Intermediate model structures for simplicial presheaves J.F. Jardine November 1, 2002 Suppose that C is a small Grothendieck site, and let s Pre(C) denote the category of simplicial presheaves on the site C. It has been known for some time [5] that the category of simplicial presheaves carries a proper closed simplici* *al model structure for which a cofibrations is an inclusion of simplicial presheav* *es, the weak equivalences are the local weak equivalences, and the fibrations are the global fibrations. In the presence of an adequate supply of stalks, a local weak equivalence is a map f : X ! Y of simplicial presheaves which induces a weak equivalence f* : Xx ! Yx of simplicial sets in all stalks. Alternatively, f is a local weak equivalence if 1)the induced map f* : ~ß0X ! ~ß0Y of sheaves of path components is an isomorphism, and 2)the comparison diagram f* ~ßnX ____//_~ßnY | | | | fflffl| fflffl| ~X0__f___//_~Y * 0 of sheaves of group objects is a pullback, for n 1. A global fibration is a map which has the right lifting property with respect t* *o all morphisms which are simultaneously cofibrations and local weak equivalences. The indicated structure has become the standard tool for calculational ap- plications of the homotopy theory of simplicial presheaves. Recently, however, it has become clear that there is another useful model structure on s Pre(C) which has the same weak equivalences and hence describe the same homotopy category, but has different cofibrations and fibrations. This is the local projective model structure, which was introduced by Blander [1] as a tool for investigating some of the models for the motivic stable category. Specifically, a cofibration i : A ! B is said to be projective if it has the* * left lifting property with respect to all maps p : X ! Y which are trivial fibrations 1 of simplicial sets f : X(U) ! Y (U), U 2 C, in all sections. A map p : X ! Y is then said to be a local projective fibration if it has the right lifting pro* *perty with respect to all maps which are simultaneously projective cofibrations and local weak equivalences. Blander [1] proves the following: Theorem 1. The category of simplicial presheaves s Pre(C) on a small Grothen- dieck site C, with the classes of projective cofibrations, local weak equivalen* *ces and local projective fibrations, satisfies the axioms for a proper closed simpl* *icial model category. The closed model structure given by this result is the local projective model structure for the category s Pre(C) of simplicial presheaves on C. Say that a map f : X ! Y of simplicial presheaves is a sectionwise weak equivalence (respectively sectionwise fibration) if all of its component maps f : X(U) ! Y (U), U 2 C, are weak equivalences (respectively fibrations) of simplicial sets. One sometimes sees these objects referred to as pointwise weak equivalences or pointwise fibrations in the literature [7]. The special case of Theorem 1 corresponding to the trivial (or chaotic) topo* *l- ogy on C, for which the local projective fibrations are the sectionwise fibrati* *ons, was proved long ago by Bousfield and Kan [2]. Explicitly, the chaotic topology on a site C is the topology for which every covering family of an object U con- tains the identity map 1 : U ! U. Every presheaf is a sheaf for the chaotic topology. Observe that the definition of projective cofibration makes no reference to * *an underlying topology, and thus coincides with the notion arising in the Bousfiel* *d- Kan result. That result is easy to prove: if LU denotes the left adjoint to the U-sections functor s Pre(C) ! S taking values in simplicial sets, then the cofi- brations LU (@ n) ! LU ( n) generate the projective cofibrations in the usual sense, and the while the maps LU ( nk) ! LU ( n) generate the acyclic projec- tive cofibrations for the Bousfield-Kan theory. The function spaces hom (X, Y ) have the form that we expect, and there is a minor amount of fussing (in the proof of the simplicial model axiom) to show that if i : A ! B is a projective cofibration and j : K ! L is an inclusion of simplicial sets then the induced map (i, j)* : (B x K) [(AxK) (A x L) ! B x L is a projective cofibration. One resolves this, however, by showing that the collection of all inclusions i for which the cofibration (i, j)* is projective * *is sat- urated, and includes all LU (@ n) ! LU ( n). The point of this note is to show that there are other intermediate model structures on the simplicial presheaf category s Pre(C) which are intermediate between the local projective and standard theories. Sufficient conditions are then given for these structures to be cofibrantly generated. These conditions apply in particular to the local projective structu* *re, giving a new proof of a result of Blander. It remains to be seen, however, whet* *her any of the new intermediate structures outside of either the standard or local projective cases are cofibrantly generated. 2 To start the discussion, let CP denote the class of projective cofibrations, and let C denote the full class of cofibrations in s Pre(C). Then there is clea* *rly an inclusion relation CP C. Recall that CP has a generating family of cofibrations S0 = {LU (@ n) ! LU ( n)}. Suppose that S = {Aj ! Bj} is some other set of cofibrations which contains S0, and let CS denote the saturation of the set of cofibrations (Bj x @ n) [(Ajx@ n)(Aj x n) Bj x n where n 0, and the cofibrations Aj ! Bj belong to the set S. Note that the case n = 0 reduces to maps Aj ! Bj in the generating set for CS. The class CS will also be called the class of S-cofibrations. If S is a generating set fo* *r the full class of cofibrations, then CS = C and in that case the set of S-cofibrati* *ons is the full set of cofibrations. Say that a map p : X ! Y is a local S-fibration if p has the right lift- ing property with respect to all maps which are S-cofibrations and local weak equivalences. Theorem 2. With these definitions, the category s Pre(C) and the classes of S-cofibrations, local weak equivalences and local S-fibrations together satisfy* * the axioms for a proper closed simplicial model category. Proof. Every map f : X ! Y of simplicial presheaves has a factorization f X ____________//_@@Y??~ @@ ~~~ j @@__@~~~p Z where the map j is a member of CS and p has the right lifting property with respect to all morphisms of CS. It follows that p has the right lifting property with respect to all projective cofibrations, so that p is a projective fibration and a sectionwise (hence local) weak equivalence. The map p is also a local S-fibration. The map f : X ! Y can be factored f X _____________//BBY>>" BBB """" i BB__B"""q W where q is a global fibration (hence a local S-fibration) and i is a cofibration and a local weak equivalence. The map i has a factorization X ______i_____//_@@W>>" @@ """ j @@__@"""p" Z 3 where p is a local S-fibration and a local weak equivalence and j is an S- cofibration. Then j must also be a local weak equivalence, and the composite qp is a local S-fibration. Suppose that the map p : X ! Y is a local S-fibration and a local weak equivalence. Then p has a factorization p X ____________//_@@Y??~ @@ ~~~ j @@__@~~~q Z where j is an S-cofibration and q has the right lifting property with respect to all S-cofibrations. As before, q is therefore a local weak equivalence as well * *as a local S-fibration. The S-cofibration j is thus a local weak equivalence, and * *so the indicated lifting exists in the diagram X __1__//X>>___ ____ j|| _____|p|__ fflffl|_fflffl|____ Z __q__//Y The map p is therefore a retract of q, and thus has the right lifting property with respect to all S-cofibrations. The function complex is the standard one, and the model structure satisfies Quillen's axiom SM7, because the class CS was cooked up so that it would do so: it includes all maps (Bix @ n) [(Aix@ n)(Aix n) Bix n All local S-fibrations are sectionwise fibrations and pullback along section- wise fibrations preserves local weak equivalences, by a Boolean localization ar* *gu- ment. üD ally", all S-cofibrations are cofibrations and the standard structure is proper, so that pushout along S-cofibrations preserves local weak equiva-_ lences. |__| The case S = S0 for Theorem 2 is the Blander result Theorem 1, and the proof of Theorem 2 is an abstraction of Blander's proof. The meaning of the term öl cal" in the statement of Theorem 2 can also vary wildly. In particular, if the ambient topology is the chaotic topology, then local weak equivalences are sectionwise weak equivalences. In this case the corresponding local S-fibrations will just be called S-fibrations; thus an * *S- fibration is a map which has the right lifting property with respect to all maps which are S-cofibrations and sectionwise weak equivalences. Corollary 3. With these definitions, the category s Pre(C) and the classes of S-cofibrations, sectionwise weak equivalences and S-fibrations together satisfy the axioms for a proper closed simplicial model category. 4 It's a rather silly way to prove the result, but the Bousfield-Kan theorem is the case S = S0 of Corollary 3. Remark 4. Some question remains about whether or not the S-structure just coincides with either the projective of the standard structure, in the absence * *of good examples. Remark 5. If one is willing to forego the simplicial model structure, one can define CS to simply be the saturation of the class CP [ S in the full class of cofibrations C. Then one just gets back a proper closed model structure, by the same argument. A condition, most properly expressed in Proposition 9 below, for the closed model structure given by Theorem 2 to be cofibrantly generated will now be developed. The problem is to find a generating set for the class of maps which are simultaneously S-cofibrations and local weak equivalences, and Proposition 9 says that one can solve this problem for any given topology on the site C if one can first solve it for the chaotic topology. The method of proof of Proposition 9 involves an application of a bounded approximation technique which appears here as Lemma 7 and Corollary 8. This technique made its first appearance in [8]. The S-cofibrant replacements of local trivial cofibrations have a special ro* *le in this development. Explicitly, suppose that i : A ! B is a cofibration. Then an S-cofibrant replacement for i is a commutative diagram ~A_ßA__//A ~i|| |i| fflffl|fflffl| ~B_ßB__//B where ßA and ßB are sectionwise weak equivalences, and ~iis an S-cofibration. We require further that the simplicial presheaf ~Ais S-cofibrant. We can further assume that the maps ßA and ßB are trivial S-fibrations, and hence are sec- tionwise fibrations as well as sectionwise weak equivalences _ this is how one shows that all cofibrations have S-cofibrant replacements. There is a recognition principle for local S-fibrations which is based on the globally fibrant replacement for the chaotic topology. In general (for any topo* *l- ogy) a globally fibrant replacement for a map p : X ! Y of simplicial presheaves is a commutative diagram jX __ X _____//X (1) p|| |_p| |fflffl_fflffl|_ Y _jY__//Y __ _ where Y is globally fibrant, pis a global fibration, and the maps jX and jY are local weak equivalences. 5 Lemma 6. Suppose that the map p : X ! Y is an S-fibration and that in some globally fibrant replacement (1) for the chaotic topology the map _pis a local * *S- fibration. Then p is a local S-fibration. In particular, if _pis a global fibra* *tion for the topology on C, then p is a local S-fibration. Proof. Recall that the maps jX and jY are sectionwise weak equivalences. It follows from properness_for the simplicial set category that the induced map (p, jX ) : X ! Y x__YXis_a sectionwise weak equivalence. One also sees that the induced map ~p*: Y x__YX! Y has the right lifting property with respect to all maps which are S-cofibrations and local weak equivalences. Find a factorization X ____i___//GZ GG GGG |q| (p,jXG)##GGfflffl|_ Y x__YX where q is a sectionwise weak equivalence and an S-fibration, and i is an S-cofibration. Then q has the right lifting property with respect to all S- cofibrations, and i is sectionwise weak equivalence as well as an S-cofibration. The indicated lifting therefore exists in the diagram X __1__//X>>___ ____ i|| _____|p|___ fflffl|_fflffl|___ Z _~p*q//_Y It follows that the map p is a retract of a map ~p*q which has the right lifting property with respect to all maps which are both Scofibrations and local weak_ equivalences. The map p is therefore a local S-fibration. |__| Pick an S-cofibrant replacement ~i: ~A! ~Bfor each i : A ! B in a set of ff-bounded local trivial cofibrations. The cardinal ff is chosen sufficiently l* *arge such the set of all such maps i generates the class of locally trivial cofibrat* *ions for the standard model structure on the simplicial presheaf category s Pre(C). Let Sp denote the set of all such S-cofibrant replacements. We have the following simple consequences of the bounded cofibration con- dition [3] that will be used below. Lemma 7. Suppose given cofibrations A ! B ! X such that the composite A ! X is a weak equivalence and B is ff-bounded. Then there is a trivial cofibration C X such that B C and C is ff-bounded. Proof. This result is a consequence of the bounded cofibration condition. Under the stated assumptions, and for the diagram of cofibrations A | | fflffl| B _____//X 6 there is an ff-bounded object C with B C X and such that the induced cofibration C \ A ! C is a weak equivalence. Then A = C \ A because_ A C. |__| Corollary 8. Suppose given a diagram of trivial cofibrations X | | fflffl| A _____//Y such that A is ff-bounded. Then there is an ff-bounded subobject C Y such that the inclusions A C and C \ X ! C are trivial cofibrations. Proof. By the bounded cofibration condition, there is an ff-bounded subobject D1 Y with A D1 and such that D1 \ X ! D1 is a trivial cofibration. By the second lemma, there is a trivial cofibration C1 Y with C1 ff-bounded and such that D1 C1. Inductively form the string of cofibrations A D1 C1 D2 C2 . . . such that all objects are ff-bounded, all maps A ! Ci and Di\ X ! Di are trivial cofibrations. Let C = lim-!Ci Y . Then C is ff-bounded, the map A ! C * *__ is a trivial cofibration, and C \ X ! C is a trivial cofibration. |* *__| Proposition 9. Suppose that a map p : X ! Y is an S-fibration which has the right lifting property with respect to all maps ~i: ~A! ~Bappearing in the set * *Sp. Then p is a local S-fibration. __ __ Proof. Suppose that _p: X ! Y is a global fibrant replacement for p in the chaotic topology as above. We want to show that the map _pis a global fibration for the original topology on C, and then invoke Lemma 6. We have to solve all lifting problems __ A _____//X??__ (2) _____ i|| _____|_p|___ fflffl|_fflffl|____ B _____//Y where i : A ! B is an ff-bounded locally trivial cofibration. This amounts to showing that the vertex in the base of the simplicial set fibration __ __ __ (i*, _p*) : hom (B, X) ! hom (A, X) xhom (A,__Y)hom(B, Y) represented by the commutative square is the image of a vertex of the total space. The strategy is to replace the map i : A ! B by another cofibration up to weak equivalence for which the lifting exists, and then compare fibrations. 7 Factorize the commutative square (2) as the composite diagram iA pA __ A ____//_A1____//X i|| |i1| |_p| fflffl| fflffl|_fflffl|_ B _iB_//_B1pB__//Y where iA and iB are sectionwise trivial cofibrations, pA and pB are sectionwise fibrations, and i1 is a cofibration. To do this, factor the map B ! Y as a composite B j-!U -q!Y where q is a sectionwise fibration and j is_a sectionwise trivial projective co* *fi- bration. Factor the induced map A ! U x__YX! U as a composite __ A jA-!A1 ß-!U x__YX, where iA is a sectionwise trivial projective cofibration and ß is a sectionwise fibration. There is a cofibration i1 and a trivial global fibration q0such that* * the diagram __ A1 ___ß_//U x__YX i1|| || fflffl| fflffl| B1 ___q0__//_U commutes. Then the lifting iB in the diagram i1jA A _____//B1>>___ ____ i||iB____q0||____ fflffl|_fflffl|___ B ___j__//U The the map iB is a cofibration since j is a cofibration. Also the composites __ __ A1 ß-!U x__YX! X and 0 __ B1 q-!U -q!Y are sectionwise fibrations and are therefore candidates for pA and pB , respec- tively. Pull back the chaotic cofibrant replacement diagram (1) along the diagram pA __ A1 _____//X i1|| _p|| fflffl|_fflffl|_ B1 __pB_//Y 8 to obtain the commutative cube A2 _____________//X | BBB | ??? | BffAB | ?jX? | BBB | ??? | __ pA | ØØ_ | A1 ______|______//X | | |p | | | | | | i1| | | fflffl| | |fflffl | B2 _______|_____//_Y |_p BB | ?? | BBB | ??? | ffBBB | j ?? | B__fflffl| Y ?Øfflffl|Ø_ B1 _____pB______//Y Then the induced cofibrations ffA and ffB are sectionwise weak equivalences, since they are obtained by pulling back the sectionwise weak equivalences jX and jY along the sectionwise fibrations pA and pB , respectively. I claim that there are ff-bounded objects A3 and B3 such that there is a diagram of cofibrations A _____//A3____//_A1 | | | | | | fflffl| fflffl| fflffl| B _____//B3____//_B1 and such that the induced maps A3 \ A2 ! A3 and B3 \ B2 ! B2 are sec- tionwise weak equivalences, and such that the maps A ! A3 and B ! B3 are sectionwise weak equivalences. In effect, A3 is found by Corollary 8 applied to the sectionwise trivial cofibration ffA : A2 ! A1 and the ff-bounded subobject A A1. Then B3 is found by applying Corollary 8 to the sectionwise trivial cofibration ffB : B2 ! B1 and an ff-bounded subobject D of B1 which contains i1(A3) [ B and is sectionwise equivalent to B _ the object D exists by Lemma 7. Observe that the composite A2 ffA--!A1 i1-!B1 is a cofibration, so that the map A2 ! B2 is a cofibration. It also follows that the induced map A3 \ A2 ! B3 \ B2 is a cofibration, which is of course ff - bounded, and is a local weak equivalence because it's sectionwise equivale* *nt to the local weak equivalence i : A ! B. Suppose that the map ~i: ~A! ~Bis a projective cofibrant replacement for the ff-bounded locally trivial cofibration A3\A2 ! B3\B2. Then by construction, the fibration __ __ __ (i*, _p*) : hom (B, X) ! hom (A, X) xhom (A,__Y)hom(B, Y) is weakly equivalent to the fibration __ __ __ (~i*, _p*) : hom (B~, X) ! hom (A~, X) xhom (A~,__Y)hom(B~, Y) 9 and the element in __ __ ß0(hom (A, X) xhom (A,__Y)hom(B, Y)) corresponding the original diagram (2) maps to an element of __ __ ß0(hom (A~, X) xhom (A~,__Y)hom(B~, Y)) which lifts to an element of ß0hom (B~, X) by the assumption_for_the sectionwis* *e_ fibration p : X ! Y , and hence to an element of ß0hom (B~, Y). |__| Corollary 10. Suppose that the class consisting of all S-cofibrations and sec- tionwise weak equivalences has a set of generators. Then the closed model struc- ture given by Theorem 2 is cofibrantly generated. Proof. The set of S-cofibrations already has a set of generators, namely S, by construction. According to Proposition 9 a map p : X ! Y is a local S-fibration if and only it has the right lifting property with respect to all trivial local S-cofibrati* *ons appearing in the set SP and is an S-fibration. According to the assumption, p is an S-fibration if and only if it has the right lifting property with respect* * to some set S0 of sectionwise trivial S-cofibrations. Thus, p is a local S-fibratr* *ion if and only if it has the right lifting property with respect to the S-cofibrat* *ions_ in the set SP [ S0. |__| Corollary 11. The local projective model structure for simplicial presheaves on a small Grothendieck site C is cofibrantly generated. Proof. The S0-fibrations are the sectionwise fibrations, and a map p : X ! 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