Boolean Localization, in Practice

A new proof of the existence of the standard closed model structure for the category of simplicial presheaves on an arbitrary Grothendieck site is given. This proof uses the principle of Boolean localization.


Introduction
This paper is an exposition on the use of the topos theoretic principle of Boolean localization in demonstrating the existence of closed model structures for the categories of simplicial sheaves and presheaves on a Grothendieck site C.
Explicitly, a closed model category is a category M equipped with three classes of maps, called co brations, brations and weak equivalences, such that the following list of axioms is satis ed: CM1: M is closed under all nite limits and colimits.CM2: Suppose that the following diagram commutes in M: If any two of f, g and h are weak equivalences, then so is the third.CM3: If f is a retract of g and g is a weak equivalence, bration or co bration, then so is f.
CM4: Suppose that we are given a commutative solid arrow diagram U w u i X u p V w i i i i j Y where i is a co bration and p is a bration.Then the dotted arrow exists, making the diagram commute, if either i or p is also a weak equivalence.
CM5: Any map f : X !Y may be factored: (a) f = p i where p is a bration and i is a trivial co bration, and (b) f = q j where q is a trivial bration and j is a co bration.
Here, and as usual, one says that a map is a trivial co bration (respectively trivial bration) if it is both a co bration (respectively bration) and a weak equivalence.
The fundamental example of a closed model category is the category S of simplicial sets 11], 12], 2]: the co brations of S are the monomorphisms, the weak equivalences are the maps which induce isomorphisms in all possible homotopy groups of associated realizations, and the brations are the Kan brations.Recall that a Kan bration is a map q : X !Y of simplicial sets which has the \right lifting property" with respect to all inclusions n k n of horns in simplices.Here, the k th horn n k is the subcomplex obtained from the boundary @ n of the standard n-simplex by deleting the k th face from its list of generators.This paper addresses the various avours of homotopy theory that arise from contravariant simplicial set-valued diagrams, or presheaves of simplicial sets, de ned on small categories equipped with Grothedieck topologies.The list of all possible Grothendieck topologies includes the option of having no topology at all, so the theory includes that of ordinary small diagrams of simplicial sets.
There are both local and global homotopy theories for simplicial presheaves.The local theory is a theory of local weak equivalences and local brations.In particular, if one is working in a context so civilized as the category of simplicial presheaves on the category of open subsets of a topological space X, then a map (ie.natural transformation) f : Y !Z is a local bration if each of the induced maps f x : Y x !Z x , x 2 X, in stalks is a Kan bration of simplicial sets.Similarly, a local weak equivalence in this case is a map which induces weak equivalences in all stalks.One uses the same notion of local weak equivalence in the global theory (so that the two theories induce equivalent homotopy categories), along with co brations, or monomorphisms of simplicial presheaves, and then global brations are de ned by a lifting property.There is a di erence between the two theories: the Eilenberg-Mac Lane objects K(A; n) associated to sheaves of abelian groups A are certainly locally brant, but almost never globally brant.A globally brant model of K(A; n) is most properly thought of as a type of injective resolution of the abelian sheaf A, up to a degree shift.
The main results of this paper (Theorems 18, 27) together assert that the cobrations, local weak equivalences and global brations determine closed model structures on the categories of simplicial presheaves and simplicial sheaves on an arbitrary Grothendieck site, and that the homotopy categories associated to simplicial presheaves and sheaves on any such site are equivalent.In all of this, one of the main technical di culties is to arrange for a de nition of local weak equivalence which specializes to the stalkwise notion in cases where the underlying topos has enough points.Historically, this was done for simplicial presheaves in a somewhat ad hoc way 4], by using sheaves of homotopy groups for associated presheaves of Kan complexes.Here, one nds an alternative de nition of local weak equivalence and proofs of the main results which are based on the method of Boolean localization.The proof in the simplicial sheaf case is roughly what Joyal had in mind in his letter to Grothendieck 7] of 1984, except that it's been somewhat reverse engineered so that the relationship between sheaves of homotopy groups and weak equivalences comes out only after the fact.
Stated bluntly, the Boolean localization principle asserts that every Grothendieck topos can be faithfully imbedded in a topos that satis es the axiom of choice.The applicability of Boolean localization in homotopy theory was rst noticed by Van Osdol 14] in the 1970's, in his proof of what was then called the Illusie conjecture 3], but the descriptions of the underlying topos theory in the literature remained fragmentary until the appearance of the Mac Lane-Moerdijk book 9] in 1992.Even so, the principle as stated in 9] has to be reinterpreted somewhat to achieve the form that is used in this paper.This is done in the rst section below.This reinterpretation is trivial for a topos theorist, but quite opaque to almost everybody else.
The reader who is familiar with the \Simplicial presheaves" paper 4] will notice minor technical improvements here and there, particularly in the statement and proof of Lemma 12, and in the proof of Lemma 14, along with a more aggressive use of Kan's Ex 1 functor throughout.The basic thrust of using a trans nite small object argument to prove the factorization axiom CM5 survives, and the local bration concept continues to be an essential building block of the theory.
The idea appearing in the third section, that homotopy groups should really be bred group objects, is due to Joyal as far as I can tell.Such objects, combinatorially de ned, are exactly the right kind of thing to feed to a Boolean localization functor.They also have other uses: in particular, bred homotopy group objects appear implicitly (the -Kan condition) in the proof of the Bous eld-Friedlander theorem 1], 2] that recognizes homotopy cartesian diagrams of bisimplicial sets.One can also express the theory of long exact sequences for brations in these terms.
The writeup that follows assumes that the reader knows the basic exactness properties of a topos, and is familiar with the nuts and bolts of the associated sheaf construction.In this connection, there is one notational oddity: I use the notation L 2 F to denote the associated sheaf of a presheaf F. There is some precedent for this in the literature { see 13], for example.The notation is used in order to avoid the repeated appearance of some rather ugly very wide tildes.It is also assumed that the reader is familiar with the ordinary homotopy theory of simplicial sets 10], 2].Suppose that C is an arbitrary small Grothendieck site, and let E denote the sheaf category Shv(C) on the site C. A Boolean localization of E is a complete Boolean algebra B and a geometric topos morphism } : Shv(B) !E, such that the inverse image functor } : E !Shv(B) is faithful.
The de nition is a bit of a mouthful.A complete Boolean algebra B can be characterized as a poset having at least a terminal object 1 and an initial object 0 such that 0 6 = 1.Furthermore, B is required to have all limits (meets) and all colimits (joins), such that (1) B is complemented in the sense that every element x has a complement :x satisfying x _ :x = 1 and x ^:x = 0; and Documenta Mathematica 1 (1996) 245{275 (2) B satis es the distributive law x ^(y _ z) = (x ^y) _ (x ^z): The word \complete" refers to the fact that B is required to have all meets as opposed to all nite meets.Complete Boolean algebras also satisfy the in nite distributive law: x We shall discuss the homotopy theoretic consequences of the existence of Boolean localizations here, and defer to the Mac Lane-Moerdijk text for its proof.The applications depend explicitly on the fact that the topos Shv(B) satis es the axiom of choice in the sense that every epimorphism in Shv(B) has a section; we begin by giving an explicit proof of this result (Proposition 2).contradicting the maximality of the lifting s.
Generally, a map f : X !Y of presheaves on a Grothendieck site C is said to be a local epimorphism if for all sections y 2 Y (U), U 2 C, there is a covering sieve R hom( ; U) and elements x 2 X(V ) for each morphism : V !U in R, such that y lifts to X along R in the sense that (y) = f(x ) in Y (V ) for all 2 R, as in the picture In cases where there is an adequate notion of stalk, local epimorphisms are stalkwise epimorphisms: the point is that all sections should be \liftable" up to local re nement.
Examples of local epimorphisms of presheaves include all sheaf epimorphisms and the associated sheaf map : X !L 2 X.It's easy to show that local epimorphisms are closed under composition and that a map f : X !Y is a local epimorphism if and only if the induced map f : L 2 X !L 2 Y is an epimorphism of sheaves.
There is a dual notion of local monomorphism: a map g : A ! B of presheaves is a local monomorphism if for all x; y 2 A(U), U 2 C, g(x) = g(y) implies that there is a covering sieve R hom( ; U) such that (x) = (y) 2 A(V ) for all maps : V !U in R. Again, the associated sheaf map : X !L 2 X is a local monomorphism, local monomorphisms are closed under composition, and a map g is a local monomorphism if and only if the induced map g of associated sheaves is a monomorphism of sheaves.Now suppose that } : Shv(B) !E is a xed Boolean localization, where E = Shv(C).This means, in particular, that the inverse image functor } : E !Shv(B) is faithful.The functor } also preserves nite limits and all colimits { this is part of the de nition of a geometric morphism.The combination of these properties for } , together with basic exactness properties of Grothendieck topoi, has the following rather powerful consequence: Lemma 3. Suppose that } : F ! E is a geometric morphism of Grothendieck topoi.
Then the following are equivalent: (1) The inverse image functor } : E !F is faithful.
Proof: Suppose that } is faithful.This means that } (f 1 ) = } (f 2 ) for f 1 ; f 2 : Similarly, } re ects epimorphisms.A morphism of E is an isomorphism if and only if it is both a monomorphism and an epimorphism, so it follows that } re ects isomorphisms.
To see that (3) implies (1), observe that the maps f 1 ; f 2 : A ! B coincide if and only if their equalizer m : C !A is an isomorphism.Suppose that } (f 1 ) = } (f 2 ).Then } (m) is the equalizer of } (f 1 ) and } (f 2 ), by exactness of } , so that } (m) is an isomorphism.Thus, by assumption, m is an epimorphism and hence an isomorphism, so that f 1 = f 2 .Statement (4) implies statement (1) by a dual argument.

Closed model structures.
In this section, we show that any xed Boolean localization } : Shv(B) !Shv(C) determines a class of local weak equivalences of simplicial presheaves on the site C.
We further show that this class, along with the co brations (or monomorphisms) of simplicial presheaves, creates closed model structures for both simplicial presheaves and simplicial sheaves on C, in such a way that the associated homotopy theories are equivalent (Theorem 18).These closed model structures are seen to be independent of the choice of Boolean localization } in the next section.
The de nition of local weak equivalence is based on universally de ned notions of local bration and trivial local bration for simplicial presheaves on arbitrary sites, which specialize to Kan brations (respectively trivial Kan brations) in all sections in the case of morphisms of simplicial sheaves on a complete Boolean algebra B, via the axiom of choice.With co brations and local weak equivalences in hand, one de nes global brations by a right lifting property with respect to all maps which are both co brations and local weak equivalences, thus e ectively forcing the factorization axiom CM5 to be the non-trivial part of Theorem 18.To prove it, one shows that a map is a global bration if and only if it has the right lifting property with respect to some set of trivial co brations (Lemma 15).These are the -bounded trivial co brations, de ned with respect to a cardinal number which is su ciently large (and in particular larger than the cardinality of the set of morphisms of C).The most interesting part, technically, is the proof of Lemma 12.
Suppose that K is a nite simplicial set, and that Y is a simplicial presheaf on the Grothendieck site C. Write Y K for the presheaf de ned by simplicial set morphisms in sections via the formula Observe that Y K is a sheaf if Y is a simplicial sheaf, and that any exact functor preserves this de nition, so that, for example, the sheaf associated to Y K is canonically isomorphic to (L 2 Y ) K .Also, any geometric topos morphism preserves this construction.
One says that a map p : X !Y of simplicial presheaves is a local bration if the induced maps are local epimorphisms of presheaves for n > 0. Implicitly, a map p : Z !W of simplicial sheaves is a local bration if and only if the maps (1) are sheaf epimorphisms.More than this is true in the Boolean topos setting: One says that a map f : X !Y of simplicial presheaves has the local right lifting property with respect to the simplicial set inclusions @ n n if all of the maps X n (i ;f ) ! X @ n Y @ n Y n are local epimorphisms.One can speak, as well, about local right lifting properties with respect to more general collections of inclusions K L of nite simplicial sets.
In particular, a local bration is a map which has the local right lifting property with respect to all inclusions n k n .
Suppose that X is a simplicial presheaf.The simplicial presheaf Ex m X has nsimplices de ned by (Ex m X) n = hom(sd m n ; X): with simplicial structure maps induced by precomposition with the induced simplicial set maps sd m k !sd m n .The subdivision functor that we use here is the classical one: the subdivision sd n is the nerve of the poset of non-degenerate simplices of n , and the subdivision sdK of a simplicial set K is a colimit of simplicial sets sd m , indexed over the simplices m !K of K.The collection of \last vertex" maps sd m !m , m 0, induce a natural map X !ExX, and iteration of the construction produces a sequence of simplicial presheaf maps X !ExX !Ex 2 X !Ex 3 X !: : : The simplicial presheaf Ex 1 X is de ned to be the ltered colimit of these maps in the simplicial presheaf category.Write : X !Ex 1 X for the canonical map.
To put it a di erent way, Kan's original Ex 1 -construction 8], 2] is natural, so that it certainly applies to simplicial presheaves, and that's all we're doing here.In particular, the map : X !Ex is a pointwise weak equivalence.
Remark 5.All of the decorations that appear in the de nition of local weak equivalence are necessary.The categories of simplicial presheaves and sheaves on the site de ned by the power set P(I) of an in nite set I are very good examples to keep in mind.The power set P(I) is, of course, a complete Boolean algebra, so that the Boolean localization } can be taken to be the identity in this case.A simplicial presheaf X on P(I) is nothing more than a contravariant functor de ned on the category of subsets of I, and taking values in simplicial sets.The stalks of the simplicial presheaf X are the simplicial sets X i = X(fig) corresponding to sections in the various singleton subsets of I, and the associated sheaf L 2 X is de ned in sections at a subset U of I by L 2 X(U) = Y i2U X i : One says that a map f : X !Y of simplicial sheaves on P(I) is a stalkwise weak equivalence if all induced maps f i : X i !Y i , i 2 I are weak equivalences of simplicial sets.Observe that all induced maps in sections for the simplicial sheaf map f have the form for U I. It is known that in nite products do not necessarily preserve weak equivalences (see the next paragraph), so that a stalkwise weak equivalence f of simplicial sheaves may not induce weak equivalences of simplicial sets in all sections.In nite products do, however, preserve weak equivalences when all of the spaces X i and Y i are Kan complexes.The assertion that all of the X i , i 2 I, are Kan complexes is exactly what it means for the simplicial sheaf (or presheaf) X on P(I) to be locally brant.Thus, local weak equivalences as de ned above coincide with stalkwise weak equivalences for simplicial sheaves and presheaves de ned on P(I), and the implicit passage to locally brant models is fundamental.Example 6. Here's an example of a countable collection of contractible simplicial sets X n , n 0, such that the product Q i 0 X i is not contractible.Let X n be the subcomplex of n which is the union of the 1-simplices 1 n de ned by pairs of vertices (i; i + 1).The sequence of simplicial sets can therefore be identi ed with the graphs 0; 0 !1; 0 ! 1 !2; : : : with no compositions allowed.Then the vertices (0; 0; 0; 0; : : :) and (0; 1; 2; 3; : : :) cannot be in the same path component of the product Q n 0 X n .This observation can be expanded to a calculation of the homotopy type of the product: its path components are contractible, and two vertices x = (x n ), y = (y n ) of Q n 0 X n are in the same path component if and only if the list of combinatorial distances d(x n ; y n ) = jy n x n j (ie.number of 1-simplices between them in X n ) has a nite uniform bound.Lemma 7. Suppose, for a map f : X !Y of SPre(C), the preheaf maps X n (i ;f ) ! X @ n Y @ n Y n are local epimorphisms for n 0. Then f is a local weak equivalence and a local bration.

Documenta Mathematica 1 (1996) 245{275
Proof: If f has the local right lifting property with respect to all @ n n , then f has the local right lifting property with respect to all inclusions of nite simplicial sets K L. In e ect, the morphisms } L 2 X n (i ;f ) ! } L 2 X @ n } L 2 Y @ n } L 2 Y n are sheaf epimorphisms in Shv(B), and hence pointwise epimorphisms, so that all maps } L 2 X(b) !} L 2 Y (b) in sections are trivial Kan brations.But this means that the sheaf maps are pointwise epimorphisms by standard nonsense about trivial Kan brations, and are therefore sheaf epimorphisms.It follows that the maps are local epimorphisms.In particular, the map f is a local bration.
Also, if f has the local right lifting property with respect to all @ n n , then f has the local right lifting property with respect to all induced inclusions sd m @ n sd m n , so that Ex m f : Ex m X !Ex m Y has the local right lifting property with respect to all @ n n .But then Ex 1 f has the same local lifting property, and so does } L 2 Ex 1 f.In particular, } L 2 Ex 1 f is a pointwise trivial bration of simplicial sheaves on B, and is therefore a weak equivalence.Corollary 8.For any simplicial presheaf X, the canonical map : X !L 2 X has the local right lifting property with respect to all inclusions @ n n , and is therefore a local bration and a local weak equivalence.Lemma 9. Suppose that a map f : X !Y of simplicial presheaves on C is a pointwise weak equivalence in the sense that all maps in sections f : X(U) !Y (U); U 2 C; are weak equivalences of simplicial sets.Then f is a local weak equivalence.
Proof: The canonical map : X !Ex 1 X is a pointwise weak equivalence of simplicial presheaves, so it's enough to assume that f : X !Y is a pointwise weak equivalence of presheaves of Kan complexes, and then deduce that the map } L 2 f : of f in the simplicial presheaf category SPre(C), where p is a map which is a pointwise Kan bration and a pointwise weak equivalence, and the map i is right inverse to a map which is a pointwise Kan bration and a pointwise weak equivalence.The maps p and have the local lifting property with respect to all inclusions @ n n , so both maps are local brations and local weak equivalences by Lemma 7.
In particular, the maps } L 2 i and } L 2 p are pointwise weak equivalences, so that } L 2 f = (} L 2 p)(} L 2 i) is a pointwise weak equivalence.
Corollary 10.A map f : X !Y is a local weak equivalence of SPre(C) if and only if } L 2 f : } L 2 X !} L 2 Y is a local weak equivalence of SShv(B).
Proof: Observe that (by de nition, and with respect to the Boolean localization 1 : Shv(B) !Shv(B)) a map g : Z !W of SShv(B) is a weak equivalence if and only if the map L 2 Ex 1 g : L 2 Ex 1 Z !L 2 Ex 1 W is a pointwise weak equivalence of SShv(B) Also notice that there are natural isomorphisms L 2 Ex 1 } L 2 X = } L 2 Ex 1 X for X 2 SPre(C).Thus, L 2 Ex 1 } L 2 f is a pointwise weak equivalence if and only if } L 2 Ex 1 f is a pointwise weak equivalence.
Generally, for a xed property P of simplicial sets, one says that a map f : X !Y has the property P pointwise if each of the simplicial set maps f : X(U) !Y (U), U 2 C, in sections has the property P. The class of pointwise weak equivalences appearing in the statement of Lemma 9 is a common example.Pointwise (Kan) brations and pointwise trivial brations also occur frequently: a map f : X !Y of simplicial presheaves is a pointwise bration (respectively pointwise trivial bration) if all of the maps f : X(U) !Y (U), U 2 C, are brations (respectively trivial brations) of simplicial sets.We have already met such maps in the context of simplicial presheaves on a complete Boolean algebra B.
We shall also need the following partial converse to Lemma 7: Lemma 11.Suppose that X and Y are locally brant simplicial presheaves on C, and that the map q : X !Y is a local bration and a local weak equivalence.Then q has the local right lifting property with respect to all inclusions @ n n .
Proof: It su ces to assume that X and Y are locally brant simplicial sheaves, since the associated sheaf functor L 2 preserves local brations and local weak equivalences, and re ects the desired local right lifting property.
The induced map } L 2 Ex 1 q : } L 2 Ex 1 X !} L 2 Ex 1 Y is a pointwise weak equivalence of simplicial sheaves on B, since q is assumed to be a local weak equivalence.There is a natural isomorphism } L 2 Ex 1 = L 2 Ex 1 } , so the map L 2 Ex 1 } q : L 2 Ex 1 } X !L 2 Ex 1 } Y is a pointwise weak equivalence.The simplicial sheaf } X is a presheaf of Kan complexes on B, as is the object Ex 1 } X.Furthermore, the natural map L 2 : Documenta Mathematica 1 (1996) 245{275 } X !L 2 Ex 1 } X can be identi ed with the e ect of applying the associated sheaf functor L 2 to the canonical pointwise weak equivalence : } X !Ex 1 } X.If we can prove that the associated sheaf functor on B preserves pointwise weak equivalences between presheaves of Kan complexes, then we'd be done, since then L 2 would be a pointwise weak equivalence, and so the map } q : } X !} Y would be a pointwise property with respect to the maps @ n n , and of course Shv(B) satis es the axiom of choice, so that the maps L 2 and L 2 0 are pointwise trivial brations, and so L 2 f is a pointwise weak equivalence.
Pick some in nite cardinal such that is strictly larger than the cardinality of the set of morphisms of the site C. A simplicial presheaf X on C is said to be -bounded if jX n (U)j < for all n 0 and all objects U of C. Standard cardinal arithmetic implies that if X is -bounded, then so is its associated simplicial sheaf L 2 X.
Suppose that K is a simplicial set and U is an object of C. Then the simplicial presheaf L U K is de ned for V 2 C by Observe that morphisms of simplicial presheaves L U K !X are in one to one correspondence with simplicial set maps of the form K !X(U).If the simplicial set K is -bounded in the sense that jK n j < for n 0, then the simplicial presheaf L U K is -bounded.
Lemma 12. Suppose that f : X !Y is a local weak equivalence of simplicial presheaves on C, and that pullback along f preserves -bounded subcomplexes in the sense that if T is an -bounded subcomplex of Y then T Y X is an -bounded subcomplex of X. Suppose that there is a simplicial presheaf monomorphism i : Z , !Y where Z is -bounded.Then i has a factorization Z W Y such that W is -bounded and such that the projection map f : W Y X !W is a local weak equivalence.
Proof: First of all, one sees that any map of simplicial presheaves f : X !Y has a factorization X i u f X 4 4 4 4 7 p Y such that p is a pointwise Kan bration and i is a pointwise weak equivalence, and that this factorization is natural and preserves ltered colimits in f.In e ect, take the standard factorization Note nally that if X and Y are -bounded simplicial presheaves, then so is X.
The pulled back map p has the local right lifting property with respect to all inclusions @ n n , since Lemma 9 implies that p is a local weak equivalence as well as a pointwise bration, so that Lemma 11 applies.This means explicitly that given any diagram of simplicial set maps of the form @ n w a y u there is a covering sieve R hom( ; U) such that for each : V !U in R, there is a Then X is a ltered colimit of simplicial presheaves of the form T Y X, where T is an -bounded subobject of Y containing Z.It follows that there is anbounded S containing Z such that all the liftings x corresponding to the outer square live in S Y X.
where i is a pointwise weak equivalence.The map i is therefore a local weak equivalence by Lemma 9, so that f is also a local weak equivalence.
Corollary 13.Suppose that f : X !Y is a local weak equivalence of simplicial sheaves which satis es the boundedness condition of Lemma 12, and that there is a simplicial sheaf monomorphism i : Z , !Y where Z is -bounded.Then i has a factorization Z W Y such that W is -bounded and such that the projection map f : W Y X !W is a local weak equivalence.
Proof: Apply the associated sheaf functor to the output of Lemma 12.
A co bration of simplicial presheaves is a monomorphism A , !B of simplicial presheaves.A map of simplicial presheaves which is both a co bration and a local weak equivalence is called a trivial co bration.A global bration is a morphism p : X !Y of simplicial presheaves which has the right lifting property with respect to all trivial co brations.Finally a map which is simultaneously a global bration and a local weak equivalence is said to be a trivial global bration.Lemma 14. (1) Trivial co brations of simplicial presheaves are closed under pushout.
(2) Suppose that is an limit ordinal, thought of as a poset, and that there is a functor X : !SPre(C) such that for each morphism i j of , the induced map X(i) !X(j) is a trivial co bration.Then the canonical maps are trivial co brations.
(3) Suppose that the morphisms f i : X i !Y i are local weak equivalences for i 2 I. is a pushout of simplicial presheaves on B, where i is a co bration and a local weak equivalence.To show that i is a local weak equivalence, it su ces to show that the map i 0 in the pushout diagram of simplicial presheaves is a local weak equivalence.To see this, one invokes the ordinary patching lemma for simplicial sets and Corollary 8.But then the map i 0 is a pointwise weak equivalence since L 2 Ex 1 i is a pointwise weak equivalence, so we're done.For (2), let X : !SPre(B) be a functor as in the statement, and form a new functor Ex 1 X with Ex 1 X(i) de ned in the obvious way for i 2 , and consider the natural transformation : X !Ex 1 X arising from the pointwise weak equivalences : X(i) !Ex 1 X(i); i 2 : Then each morphism i j of induces a trivial co bration Ex 1 X(i) !Ex 1 X(j) by Lemma 9, and there is a commutative diagram where the ltered colimits are formed in the presheaf category, so that is a pointwise weak equivalence.It follows from Lemma 9 that one instance of i in the diagram is a local weak equivalence if and only if the other is, so it su ces to assume that each of the simplicial presheaves X(i) is a presheaf of Kan complexes.Now suppose that X is a diagram of presheaves of Kan complexes, and form the diagram which is induced the comparison transformation : X !L 2 X induced by the associated sheaf construction.The induced morphisms L 2 X(i) !L 2 X(j) are local weak equivalences of locally brant simplicial sheaves on the complete Boolean algebra B, and are therefore pointwise weak equivalences, so that the simplicial presheaf maps i : L 2 X(i) !lim !L 2 X(i) are pointwise weak equivalences and therefore local weak equivalences, by Lemma 9.The associated sheaf maps are local weak equivalences by Corollary 8, so that the original maps i : X(i) !lim !X(i) are local weak equivalences as well.
In the case of statement (3), the Ex 1 construction preserves disjoint unions of simplicial sets, so it su ces to assume that the simplicial presheaves X i and Y i are presheaves of Kan complexes.In that case, the induced morphisms L 2 f i : L 2 X i !L 2 Y i are local weak equivalences of locally brant simplicial sheaves on B, so that they are all pointwise weak equivalences.It follows that the induced morphism are pointwise and hence local weak equivalences.One nishes by observing that there where p is a global bration and i is a trivial co bration.
Proof: The proof is a trans nite small object argument.
Take a cardinal > 2 , where is the cardinality of the set of morphisms of the site C. We de ne a functor F : !SPre(C) # Y by rst setting F(0) = f : X !Y .
We let X( ) = lim !< X( ) for limit ordinals .Finally, the map X( ) ! X( + 1) is de ned by taking the set of all diagrams such that i D is an -bounded trivial co bration, and then forming the pushout Then i is a trivial co bration, by Lemma 14, as is the map i in the resulting diagram X where X( ) = lim !< X( ), and F( ) is induced by all maps F( ).In any diagram U w Y where i is a trivial -bounded co bration, the simplicial presheaf U is -bounded, so that g must factor through some subcomplex X( ) X( ) with < .It follows Lemma 17.Any simplicial presheaf map f : X !Y has a factorization X w f h h h j j Y W ' ' ' ) q where q is a trivial global bration and j is a co bration.
Proof: First of all, if a map f : X !Y has the right lifting property with respect to all morphisms of the form A L U n , then f is a global bration and a local weak equivalence.In e ect, f has the right lifting property with respect to all co brations by an argument similar to that of Lemma 15, so that f is a global bration, and f has the right lifting property with respect to all co brations of the form L U @ n L U n , so that f is a pointwise weak equivalence and hence a local weak equivalence by Lemma 9.
The existence of the required factorization is now a consequence of a trans nite small object argument similar to that given for Lemma 16. of the associated homotopy categories.
Proof: The only non-trivial parts of the respective demonstrations are the factorization axiom and CM4, for both simplicial presheaves and simplicial sheaves.But the factorization axioms follow from Lemma 16 and Lemma 17, and their simplicial sheaf counterparts (which have the same arguments), and CM4 is a consequence of the assertion that every trivial global bration has the right lifting property with respect to all co brations, for both categories.
For the latter, observe that if p : X !Y is a global bration and a local weak equivalence, then the proof of Lemma 17 shows that p has a factorization X w p h h h j j Y W; ' ' ' ) q where j is a co bration and q has the right lifting property with respect to all cobrations and is a local weak equivalence.But then j is a trivial co bration, so that there is a commutative and so p is a retract of q.
The equivalence of categories Ho(SShv(C)) ' Ho(SPre(C)) is induced by the inclusion SShv(C) SPre(C) and its left adjoint, the associated sheaf functor L 2 : SPre(C) !SShv(C).Both of these functors preserve local weak equivalences, and the canonical simplicial presheaf map X !L 2 X is a weak equivalence, by Corollary 8.
Suppose that X is a simplicial presheaf and that K is a simplicial set.There is a simplicial presheaf hom(K; X), which is de ned in sections by hom(K; X)(U) = hom(K; X(U)); U 2 C; where hom(K; X(U)), denotes the ordinary function space object in the category of simplicial sets.The simplicial presheaf hom( 1 ; X) is the path object that was used in the proof of Lemma 9.
The ordinary exponential law for simplicial sets induces a natural isomorphism of the form hom(X; hom(K; Y )) = hom(X K; Y ); where the indicated morphisms are in the category of simplicial presheaves, and X K is the simplicial presheaf de ned in sections by The main homotopical result about function spaces of this type is the following: Lemma 19.Suppose that q : X !Y is a local bration of simplicial presheaves on C, and that i : K , !L is a co bration of simplicial sets where L is nite in the sense that it has only nitely many non-degenerate simplices.Then the induced simplicial presheaf map hom(L; X) (i ;q ) ! hom(K; X) hom(K;Y ) hom(L; Y ) is a local bration, and this map is a local weak equivalence if q is a local weak equivalence or if i is a trivial co bration of simplicial sets.
Proof: There is a natural isomorphism } L 2 hom(K; X) = hom(K; } L 2 X) for all nite simplicial sets K and simplicial presheaves X, since the associated sheaf functor L 2 and the Boolean localization functor } both preserve nite limits.The map } L 2 q : } L 2 X !} L 2 Y is a pointwise Kan bration, so that the map hom(L; } L 2 X) is a pointwise Kan bration, which is a pointwise weak equivalence if i is a trivial co bration or if } L 2 q is pointwise trivial.
One should know rst that the functor X 7 !X K preserves local weak equivalences in simplicial presheaves X, for all simplicial sets K.For this, there are natural The functor X 7 !} L 2 Ex 1 X K takes local weak equivalences to pointwise weak equivalences, and so the desired result follows from Corollary 10.
It follows that, in the diagram the maps i n and i are trivial co brations.
There is a corresponding statement about simplicial sheaves, which is an immediate corollary of Lemma 22.
Corollary 23.The simplicial presheaf category SPre(C) and the simplicial sheaf category SShv(C) are both closed simplicial model categories.
One says that a closed model category is proper if weak equivalences are preserved by pullback along brations and by pushout along co brations.Traditionally, weak equivalences of simplicial sheaves and presheaves have been dened via sheaves of homotopy groups, which we haven't even mentioned yet.We have so far used a de nition of weak equivalence that appears to depend on a xed Boolean localization } : Shv(B) !E = Shv(C).In this section we will show that this apparent dependence on } can be removed by introducing a notion of bred homotopy group objects which is preserved by the inverse image functor } and specializes to the standard homotopy groups for ordinary simplicial sets over all vertices (but see also Remark 28 below).These homotopy group objects are made up of sheaves of homotopy groups in the usual sense, and our de nition of weak equivalence is seen to coincide with the familiar one.
Suppose that X is a Kan complex, with base point x.The set underlying the homotopy group n (X; x) can be identi ed with the set of path components 0 F n;x X, where the F n;x X is de ned by the pullback diagram F n;x X w u hom( n ; X) u i w x hom(@ n ; X) and i is the bration induced by the inclusion i : @ n , ! n .Note, in particular, that F n;x X is a Kan complex, so that 0 F n;x X can be identi ed with a set of homotopy classes of vertices.
To collect all such de nitions together, use the notation X 0 for the discrete simplicial set F x2X0 on the set of vertices of X as well as for the set of vertices itself, and form the pullback F n X w u hom( n ; X) u i X 0 w hom(@ n ; X); where the map X 0 !hom(@ n ; X) takes the vertex x to the map x : @ n !X which factors through x.The simplicial set X 0 is a Kan complex, so that F n X = G x2X0 F n;x X is a Kan complex bred over X 0 , and we write There is a canonical function n X = 0 F n X !X 0 which gives n X a bred structure over the set of vertices X 0 .
To see the group multiplication, let 0;n 2] n+1 be the subcomplex which is generated by the simplices d i : n !n + 1, 0 i n 2, and write K n = 0;n 2] sk n 1 n+1 let j denote the inclusion K n n+1 .Form the pullback diagram G n X w u hom( n+1 ; X) u j X 0 w hom(K n ; X) in the category of simplicial sets.The maps d i : n !n+1 induce morphisms d i : G n X !F n X of spaces bred over X 0 for n 1 i n + 1.Furthermore, the induced map (d n 1 ; d n+1 ) : G n X !F n X X0 F n X is surjective, since it is induced by pulling back a trivial bration hom( n+1 ; X) !hom( n n 1 n ; X).By looking at vertices and taking path components one sees, via the standard constructions, that there is a unique map m : 0 F n X X0 0 F n X !0 F n X of objects bred over X 0 making the following diagram commute: Observe that the map m can be identi ed with the map G x2X0 n (X; x) n (X; x) !G x2X0 n (X; x) that one obtains by collecting all of the ordinary homotopy group multiplication maps together.
The group inverse : n X !n X is de ned as a bred map over X 0 by letting n+1 n 1;n+1 be the subcomplex of n+1 generated by the simplices d i for i 6 = n 1; n+1, and forming the pullback which consists of the group inverses for the regular homotopy groups.
The identity e : X 0 !n X is the section of the structure map n X !X 0 which is induced by the canonical section of the simplicial set map F n X !X 0 .Of course e specializes to the map !n (X; x) which picks out the identity map of the group n (X; x) over each summand of X 0 .The de ning axioms for the group structures on the various n (X; x) can now be used to show that the bred objects n X !X 0 , together with the multiplication map m, the inverse map and the identity section e, give n X the structure of a group object in the category of sets bred over X 0 .This group object is abelian if n 2. The existence of the group object isn't news by itself, but the descriptions of the maps m, and e are combinatorial and functorial, and are therefore more broadly applicable.
Observe that a map f : X !Y of Kan complexes is a weak equivalence if and only if (1) the induced map f : 0 X !0 Y of path components is a bijection, and The bottom sequence is a coequalizer in the sheaf category, while the top sequence is a coequalizer in the presheaf category.
The sheaf epimorphism L 2 c is a pointwise epimorphism, by the axiom of choice (Proposition 2), so that the canonical presheaf map : p 0 X !0 X is also a pointwise epi.The composite map displayed by the dotted arrow can be identi ed with the sheaf map associated to the presheaf epimorphism (d 0 ; d 1 ), so it's a sheaf epi and hence a pointwise epi, again by the axiom of choice.If L 2 c(x) = L 2 c(y) in 0 X, then (x; y) de nes an element of X 0 0X X 0 , and so there is a section z of X 1 which maps to (x; y) under the dotted composite.But then x = d 1 z and y = d 0 z, so that x and y represent the same element of p 0 X.The associated sheaf map : p 0 X !0 X is therefore pointwise monic as well as pointwise epi.
Suppose that X is a locally brant simplicial sheaf on the site C.The homotopy group sheaves n X !X 0 are de ned as sheaves bred over the sheaf of vertices X 0 by letting F n X be the locally brant simplicial sheaf de ned by the pullback diagram F n X w u hom( n ; X) u i X 0 w hom(@ n ; X); and then by de ning n X = 0 F n X, where the latter denotes the sheaf of path components of the simplicial sheaf F n X, as above.The group object multiplication m : n X X0 n X !n X is de ned by analogy with the group object multiplication

Lemma 1 .
Suppose that F is a sheaf (of sets) on a complete Boolean algebra B. Then the category Sub(F) of subobjects of F is a complete Boolean algebra.Proof: The category Sub(F) has all meets and joins, and satis es the in nite distributive law, by an argument on the presheaf level.Given G 2 Sub(F), de ne clear that G ^:G = ;; the interesting bit is to show that G _ :G = F. Documenta Mathematica 1 (1996) 245{275

Lemma 4 .
A map p : Z !W of simplicial sheaves on a complete Boolean algebra B is a local bration if and only if the induced maps in sections p : Z(b) !W(b) are Kan brations of simplicial sets for all b 2 B. Proof: The sheaf epimorphisms Z n (i ;p ) ! Z n k W n k W n have sections, by Proposition 2, so that the maps Z n (b) (i ;p ) ! Z n k (b) W n k (b) W n (b) in sections are surjective, for all b 2 B.
a pointwise weak equivalence of simplicial sheaves on B.Since X and Y are presheaves of Kan complexes, the classical method of replacing a map by a bration gives a factorization Kan bration and a pointwise weak equivalence, and would therefore have the local right lifting property with respect to all inclusions @ n n .Finally, our faithful functor } re ects this local right lifting property.Suppose that f : Z !W is a pointwise weak equivalence between presheaves of Kan complexes on B, and form a diagram of simplicial presheaf maps a pointwise trivial bration and i is right inverse to a pointwise trivial bration 0 : Z !Z.The associated sheaf functor L 2 preserves the local right lifting weak equivalence.Proof: It su ces, by Corollary 10 and Corollary 8, to prove all three statements for the category SPre(B) of simplicial presheaves on the complete Boolean algebra B.For statement(1), suppose that the diagram Theorem 18.With respect to the de nitions of co bration, weak equivalence and global bration given above, (1) the category SPre(C) of simplicial presheaves is a closed model category, (2) the category SShv(C) is a closed model category, (3) the inclusion SShv(C) SPre(C) induces an equivalence Ho(SShv(C)) ' Ho(SPre(C)) n 1 and d n+1 induce functions d n 1 ; d n+1 : H n X !F n X of spaces bred over X 0 , and both of these maps are surjective because they are induced by trivial brations of the form (d j ) : hom( n+1 ; X) !hom( n ; X).Then is the unique map of sets bred over X 0 which makes the following diagram commute:

1 (
veri ed, given that the displayed group objects consist of ordinary homotopy groups.Suppose that Y is a simplicial presheaf, and de ne a presheaf p 0 category.Let 0 Y denote the associated sheaf for p 0 ; one oftens says that 0 Y is the sheaf of path components of Y .Observe that the canonical map Y !L 2 Y from Y to its associated sheaf induces an isomorphism 0 Y = 0 L 2 Y .Documenta Mathematica 1 (1996) 245{275Lemma 25.Suppose that X is a locally brant simplicial sheaf on a complete Boolean algebra B. Then the associated sheaf map : p 0 X !0 X is an isomorphism of presheaves.Proof: The locally brant simplicial sheaf X is a presheaf of Kan complexes, by Lemma 4. It follows that the canonical presheaf map X

for
Kan complexes: de ne a locally brant simplicial sheaf G n X by requiring that :Sets !E of geometric morphisms such that two maps f; g : F ! G of E coincide if and only if x i f = x i g for all i 2 I.The set category Sets is equivalent to the i2I y i = x.The in nite distributive law guarantees that the covering families of B satisfy the axioms for a pretopology, and hence give rise to a category of sheaves Shv(B).Boolean localizations exist for all Grothendieck toposes E: this is a major theorem of topos theory (Mac Lane and Moerdijk call it Barr's Theorem 9, p.513], but a result of Diaconescu plays a major part { see 9, p.511]).It's also important to know, so we don't leave the realm of small sites, that the construction doesn't blow up: if the cardinality of the set of morphisms of the underlying site C is bounded by some in nite cardinal , then jBj < .Boolean localization is a vast generalization of what it means for a topos to have enough points.Speci cally, the topos E has enough points if there is a collection x i Taking the union of all such subcomplexes S over thebounded set of diagrams of the form (2) gives an -bounded subcomplex Z 1 of Y such that Z Z 1 Y , and such that all local lifting problems (2) are solved in Z 1 Y X. Repeat the construction to obtain a sequence of -bounded subobjects Z = Z 0 Z 1 Z 2 : : : such that all local lifting problems Theorem 24.The simplicial presheaf category SPre(C) and the simplicial sheaf category SShv(C) are proper closed simplicial model categories.with q a global bration and g a local weak equivalence.To show that g is a local weak equivalence, it su ces, by Corollary 8 and Corollary 10, to assume that X, Y , and Z are simplicial sheaves on the complete Boolean algebra B. By Corollary 8 and exactness, applying the composite functor L 2 Ex 1 doesn't change the problem, so it su ces to assume that X, Y and Z are locally brant simplicial sheaves on B. But then g is a pointwise weak equivalence, so that g is a pointwise, i a co bration and f a local weak equivalence.By the patching lemma for simplicial sets, Corollary 8 and Corollary 10, it su ces to assume that A, B and X are locally brant simplicial sheaves on B and that the pushout is formed in the category of simplicial presheaves on B. In that case, f is a pointwise weak equivalence, so that f is a pointwise hence local weak equivalence, by Lemma 9.