is a fundamental and well known result of Quillen [11], [4, IV.5.7] that each pullback square Xi _____//holim---!IX (1) | |c | | fflffl| fflffl| * ____i___//BI is homotopy cartesian in the sense that the simplical set Xiis weakly equivalent to the homotopy fibre of the canonical map c over the vertex i of the nerve BI * *of the category I. Quillen's Theorem B, which is the original homotopy theoretic foundation for much of algebraic K-theory, is an easy consequence of this resul* *t. The category SI of I-diagrams of simplicial sets has a model structure (orig- inally introduced by Bousfield and Kan [1]) for which a natural transformation of f : X ! Y of I-diagrams is a weak equivalence (respectively fibration) if all constituent maps f : Xi! Yi are weak equivalences (respectively fibrations) of simplicial sets. Let I=i be the category of morphisms j ! i in I. Then the functor I=i ! I which sends j ! i to j determines a pullback square of simplicial set maps pb (holim---!IX)i_//holim---!IX (2) | | | |c fflffl| fflffl| B(I=i)__________//_BI which is equivalent to the square (1), so that Quillen's result could be rephra* *sed to assert that the diagram (2) is homotopy cartesian if X is a diagram of equiv- alences. 1 The category S=BI of simplicial sets over BI has for objects all simplicial set maps X ! BI, and the commutative triangles f X _______//3Y 33 ffff 33 ffff 3ssssffff BI are its morphisms. It is an elementary observation that S=BI inherits a model structure from simplicial sets for which a morphism as above is a weak equiva- lence (respectively fibration, cofibration) if and only if the map f : X ! Y is* * a weak equivalence (respectively fibration, cofibration) of simplicial sets. It is a consequence of the fact that the model structure for simplicial sets* * is proper that if p : Z ! BI is a fibration, and the I-diagram pb(Z) is defined by the pullback pb(Z)i _____//_Z | | | |p fflffl| fflffl| B(I=i) ____//_BI then pb(Z) is a diagram of equivalences. Write Ho (SI)e for the full subcategory of the homotopy category Ho (SI) arising from the Bousfield-Kan model structure on the diagrams of equivalences. It follows in the standard way that the pullback functor pb induces a homotopy derived functor R pb: Ho(S=BI) ! Ho(SI)e. In effect, R pb(X) is the I-diagram pb(Z) where Z ! BI is a (natural) fibrant replacement of X ! BI. The homotopy colimit functor holim---!Ipreserves weak equivalences, and hence induces a functor holim---!I: Ho(SI)e ! Ho(S=BI), and it is essentially a consequence of Quillen's result (ie. that the diagrams (2) are homotopy cartesian) that the derived pullback and homotopy colimit functors together determine an equivalence of categories Ho(S=BI) ' Ho(SI)e. (3) There's a new concept that can now be introduced: say that an I-diagram X is an I-torsor if X is a diagram of equivalences and the canonical simplicial set map holim---!IX ! * is a weak equivalence. Write I -Tors for the category of I-torsors, and let ss0(I -Tors ) for its class of path components. Write Triv=BI for the subcategory of S=BI on those objects Z ! BI such that Z is weakly equivalent to a point, and let ss0(Triv=BI) be its class of path components. Then the construction giving the equivalence of categories (3) specializes to a bijection ss0(Triv=BI) ~=ss0(I - Tors). 2 At the same time, we know from [9] that there is a bijection ~= ss0(Triv=BI) -! [*, BI] which is defined by taking the diagram * -' Z ! BI to the obvious map * ! BI in the homotopy category of simplicial sets. We have, in other words, a natural identification ss0(I - Tors) ~=[*, BI] (4) relating the set of morphisms from * to BI in the homotopy category with path components of a suitably defined category of I-torsors. The essential point of this paper is that all of the foregoing can be greatly generalized: simplicial sets can be replaced by the category s Pre(C) on an arbitrary small Grothendieck site C, and the category of I-diagrams above can be replaced by the category s Pre(C)A of enriched A-diagrams on a presheaf of small categories A which is enriched in simplicial sets. One can go even further, and replace the standard model structure on the category of simplicial presheaves by an f-local theory for a set of cofibrations f : A ! B, provided that the corresponding f-local model structure is proper. The main results are Theorem 17 and Theorem 18 in the f-local case, and Theorem 20 and Theorem 21 for the ordinary (ie. "unlocalized") model structure on enriched diagrams of simplicial presheaves. Theorems 17 and 20 are analogues of the equivalence (3) while Theorems 18 and Theorem 21 give analogues of (4) which identify path components of suitably defined categories of A-torsors with morphisms [*, dBA] in the respective homotopy categories. Here, dBA is the diagonal of the bisimplicial nerve BA which is canonically associated to the simplicial category A. Theorems 20 and 21 are consequences of the arguments for Theorems 17 and 18, respectively, but the unlocalized case is important in its own right. In particular, the concept of A-torsor specializes to the definition of G-torsor f* *or a presheaf of groupoids enriched in simplicial sets given in [9], which itself is* * the higher analogue of the classical notion of torsor for a sheaf of groups. The ar* *gu- ments of [9] are given within the framework of a model structure for presheaves of groupoids enriched in simplicial sets whose associated homotopy category is equivalent to the standard homotopy category of simplicial presheaves. I don't know of a corresponding model structure for presheaves of categories enriched in simplicial sets, but it doesn't matter: the argument for Theorem 21 which is presented here requires no such ambient model structure. As such, the result was unexpected. The localized structure theorems Theorem 17 and Theorem 18 specialize to the case of motivic homotopy theory. Thus, one can now speak of motivic A- torsors for a presheaf of categories enriched in simplicial sets on the smooth * *Nis- nevich site of a scheme, and identify the path components of the corresponding 3 category with the set of morphisms [*, dBA] in the motivic homotopy category. Note that the whole question of what a motivic stack or a higher motivic stack should be has been avoided. The model structures which describe such objects can be written down, but they are irrelevant for the arguments presented here. The basic outline of the argument for the main results is fairly simple, but there are technical issues to overcome. One such issue is that fibres over poin* *ts lose their meaning in the simplicial presheaf context, and so one has to work relative to the full presheaf Ob (A) of objects of A, rather than one fibre at a time. Thus, for example, the assertion that an enriched A-diagram X is a diagram of local equivalences has to be expressed in the requirement that the diagram (6) below is locally homotopy cartesian. Also, one uses an f-local model structure for enriched A-diagrams which is bootstrapped up from an f- local model structure on the category s Pre(C)= Ob(A) in a predictable way, but the f-local model structure that is used for s Pre(C)= Ob(A) does not have a naive definition. The required structure is defined "internally" here by methods which were introduced recently by Cisinski [2], [7], although one has the option of making it by hand. 1 Diagrams of simplicial sets Suppose that A is a small category which is enriched in simplicial sets, and th* *at X is an enriched diagram on A taking values in simplicial sets. This means that X consists of set-valued functors Xn : An ! Set which respect the simplicial identities in the obvious way, or equivalently that X consists of a simplicial * *set map ss : X0 ! Ob(A) and an action X0 xs Mor(A) ___OE__//X0 | | | |ss fflffl| fflffl| Mor (A)____t___//Ob(A) which is associative and respects the composition law and identities of A. Here s, t : Mor(A) ! Ob (A) are source and target maps, respectively, and the pull- back diagram X0 xs Mor(A) _____//Mor(A) | | | |s fflffl| fflffl| X0 _____ss___//_Ob(A) defines the object X0 xs Mor(A). Natural transformations f : X ! Y between enriched A-diagrams are the 4 obvious thing, namely commutative diagrams of simplicial set maps f X0:_________//_Y0 ::: ss:oo:: ss Ob(A) which respect the actions of A on X and Y . The category of enriched A-diagrams with natural transformations between them will be denoted by sSetA . Say that a map f : X ! Y of enriched A-diagrams is a weak equivalence (respectively fibration) if the simplicial set map f : X0 ! Y0 is a weak equiv- alence (respectively fibration) of simplicial sets over Ob (A). A cofibration * *in the category of enriched A-diagrams is a map which has the left lifting property with respect to all trivial fibrations. Lemma 1. With these definitions, sSetA has the structure of a proper closed simplicial model category. Proof. It is easy to verify the claim when A is discrete in the sense that it consists entirely of identity morphisms on the elements of Ob (A). In that case, diagrams are simplicial sets defined over Ob (A) and the model structure of the statement of the Lemma is just the standard model structure for simplicial sets over Ob (A). Write j : Ob (A) ! A for the inclusion of the object set in the simplicial category A, and interpret Ob (A) to be a discrete simplicial category, so that j is a functor. Ob(A)-diagrams, from this point of view, are just simplicial set maps Z ! Ob(A). Write j*Z for the left Kan extension of the functor Z along j. Then the standard construction of the left Kan extension functor implies directly that the "object set" of j*Z has the form j*Z0 = Z xs Mor(A). over Ob(A). It follows that the left Kan extension functor preserves cofibratio* *ns and weak equivalences. The usual small object arguments give the desired_closed model structure. |__| Say that an enriched A-diagram X is a diagram of equivalences if all mor- phisms ff : i ! j of the category A0 induce weak equivalences Xi! Xj. There is a canonical morphism A0 ! A of categories enriched in simplicial sets, and X is a diagram of equivalences if and only if the restriction of X to* * A0 is a diagram of equivalences. The condition that X is a diagram of equivalences 5 is also equivalent to the requirement that the composite square X xs Mor(A0) _____//Mor(A0) | | | | fflffl| |fflffl X xs Mor(A) ______//Mor(A) OE|| |t| fflffl| |fflffl X ______ss___//Ob(A) is homotopy cartesian. The homotopy colimit holim---!AX is naturally a bisimplicial set, as is the * *clas- sifying object BA, and there is a canonical bisimplicial set map c : holim---!A* *X ! BA. In more detail, BA is the bisimplicial set consisting of the nerves BAn of the categories An, (holim---!AX)n is the nerve of the translation category for * *the functor Xn : An ! Set defined by the n-simplices of X, and the canonical map (holim---!AX)n ! BAn is induced by the canonical functor from the translation category for Xn to An. In horizontal degree n, the map c is the projection X0 xOb(A)BAn ! BAn, where the map v0 : BAn ! Ob(A) is induced by the ordinal number map 0 ! n which picks off the vertex 0. Here is the enriched diagram version of a well known result of Quillen. The first proof of this result was given by Moerdijk [10]. Lemma 2. Suppose that the enriched A-diagram X is a diagram of equivalences. Then the canonical pullback square X0 ______//_holim---!AX ss|| |c| fflffl| fflffl| Ob (A)________//BA is a homotopy cartesian diagram of bisimplicial sets. Remark 3. The claim in Lemma 2 that the diagram in the statement of the Lemma is homotopy cartesian refers to the closed model structure on the cate- gory of bisimplicial sets for which the cofibrations are the inclusions and the weak equivalences are the maps X ! Y which induce a weak equivalence d(X) ! d(Y ) of the associated diagonal simplicial sets [2], [7]. The left ad- joint d* of the diagonal functor preserves cofibrations and weak equivalences [4, IV.3.12], and it follows that a square diagram of bisimplicial sets is homo- topy cartesian if and only if the associated diagram of diagonal simplicial sets is homotopy cartesian. 6 Proof of Lemma 2. The displayed statement is equivalent to the more common assertion that X(a) = ss-1(a) is the homotopy fibre of c over a 2 BA0,0for each a 2 Ob(A), and this is what we prove. We shall assume that the Lemma holds when A is an ordinary category _ this statement has a separate proof [4, IV.5.7], and is Quillen's original result. The homotopy colimit functor preserves weak equivalences of enriched A- diagrams. In effect, the induced map holim---!AX ! holim---!AY is the comparison of fibre products X0 xOb(A)BAn ! Y0 xOb(A)BAn, in horizontal degree n, which is a weak equivalence since Ob(A) is discrete (but see also Lemma 12 below). Note also that if f : X ! Y is a weak equivalence of enriched diagrams, then X is a diagram of equivalences if and only if Y is a diagram of equivalences. It follows that we can assume that X is a fibrant enriched A-diagram. Form the pullback diagrams of bisimplicial sets c-1(oe)____//holim---!AX | |c | | fflffl| fflffl| m,n ___oe___//_BA where oe varies over the bisimplices of BA. By standard techniques [4, p.244] (and by using the case of the Lemma where A is an ordinary category) it suffices to show that all maps of bisimplices `xfl r,s8______//_8 m,n 88 o888 oe oo BA induce weak equivalences c-1(o) ! c-1(oe). Suppose that v : 0 ! s is an ordinal number map, and consider the corre- sponding diagram 1xv //r,s r,0 ______ RRoR RRR `x1|| `xfl|| l))BA55 fflffl| fflffl|oelllll m,0 1xfl(v)//_ m,n The map holim---!AX ! BA is a fibration in each horizontal degree since X is fibrant, and the map 1xv : r,0! r,sis a weak equivalence in each horizontal degree, so that pulling back holim---!AX over the maps 1 x v and 1 x fl(v) indu* *ces weak equivalences of pullbacks. The pullback of holim---!AX over the map ` x 1 is a weak equivalence, by the assumption that X is a diagram of equivalences_ over A0. |__| 7 Suppose that y 2 Ob (A), let ss : Z ! dBA be a simplicial set map, and define a simplicial set pb(Z)y by the pullback diagram pb(Z)y __c____//Z | | | |ss fflffl| fflffl| dB(A=y) ____//_dBA Here A=y is the simplicial category, which is the slice category An=y in each simplicial degree n. The simplicial set A(y, z) of homomorphisms from y to z in A determines a simplicial set map pb(Z)y x A(y, z) ! pb(Z)z, giving pb(Z) the structure of an enriched A-diagram. An n-simplex of pb(Z)y consists of a triple (x, oe : a0 ! . .!.an, ff : an ! y) where x 2 Zn, ss(x) = oe, oe is a string of arrows of length n in An, and ff is* * a morphism of An. An n-simplex of the product pb(Z)y0x A(y0, y1) x A(y1, y2) x . .x.A(ym-1 , ym ) therefore consists of a pair (x, a0 ! . .!.an ! y0 ! . .!.ym ) where the string of arrows lives in An and ss(x) is the simplex a0 ! . .!.an of BAn. The fibre over x of the bisimplicial set map holim---!Apb(Z) ! Z can therefore be identified with the simplicial set B(an=An), which is contractible. We have therefore proved the following result: Lemma 4. Suppose that ss : Z ! dBA is a simplicial set map, where A is a category enriched in simplicial sets. Then the canonical map holim---!Apb(Z) ! Z is a weak equivalence. Suppose that X is an enriched A-diagram. There is an identification pbd(holim---!AX)a ~=d(holim---!A=aX|A=a) and each category An=a has a terminal object. It follows that there is a weak equivalence pb d(holim---!AX)a ' X(a). This equivalence is realized by the canonical natural transformation holim---!A=aX|A=a ! Xa 8 which is defined in the various simplicial degrees by the simplicial set maps G (Xa0)n ! (Xa)n a0!...!an!a given in summands by the maps (Xa0)n ! (Xa)n. The corresponding maps assemble to give a natural weak equivalence _ : pbd(holim---!AX) ! X (5) of enriched A-diagrams. This map _ is split over Ob (A) by a map X0 ! pb d(holim---!AX)0 which is given by the maps Xa ! pbd(holim---!AX)a defined by the pullback diagrams Xa _____//pbd(holim---!AX)a | | | | | fflffl| fflffl| *______1a_//dB(A=a) 2 Diagrams of simplicial presheaves Suppose now that A is a presheaf of small categories on a small Grothendieck site C. An enriched A-diagram X taking values in simplicial presheaves consists of a simplicial presheaf map ss : X0 ! Ob(A) and an action X0 xs Mor(A) ___OE__//X0 | | | |ss fflffl| fflffl| Mor (A)____t___//Ob(A) which is associative and respects the composition law and identities of A. Nat- ural transformations f : X ! Y between enriched A-diagrams in simplicial presheaves are commutative diagrams of simplicial presheaf maps f X0:_________//_Y0 ::: ss:oo:: ss Ob(A) which respect the actions of A on X and Y . The category of enriched A-diagrams with natural transformations between them will be denoted by s Pre(C)A . Say that a map f : X ! Y of enriched A-diagrams is a local weak equivalence (respectively global fibration) if the simplicial set map f : X0 ! Y0 is a local weak equivalence (respectively global fibration) of simplicial presheaves over Ob (A). A cofibration in the category of enriched A-diagrams is a map which has the left lifting property with respect to all trivial fibrations. 9 Lemma 5. With these definitions, s Pre(C)A has the structure of a proper closed simplicial model category. Proof. It is easy to verify the claim when A is discrete. In that case, diagrams are simplicial presheaves defined over Ob (A) and the model structure of the statement of the Lemma is just the standard model structure for simplicial presheaves over Ob (A). Write j : Ob (A) ! A for the inclusion of the object set in the simplicial category A, and write j*Z for the left Kan extension of the functor Z along j. Then j*Z has the form j*Z0 = Z xs Mor(A). over Ob(A). It follows that the left Kan extension functor preserves cofibratio* *ns; it preserves local weak equivalences by a Boolean localization argument. The __ usual small object arguments give the desired closed model structure. |__| Say that an enriched A-diagram X is a diagram of local equivalences if the square X0 xs Mor(A0) _____//Mor(A0) (6) OE|A0|| t|| fflffl| fflffl| X ______ss___//_Ob(A) is homotopy cartesian in the category of simplicial presheaves. Then we have the following analogue of Lemma 2: Lemma 6. Suppose that the enriched A-diagram X is a diagram of local equiv- alences. Then the canonical pullback square X0 ______//_holim---!AX ss|| |c| fflffl| fflffl| Ob (A)________//BA is a homotopy cartesian diagram of bisimplicial presheaves. Remark 7. We use the closed model structure on the category of bisimplicial presheaves for which the cofibrations are the inclusions and the weak equiva- lences are the maps X ! Y which induce a local weak equivalence d(X) ! d(Y ) of the associated diagonal simplicial presheaves [7]. The left adjoint d* of th* *e di- agonal functor preserves cofibrations and local weak equivalences (the latter by a Boolean localization argument), and it follows that a square diagram of bisim- plicial presheaves is homotopy cartesian if and only if the associated diagram * *of diagonal simplicial presheaves is homotopy cartesian. Proof. Lemma 6 follows from Lemma 2, by a Boolean localization argument. |___| 10 Suppose that the enriched A-diagram X is a diagram of local equivalences, and let j : d(holim---!AX) ! F d(holim---!AX) be a choice of fibrant model in objects over dBA. Then the induced map X0 ! Ob(A) xdBA F d(holim---!AX) is a local weak equivalence _ this is just a restatement of Lemma 6. The canonical map Ob (A) ! dBA factors as a composite of canonical maps Ob(A) '-!pb(BA)0 ! BA and there are corresponding pullback diagrams X0 ________________//_pbd(holim---!AX)0__//d(holim---!AX) ' || j*|| || fflffl| fflffl| fflffl| Ob (A) xBA F d(holim---!AX)__//pb(F d(holim---!AX))0//_F d(holim---!AX) The map X0 ! pbd(holim---!AX)0 is a sectionwise weak equivalence since it splits the map induced by the weak equivalence _ of (5) over Ob (A). The map Ob (A) xBA F (holim---!AX) ! pb(F (holim---!AX))0 is the pullback of a local weak equivalence along a global fibration, and is th* *ere- fore a local weak equivalence. It follows that the induced map j* is a local we* *ak equivalence, so that we have proved the following: Lemma 8. Suppose that the enriched A-digram X is a diagram of local equiv- alences, and let j : d(holim---!AX) ! F d holim---!AX) be a fibrant model of the corresponding homotopy colimit over dBA. Then the maps X -_ pbd(holim---!AX) j*-!pb(F d(holim---!AX)). are weak equivalences of s Pre(C)A . 3 Internal localization Let f : * ! I be a map of simplicial presheaves on a site C and form the f-local theory on s Pre(C) in the style of [3] or [6]. To recall, we say that a simplicial presheaf X is f-injective (or f-fibrant)* * if X is globally fibrant and if the map X ! * has the right lifting property with respect to all maps (A x I) [A B ! B x I 11 which are induced by cofibrations A ! B. It suffices to restrict the class of cofibrations to all cofibrations of the form Y LU n, where LU denotes the left adjoint of the U-sections functor X 7! X(U). A map f : X ! Y is an f- equivalence if it induces a weak equivalence of function complexes hom (Y, Z) ! hom (X, Z) for all f-injective Z, and then f-fibrations are defined by a right lifting property with respect to all f-trivial cofibrations. An f-trivial cofib* *ration (respectively fibration) is a cofibration (respectively fibration) which is als* *o an f-equivalence. Then with these definitions and the standard simplicial category structure on s Pre(C), the category of simplicial presheaves on C has the struc- ture of a proper closed simplicial model category. The closed simplicial model structure is obtained in [3], and properness is proved in [6]. Suppose that K is a simplicial presheaf on C. It is essentially obvious that* * the category s Pre(C)=K of simplicial presheaf maps X ! K has a f-local (proper, simplicial) model structure in which a map g X _____//@Y @OO """" K is a f-equivalence (respectively cofibration, f-fibration) if and only if the m* *ap g : X ! Y is an f-equivalence (respectively cofibration, f-equivalence) of s Pre(C). We shall call this the external f-local structure. There is also an "internally defined" f-local structure on s Pre(C)=K. Say that a map g : X ! Y over K (ie. of s Pre(C)=K) is injective if it is a global fibration and has the right lifting property with respect to all maps (A xK I) [A B ! B xK I for all cofibrations A ! B over K. Here the fibre product AxK I is the product over K of the object A ! K and the projection K x I ! K. Remark 9. We have tacitly identified I with its pullback K x I ! K in s Pre(C)=K. There is an isomorphism A xK I ~=A x I over K which induces the K-structure map A x I pr-!A ! K on A x I. Note that an arbitrary composite A x I ! X ! K may not give the object A x I the product structure over K, so that maps A xK I ! X over K are somewhat special. There is a functorial injective replacement functor j : X ! LX functor which enjoys the usual properties [7, Lemma 37]. One says that g : X ! Y is an internal f-equivalence if the induced map g* : LX ! LY is a local weak equivalence of s Pre(C)=K. Equivalently g is an internal f-equivalence if and only if g* is a homotopy equivalence in s Pre(C)=K, or a sectionwise weak equivalence of s Pre(C). Cofibrations are monomorphisms, and internal f-fibrations are defined by a right lifting property with respect to all intern* *al f-trivial cofibrations. 12 Proposition 10. The definitions of internal f-equivalence, cofibration and in- ternal f-fibration give the category s Pre(C)=K of simplicial presheaves over K the structure of a cofibrantly generated proper closed simplicial model categor* *y. Proof. Write iK for the category whose objects consist of pairs (x, U) such that U is an object of the underlying site C and x : n ! K(U) is an n-simplex of K(U). A morphism (`, OE) : (x, U) ! (y, V ) of iK is a pair consisting of a morphism OE : V ! U of C and an ordinal number map ` : n ! m such that the diagram of simplicial set maps `* n _______// m x || y|| fflffl| fflffl| K(U) _OE*_//K(V ) commutes. If Y is a presheaf on iK then the family of sets G (Y*)n(U) = Y (x, U) x2Kn(U) defines a simplicial presheaf map Y* ! K. The assignment Y 7! Y* defines an equivalence of categories between presheaves of sets on the category iK and the category s Pre(C)=K of simplicial presheaves over K. The box category (see [7]) acts on s Pre(C)=K, by associating to Y ! K and the product category 1xn the object Y n = Y x ( 1)xn -pr!Y ! K. Let Ai ! Bi be a set of generating trivial cofibrations for the category of simplicial presheaves on C, and let S be the set consisting of all maps Ai! Bi! K, together with the map K (1K-,f)---!K x I ! K Then the ( , S) structure given by Theorem 46 of [7] is the required model structure. The properness of this structure is a consequence of Theorem_47 of [7]. |__| The model structure on s Pre(C)=K which is given by Proposition 10 will be called the internal f-local structure. Remark 11. 1) If K is a point (ie. the terminal simplicial presheaf), then the internal f-local structure is the standard f-local structure on s Pre(* *C). 2)Every internal f-equivalence over a simplicial presheaf K is an external f-equivalence. 13 The point of introducing the internal f-local structure over an object K is essentially the following: Lemma 12. Suppose that W is a presheaf of sets, identified with a constant simplicial presheaf on the site C, and suppose that h : K ! W is a map of simplicial presheaves. Then the pullback functor which sends X ! W to the projection K xW X ! K preserves preserves internal f-equivalences. Proof. Pulling back along h preserves local weak equivalences, by a Boolean localization argument. Pulling back along h preserves cofibrations, products and all colimits, and therefore sends a map (W xK I) [ B ! B xK I to the map ((L xK W ) xL I) [ (L xK B) ! (L xK B) xL I. It follows that if j : X ! LX is an injective replacement for X over K, then the induced map j* : L xX K ! L xK LX is an internal f-equivalence over L. If LX ! LY is a sectionwise weak equivalence of globally fibrant objects over K, then the induced map L xK LX ! L xK LY is a sectionwise weak equivalence __ of globally fibrant objects over L. |__| Remark 13. Lemma 43 of [7] asserts in the case at hand that if a map p : X ! Y is of simplicial presheaves over K is injective and the object Y ! K is injective, then p is an f-fibration. It follows that all injective objects * *are f-fibrant in the internal f-local structure of Proposition 10. Example 14. Suppose that U is an object of C, and consider the forgetful map jU : C=U ! C. If X is a simplicial presheaf on C, write X|U for the composition (C=U)op-jU!Cop X-!S Recall from [8] that the category s Pre(C)=U is equivalent to s Pre(C=U). In effect, a functor F : s Pre(C)=U ! s Pre(C=U) is defined for an object ss : X !* * U by setting F (ss)(OE : V ! U) = ss-1(OE) X(V ), while a functor G : s Pre(C=U) ! s Pre(C)=U is defined by setting G G(X)(V ) = X(OE), OE:V !U and the functors F and G are inverse to each other up to natural isomorphism. If I is a simplicial presheaf on C, then there is a natural isomorphism G(I|U )* * ~= I x U of objects over U. One can check that the equivalence of categories F : s Pre(C)=U ' s Pre(C=U) : G induces an equivalence of the internal f-local structure on s Pre(C)=U with the f|U -local structure on s Pre(C=U). 14 Say that a map g : X ! Y of enriched A-diagrams in s Pre(C) is an f- equivalence (respectively f-fibration) if the induced map g X0:_________//_Y0 ::: ssX:oo:: ssY Ob(A) is an internal f-equivalence (respectively internal f-fibration) of simplicial * *pre- sheaves over Ob (A). An f-cofibration is a map which has the left lifting property with respect to all maps which are simultaneously f-fibrations and f-equivalences. We have the following analogue of Lemma 5: Lemma 15. With the definitions of f-equivalence, f-fibration and f-cofibration given above, the category s Pre(C)A of enriched A-diagrams has the structure of a proper closed simplicial model category. Proof. Write j : Ob (A) ! A for the inclusion of the discrete subcategory of objects of A, and observe that the left Kan extension j*Z of an object Z ! Ob (A) (ie. an enriched functor Ob (A) ! s Pre(C)) has objects defined by the map Z xs Mor(A) ! Ob(A) arising from the pullback diagram Z xs Mor(A) _____//Mor(A) ____ | ____________s| | ________| fflffl| _''__fflffl| Z ___________//Ob(A) It follows from Lemma 12 that the left Kan extension functor Z 7! j*Z preserves internal f-equivalences; this functor also preserves cofibrations. The functor X 7! X0 preserves colimits. It follows that applying the left Kan extension functor j* to the generating sets of cofibrations and trivial cofibrations for * *the internal f-local structure on s Pre(C)= Ob(A) gives corresponding sets of_gener- ators for s Pre(C)A . |__| Say that an enriched A-diagram X is a diagram of f-equivalences if there is an f-fibrant replacement j : X ! Z of enriched A-diagrams such that the diagram Z0 xOb(A),sMor(A0)_______//Z0 | | | |ssZ fflffl| fflffl| Mor(A0) ____t_____//Ob(A) is a homotopy cartesian diagram of simplicial presheaves. Note that any two (internal) f-fibrant replacements of X are globally fibrant and sectionwise equ* *iv- alent over Ob (A), so that the choice of f-fibrant replacement doesn't matter. 15 Lemma 16. Suppose that X is a diagram of local equivalences in the category of enriched A-diagrams. Then X is a diagram of f-equivalences. Proof. We can suppose that X0 ! Ob(A) is a global fibration. We are presum- ing that the diagram X0 xOb(A),sMor(A0)_______//X0 | | | |ssX fflffl| fflffl| Mor(A0) ____t_____//_Ob(A) is homotopy cartesian in the category of simplicial presheaves. In particular, the induced map of global fibrations X0 xOb(A),sMor(A0) ____'____//_Mor(A0) xOb(A),tX0 MM qq MMM qqq MMM qqq MM&& xxqq Mor(A0) is a weak equivalence of globally fibrant objects over Mor (A). If X ! Z is an f-fibrant replacement for X, then the induced map Z0 xOb(A),sMor(A0) _________//_Mor(A0) xOb(A),tZ0 MM qq MMM qqq MMM qqq MM&& xxqq Mor(A0) is a comparison of internal f-fibrant replacements for the diagram above, by Lemma 12, and is therefore a weak equivalence of globally fibrant objects over_ Mor (A0). In particular, Z is a diagram of local equivalences. |__| If X is an A-diagram then the simplicial presheaf of n-simplices (holim---!A* *X)n of the homotopy colimit sits in a pullback diagram (holim---!AX)n___//_BAn | |v | | 0 fflffl| fflffl| X0 __________//Ob(A) It follows from Remark 11 and Lemma 12 that the assignment X 7! (holim---!AX)n preserves f-equivalences (which are defined internally over Ob (A).) Any f- equivalence of enriched A-diagrams X ! Y therefore induces an f-equivalence 16 of simplicial presheaves d(holim---!AX) ! d(holim---!AY ), by properness of the f-local model structure. Suppose that X is a diagram of f-equivalences with f-fibrant replacement j : X ! Z, and let d(holim---!AX) ! F d(holim---!AX) be a natural choice of external f-fibrant replacement over dBA. Then in the diagram X oo'__ pbd(holim---!AX)___//pbF d(holim---!AX) | j|| || ' |j*| fflffl| fflffl| fflffl| Z oo'___pbd(holim---!AZ)'__//_pbF d(holim---!AZ) the indicated maps are local weak equivalences of enriched A-diagrams. This follows partly from Lemma 8, which gives the equivalences on the bottom row. It follows that the natural maps X pbd(holim---!AX) ! pbF d(holim---!AX) (7) are f-equivalences of enriched A-diagrams if X is a diagram of f-equivalences. If X ! dBA is an object of s Pre(C)=dBA with external f-fibrant model j : X ! F X over dBA, then pb(F X) is a diagram of local equivalences and hence a diagram of f-equivalences by Lemma 16. There is, furthermore, a natural diagram d(holim---!Apb(X))_'__//_X (8) | j*|| j|| fflffl| fflffl| d(holim---!Apb(F X))'__//F X in which the indicated maps are local weak equivalences, and of course j is an external f-equivalence. Write Ho (s Pre(C)A )f,efor the full subcategory on diagrams of f-equiva- lences in the homotopy category Ho(s Pre(C)A )f associated to the f-local struc- ture on the category of enriched A-diagrams of simplicial presheaves. Write Ho (s Pre(C)=dBA)extf for the homotopy category associated to the external f-local structure on the category of simplicial presheaves over dBA. We have seen that the homotopy colimit functor X 7! d(holim---!AX) takes f-equivalences of enriched A-diagrams to external f-equivalences; it therefore induces a functor holim---!: Ho(s Pre(C)A )f,e! Ho(s Pre(C)=dBA)extf. On account of Lemma 16, the assignment Z 7! pbF (Z) defines a functor R pb: Ho(s Pre(C)=dBA)extf! Ho(s Pre(C)A )f,e, which, as the notation suggests, is a homotopy derived functor associated to the functor pb. 17 The natural equivalences displayed in (7) and (8) are used to prove the following result: Theorem 17. The functors pb and holim---!Adetermine an equivalence of cate- gories R pb: Ho(s Pre(C)=dBA)extfo Ho(s Pre(C)A )f,e: holim---!A. An f-local A-torsor is defined to be an diagram of f-equivalences X on A such that the canonical map d(holim---!AX) ! * is an f-equivalence. Write f - TorsA for the full subcategory of s Pre(C)A on the class of A-torsors. The homotopy colimit functor (by definition) induces a functor holim---!: f - TorsA ! Triv=dBA. where Triv=dBA is the category of objects Z ! dBA such that the map Z ! * is an f-equivalence of simplicial presheaves. Recall that the functor R pb : s Pre(C)=dBA ! s Pre(C)A is defined by R pb(Z) = pbF (Z), where j : Z ! F Z is a natural f-fibrant replacement. Then R pb(Z) is a diagram of f-equivalences, and R pb(Z) is an A-torsor if Z ! dBA is a member of Triv=BA. It follows that the functor R pbinduces a functor R pb: Triv=BA ! f - TorsA. Then Lemma 4 and Lemma 8 together imply the following: Theorem 18. The functors holim---!and R pbinduce a bijection ss0(f - TorsA) ~=ss0(Triv=dBA), and there is a bijection ~= ss0(Triv=dBA) -! [*, dBA]. In the statement of Theorem 18, [*, dBA] denotes morphisms in the f-local homotopy category for simplicial presheaves on C. The displayed map ss0(Triv=dBA) ! [*, dBA]. is induced by associating to an element * -' Z ! dBA the corresponding map * ! dBA in the f-local homotopy category. Lemma 8 of [9] (the proof of which is essentially elementary and will not be reproduced here) asserts that this map is a bijection, since the f-local structure on s Pr* *e(C) is right proper. 18 Example 19. Suppose that S is a scheme of finite dimension, and let (Sm|S)Nis be the Grothendieck site of smooth schemes over S endowed with the Nisnevich topology, as in [6]. The motivic homotopy category is obtained by localizing the category s Pre(Sm|S)Nis of simplicial presheaves on the smooth Nisnevich site at a rational point * ! A1 of the affine line over S. If A is a presheaf of categories enriched in simplicial sets then the f-local A-torsors are more prop* *erly called motivic A-torsors, and Theorem 18 identifies the path components of the category of motivic A-torsors with the maps [*, dBA] in the motivic homotopy category. The internal motivic model structure arising from Proposition 10 for objects X ! K over a fixed simplicial presheaf K is of interest in its own right. It coincides with the external motivic structure on simplicial presheaves over K in the case that K is a "discrete" object which is represented by an S-scheme U. There is nothing special about the localized model structures appearing in Theorem 17 and Theorem 18, outside of the critical underlying properness assumption for the model structures. For example, localization at the map f : * ! I could be replaced by localization at a set of maps fj : * ! Ij, and this set could be empty. In particular, the derived pullback functor R pb and the homotopy colimit functors make sense for diagrams of local equivalences. Write Ho(s Pre(C)A )e for the full subcategory of the homotopy category of enriched A-diagrams on the diagrams of local equivalences. Then we have the following: Theorem 20. The functors pb and holim---!Adetermine an equivalence of cate- gories R pb: Ho(s Pre(C)=dBA) o Ho(s Pre(C)A )e : holim---!A. Say that an enriched A-diagram X is an A-torsor if X is a diagram of equiv- alences and the canonical map d(holim---!AX) ! * is a local weak equivalence. In other words the map d(holim---!AX) ! dBA lands in the category Triv=dBA consisting of objects Z ! dBA such that the canonical map Z ! * is a local equivalence. Theorem 21. The functors holim---!and R pbinduce a bijection ss0(Tors A) ~=ss0(Triv=dBA), and there is a bijection ss0(Triv=dBA) ~=[*, dBA]. Remark 22. 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