Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids by Zhi-Ming Luo December 20, 2002 Abstract We prove that the category of presheaves of simplicial groupoids and * *the category of presheaves of 2-groupoids have Quillen closed model structures. We also * *show that the homotopy categories associated to the two categories are equivalent to t* *he homotopy categories of simplicial presheaves and homotopy 2-types, respectively. Key words: presheaves of simplicial groupoids, presheaves of 2-groupoids* *, Quillen closed model category 1 Introduction A Quillen closed model category D is a category which is equipped with three * *classes of morphisms, called cofibrations, fibrations and weak equivalences which togeth* *er satisfy the following axioms [9], [10], [3]: CM1: The category D is closed under all finite limits and colimits. CM2: Suppose that the following diagram commutes in D: g X ___________//@@Y~ @@ ~~~ h @__@@""f~~~ Z 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, fibration or cofibra* *tion, then so is f. CM4: Suppose that we are given a commutative diagram U ____//_X>>___ ____ i||_____p||___ fflffl|fflffl|____ V _____//Y 1 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 2 where i is a cofibration and p is a fibration. Then the lifting 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 fibration and i is a trivial cofibration, and (b)f = q . j where q is a trivial fibration and j is a cofibration. The central foundational theorem of simplicial homotopy theory asserts tha* *t the cat- egory S of simplicial sets has a closed model structure [9]. Mathematicians h* *ave found a large quantity of categories enjoying the closed model structures. For exampl* *e, the category of simplicial groupoids by Dwyer-Kan [2], [3] , the category of 2-groupoids b* *y Moerdijk- Svensson [8], the category of simplicial presheaves by Jardine [5], the categ* *ory of simplicial sheaves by Joyal [7] and so on. Crans [1] uses adjoint functors to prove tha* *t a kind of sheaves have closed model structures according to a well-known closed model c* *ategory. We use similar technique, basing on Jardine's paper [5], to prove that som* *e presheaves have the closed model structures. One is the category of presheaves of simpli* *cial groupoids in the section 2 and the other one is the category of presheaves of 2-groupoi* *ds in the section 3. We also show that the homotopy category associated to the first category i* *s equivalent to the homotopy category of simplicial presheaves, the homotopy category asso* *ciated to the latter category is equivalent to the homotopy category of homotopy 2-types. 2 Presheaves of simplicial groupoids Let C be a fixed small Grothendieck site. sGdPre(C) is the category of presh* *eaves of simplicial groupoids on C; its objects are the contravariant functors from C * *to the category sGd of simplicial groupoids, and its morphisms are natural transformations. Dwyer and Kan show that [2], [3], with the following definitions of weak e* *quivalence, fibration and cofibration, the category sGd of simplicial groupoids satisfies* * the axioms for a closed model category. A map f : G ! H of simplicial groupoids is said to be a weak equivalence o* *f sGd if (1) the morphism f induces an isomorphism ß0G ~=ß0H, and (2) each induced map f : G(x, x) ! H(f(x), f(x)), x 2 Ob(G) is a weak equiva* *lence of simplicial groups (or of simplicial sets). A map g : H ! K of simplicial groupoids is said to be a fibration of sGd if (1) the morphism g has path lifting property in the sense for every object x* * of H and morphism ! : g(x) ! y of the groupoids K0, there is a morphism ^!: x ! z* * of H0 such that g(^!) = !, and (2) each induced map g : H(x, x) ! K(g(x), g(x)), x 2 Ob(H) is a fibration o* *f simplicial groups (or of simplicial sets). 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 3 A cofibration of simplicial groupoids is defined to be a map which has the l* *eft lift- ing property with respect to all morphisms of sGd which are both fibrations and* * weak equivalences. Recall the adjunction_between the loop groupoid functor G : S ! sGd and the* * universal cocycle functor W [3, Lemma V.7.7 ]. By applying these functors pointwise to s* *implicial presheaves and presheaves of simplicial groupoids, one obtains functors ___ G : SP re(C) Æ sGd P re(C) : W So there is Proposition_2.1.The functor G : SP re(C) ! sGd P re(C) is left adjoint to the f* *unctor W . A_map_f :_X_! Y_in_the category sGdPre(C) is said to be a fibration if the i* *nduced map W (f) : W X ! W Y is a global fibration in the category SPre(C) in the sens* *e of [5]. A map g :_Z_! U in_the_category_sGdPre(C) is said to be a weak equivalence i* *f the induced map W (g) : W Z ! W U is a topological weak equivalence in the category* * SPre(C) in the sense of [5]. A cofibration in the category sGdPre(C) is a map of presheaves of simplicial* * groupoids which has the left lifting property with respect to all fibations and weak equi* *valences. Say that a map of presheaves of simplicial groupoids f is a trivial fibratio* *n if it is both a fibration and a weak equivalence; a map g is a trivial cofibration if it* * is both a cofibration and a weak equivalence. In [5], Jardine defines an important concept. The site C is "small", so that* * there is a cardinal number ff such that ff is larger than the cardinality of the set of su* *bsets PMor(C) of the set of morphisms Mor(C) of C. A simplicial presheaf X is said to be ff -* * bounded if the cardinality of each Xn(U), U 2 C, n 0, is smaller than ff. A map p : X ! Y in the category SPre(C) is a global fibration if and only if* * it has the right lifting property with respect to all trivial cofibrations i : U ! V s* *uch that V is ff-bounded [5, Lemma 2.4]. Then a map q : G ! H in the category sGdPre(C) is a * *fibration if and only if it has the right lifting property with respect to all maps G(i) * *: GU ! GV induced by those maps i : U ! V since there exist the adjoint diagrams: ___ GU ____//_G>>" U _____//WG== (D ) "" --- G(i)||""" q|| i|| --- __W(q)|| fflffl|"fflffl|" fflffl|-fflffl|-_ GV ____//_H V ____//_WH For each W 2 C, GV (W )n is the free groupoid on generators x 2 V (W )n+1 subje* *ct to some relations, and Ob(GV (W )) = V (W )0, so the cardinality of each Mor(GV (W )n),* * n 0 and Ob(GV (W )) is smaller than fi = max(2ff, 1). We also call the presheaf of * *simplicial groupoids GV is fi - bounded. 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 4 When G is a simplicial group there is a pullback diagram G __i_//_W G | q| | | fflffl|_fflffl|_ * __*_//_WG where q is a_fibration_of simplicial sets [3, Lemma V.4.1], G is the fibre over* * the_unique vertex * 2 W G. G is a simplicial group, so G is a Kan complex [3, Lemma I.3.4]* *. W G is a Kan complex [3, Corollary V.6.8], so is W G, then for any vertex v 2 G there ex* *ists a long exact sequence ___ @ ... ! ßn(G, v) -i*!ßn(W G, v) -q*!ßn(W G, *) -! ßn-1(G, v) ! ... ___ @ i* q* ___ ... -q*!ß1(W G, *) -! ß0(G) -! ß0(W G) -! ß0(W G) by Lemma I.7.3 in [3]. W G is contractible [3, Lemma V.4.6], so ßn(W G, v) = 0* *, n 1; and ß0(W G) = 0, since for any two vertices a, b 2 W G0 = G0, there exists a 1-* *simplex (s0b, b-1a) 2 W G1 = G1 x G0, s.t., d1(s0b, b-1a) = b, d0(s0b, b-1a) = a. Then ___ ßn(G, v) = ßn+1(W G, *), n 1 ___ ß0G = ß1(W G, *) For an ordinary groupoid H, it's standard to write ß0H for the set of path c* *omponents of H. By this one means that ß0H = Ob(H)= ~ where there is a relation x ~ y between two objects of H if and only if there i* *s a morphism x ! y in H. If now A is a simplicial groupoid, all of the simplicial structure functors * *`* : An ! Am induce isomorphisms ß0An ~=ß0Am . We shall therefore refer to ß0A0 as the set * *of path components of the simplicial groupoid A, and denote_it by ß0A. ___ When_A is a simplicial groupoid, Ob(A) = (W A)0, Mor(A0) = (W A)1, so ß0A ~= ß0(W A). Choose a representative x for each [x] 2 ß0A, the inclusion G i : A(x, x) ! A [x]2i0A is a homotopy equivalence of simplicial groupoids, and the induced map ___ ___ G ___ W (i) : W ( A(x, x)) ! W A [x]2i0A ___ ___F is a weak equivalence of simplicial sets. W preserves disjoint unions, W ( [x* *]2i0AA(x, x)) = F ___ [x]2i0AW(A(x, x)) [3, p. 303,304 ]. ___ G ___ ßn(W ( A(x, x)), x) = ßn(W (A(x, x)), *) ~=ßn-1(A(x, x), v), n 2, v 2* * A(x, x)0 [x]2i0A 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 5 ___ G ___ ß1(W ( A(x, x)), x) = ß1(W (A(x, x)), *) ~=ß0(A(x, x)). [x]2i0A so one obtains ___ ßn(A(x, x), v) ~=ßn+1(W A, x), n 1, x 2 Ob(A), v 2 A(x, x)0, ___ ß0(A(x, x)) ~=ß1(W A, x). According to the definition of topological weak equivalence of simplicial pr* *esheaves in [5] and the relations between simplicial groupoids and simplicial sets, we can give* * an explicit description of weak equivalence of presheaves of simplicial groupoids. For any presheaf of simplicial groupoids X and any object U 2 C and x 2 Ob(X* *(U)), X(U)(x, x) is a simplicial group. Associated to this presheaf of simplicial gro* *upoids X on C and * 2 X(U)(x, x)0 is a presheaf ßsimpn(X|U, x, *)(n 1) on the comma categ* *ory C # U, the presheaf of simplicial homtopy groups of X|U, based at *, which is defined * *by (C # U)op! Grp ' : V ! U 7! ßn(X(V )(xV , xV ), *V ) where xV and *V are the images of x and * in X(V ) under the map X(U) ! X(V ) w* *hich is induced by V ! U, respectively; and the simplicial homotopy group ßn(X(V )(xV ,* * xV ), *V ) exists since the simplicial group X(V )(xV , xV ) is a Kan complex [3, Lemma I.* *3.4]. Let ßn(X|U, x, *) be the associated sheaf of the presheaf ßsimpn(X|U, x, *),* * i.e., ßn(X|U, x, *) = L2ßsimpn(X|U, x, *). Then ßn(X|U, x, *) is a sheaf of groups which is abelian i* *f n 2. The sheaves ß0(X|U, x) and ß0(X) of path components are defined similarly. A map f : X ! Y of presheaves of simplicial groupoids is said to be a weak e* *quivalence if it induces isomorphisms of sheaves f* : ßn(X|U, x, *) ~=ßn(Y |U, fx, f*), n 1, U 2 C, x 2 Ob(X(U)), * 2 X(U)* *(x, x)0 f* : ß0(X|U, x) ~=ß0(Y |U, fx). f* : ß0(X) ~=ß0(Y ). In view of Proposition 1.18 in [5], the weak equivalence is just same as the* * combinatorial weak equivalence in [5]. Since the weak equivalence is defined by the isomorphi* *sms between sheaves of groups, thus, Proposition 1.11 in [5] implies ( or directly followed* * from the CM2 of category SPre(C)) Lemma 2.1. Given maps of presheaves of simplicial groupoids f : X ! Y and g : * *Y ! Z, if any two of f, g, or g O f are weak equivalences, then so is the third. ___ Lemma 2.2. The functor X 7! W G(X) preserves weak equivalences of simplicial p* *resheaves. 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 6 ___ Proof.When T is a simplicial set, the natural simplicial map j : T_!_W G(T ) is* * a weak equivalence of simplicial sets [3, Theorem V.7.8]. So the map X ! W G(X) is a p* *ointwise weak equivalence of simplicial presheaves, then it is a weak equivalence. There exists a commutative diagram ''X ___ X ____//_WG(X) | __| f|| W|G(f) fflffl|___fflffl| Y _''Y//_WG(Y ) where_both jX and jY are weak equivalences, if f : X ! Y is a weak equivalences* *, so_is W G(f) by the CM2 of the closed model category SPre(C). * * |__| Lemma 2.3. The functor G : S P re(C) ! sGd P re(C) preserves cofibrations and * *weak equivalences. Proof.The adjoint diagrams (D) imply that the functor G preserves cofibrations.* * Lemma_2.2 implies that G preserves weak equivalences. * * |__| Lemma 2.4. The category sGdPre(C) has all pushouts, and is hence cocomplete. T* *he class of cofibrations in sGdPre(C) is closed under pushout. Proof.The category sGd has all pushouts and is cocomplete, so is the category s* *GdPre(C),__ since we can take the pushout and colimit pointwise. The second statement is ob* *vious. |__| There exists a Kan Ex1 functor from SPre(C) to SPre(C), such that Ex1 X is l* *ocally fibrant for any simplicial presheaf X and the canonical map : X ! Ex1 X is a * *pointwise weak equivalence [6]. Fix a Boolean localization " : Shv(B) ! E, and consider the functors 2 "* SP re(C) L-!SE -! SShv(B) relating the categories of simplicial presheaves on C and the categories of sim* *plicial sheaves and the categories of simplicial objects in the categories of sheaves Shv(B), w* *here L2 is the associated sheaf functor. In [6] Jardine proves that the topological weak * *equivalence between simplicial presheaves [5] coincides with the local weak equivalence [6]* *, i.e., a map f : X ! Y of simplicial presheaves on C is a topological weak equivalence if th* *e induced map "*L2 : "*L2Ex1 X ! "*L2Ex1 Y is a pointwise weak equivalence. Notice that there is a commutative diagram 2 "* sGd P re(C)_L__//_sGd_E__//_sGd Shv(B) __W| |__ __| | |W W| fflffl| fflffl| fflffl| SP re(C)______//SE___*___//SShv(B) L2 " ___ W G is locally fibrant simplicial presheaf for any presheaf of simplicial group* *oid G, so a map f : G ! H of presheaves of simplicial groupoids on C is a weak equivalence if t* *he induced map "*L2 : "*L2G ! "*L2H is a pointwise weak equivalence. 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 7 Proposition 2.2.Trivial cofibrations of presheaves of simplicial groupoids are * *closed under pushout. Proof.Suppose that G ____//_C i|| |i0| fflffl|fflffl| H ____//_D is a pushout in the category sGdPre(C). i is a trivial cofibration, then i0is a* * cofibration by Lemma 2.4. The heart of the matter for this proof is the weak equivalence. Both L2 and * *"* are left adjoint functors, so the functor "*L2 preserves the pushout "*L2G ____//_"*L2C "*L2(i)|| |"*L2(i0)| fflffl| |fflffl "*L2H ____//_"*L2D the map "*L2(i) is a pointwise weak equivalence and pointwise cofibration, so f* *or any U 2 B, the diagram "*L2G(U) ____//_"*L2C(U) "*L2(i)|| |"*L2(i0)| fflffl| fflffl| "*L2H(U) ____//_"*L2D(U) is a pushout in the category sGd. The category sGd is a closed model category,* * then the map "*L2(i) is a trivial cofibration, so is "*L2(i0) : "*L2C(U) ! "*L2D(U).* * Then "*L2(i0) : "*L2C ! "*L2D is a pointwise weak equivalence, so i0 : C ! D is a we* *ak__ equivalence in the category sGdPre(C). * * |__| Given a trivial cofibration i : A ! B in the category SPre(C), suppose that GA _____//C G(i)|| i0|| fflffl| fflffl| GB _____//D is a pushout in the category sGdPre(C). The map G(i) is a trivial cofibration b* *y Lemma 2.3, then the map i0is a trivial cofibration. Lemma 2.5. Every map f : X ! Y of presheaves of simplicial groupoids may be fa* *ctored f X ___________//@@Y??~ @@ ~~~ i@@__@~~p~ Z where i is a trivial cofibration and p is a fibration. 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 8 Proof.We use transfinite small object argument. Choose a cardinal number fl > 2* *fi, and define a functor F : fl ! sGd P re(C) # Y on the partially ordered set fl by se* *tting F (0) = f : X ! Y, F (s) : X(s) ! Y such that (1) X(0) = X, (2) X(t) = lim-!X(s) for all limit ordinals t < fl, and s>" "" i||s"" |p| fflffl|fflffl|"" V ____//_Y where i is a trivial cofibration and p is a fibration in the caregory sGdPre(C)* *, there exists a lifting s. Proof.Suppose that i : U ! V is a trivial cofibration. Then i has a factorizati* *on j U ____//_W "" i||""q"" fflffl|~~"" V where q is a fibration and j is a trivial cofibration which has the left liftin* *g property with respect to all fibrations by the construction in the proof of Lemma 2.5. But t* *hen q is a trivial fibration, and so the lifting exists in the diagram j U ____//_W>>" "" i|| """ q|| |fflffl"fflffl|" V _1V_//_V It follows that i is a retract of j, so that i has the left lifting property wi* *th_respect to all fibrations. |* *__| 2 PRESHEAVES OF SIMPLICIAL GROUPOIDS 10 Theorem 2.1. The category sGdPre(C), with the classes of fibrations, weak equi* *valences and cofibrations as defined above, satisfies the axioms for a closed model cate* *gory. Proof.The category sGd is closed under all finite limits and colimits, we can t* *ake the limits and colimits pointwise, so the category sGdPre(C) is also closed under all fini* *te limits and colimits. This is CM1. CM2 is the Lemma 2.1. CM3 is trivial. The first part of * *CM4 is the Lemma 2.7, the second part is the definition of cofibration. CM5(1) is the_* *Lemma 2.5, CM5(2) is the Lemma 2.6. |__| Remark 2.1. The fibration (trivial fibration) in the category sGdPre(C) has th* *e right lifting property with respect to all maps G(i) : GU ! GV induced by the maps i * *: U ! V where i is a trivial cofibration (cofibration) in the category SPre(C) and V is* * ff-bounded. So the category sGdPre(C) is cofibrantly generated. ___ Lemma 2.8. (1)The functor W : sGd P re(C) ! SP re(C) preserves fibrations a* *nd weak equivalences. ___ (2) A map K ! W X 2 SP re(C) is a weak equivalence if and only if its adjoint * *GK ! X 2 sGd P re(C) is a weak equivalence. Proof.(1) This is implied by the definitions of fibration and weak equivalence. (2) There is a commutative diagram ___ K _____________//_CWGK CC vvv CCC vvvv CC!!_zzvv W X ___ where the map_K_! W GK is a pointwise weak equivalence [3, Theorem_V.7.8(3)]._ * *So the map K ! W X is a weak equivalence if and only if the map W GK ! W X is a we* *ak __ equivalence, i.e., the map GK ! X is a weak equivalence. * * |__| ___ Corollary 2.1.The functor G and W induce an equivalence of homotopy categories Ho (sGd P re(C)) ' Ho(SP re(C)) ___ ___ Proof.Lemma 2.8 implies that the natural maps " : GW K ! K and j : X ! W GX are weak equivalences for all presheaves of simplicial groupoids K and simplicial_p* *resheaves X. |__| Suppose that C and D are two closed model categories. 1. We call a functor F : C ! D a left Quillen functor if F is a left adjoint * *and preserves cofibrations and trivial cofibrations. 2. We call a functor U : D ! C a right Quillen functor if U is a right adjoin* *t and preserves fibrations and trivial fibrations. 3 PRESHEAVES OF 2-GROUPOIDS 11 3. Suppose that (F, U, ') is an adjunction from C to D. That is, F is a funct* *or C ! D, U is a functor D ! C, and ' is a natural isomorphism D(F C, D) ! C(C, UD) ex* *pressing U as a right adjoint of F . We call (F, U, ') a Quillen adjunction if F is* * a left Quillen functor (cf. [4]). A Quillen adjunction (F, U, ') : C ! D is called a Quillen equivalence if an* *d only if, for all cofibrant X in C and fibrant Y in D, a map f : F X ! Y is a weak equiva* *lence in D if and only if '(f) : X ! UY is a weak equivalence in C (cf. [4]). ___ Corollary 2.2.The adjunction G : SP re(C) Æ sGd P re(C) : W is a Quillen equiv* *alence. Proof.It's obvious from Theorem 2.1, Proposition 2.1, Lemma 2.3, Lemma 2.8 and * *above_ definitions. |* *__| 3 Presheaves of 2-groupoids 2-GpdPre(C) is the category of presheaves of 2-groupoids on C; its objects are * *the con- travariant functors from C to the category 2-Gpd of 2-groupoids, and its morphi* *sms are natural transformations. Moerdijk and Svensson show that [8], with the following definitions of weak * *equivalence, fibration and cofibration, the category 2-Gpd of 2-groupoids satisfies the axio* *ms for a closed model category. A map ' : A ! B of 2-groupoids is said to be a weak equivalence of 2-Gpd if (1) for every object b of B there exists an object a of A and an arrow '(a) ! * *b; (2) for any two objects a, a0in A, ' induces an equivalence of categories (gro* *upoids) 'a,a0: HomA (a, a0) ! HomB('(a), '(a0)). A map _ : B ! A of 2-groupoids is said to be a (Grothendieck) fibration of 2* *-Gpd if for any arrow f : b1 ! b2 in B and any arrows g : a0 ! _(b1) and h : a0 ! _(* *b2), any deformation ff : h ) _(f) O g can be lifted to a deformation ~ff: ~h) f O ~gin * *B (in the sense that _(~ff) = ff, _(~h) = h and _(~g) = g ). A cofibration of 2-groupoids is defined to be a map which has the left lifti* *ng property with respect to all morphisms of 2-Gpd which are both fibrations and weak equiv* *alences. Recall the adjunction [8]: ___ G : S Æ 2 - Gpd : W ___ where the functor W is the functor N in [8] and the functor G is the Whitehead * *2-groupoid functor W in [8]: W (X) = W (|X|, |X(1)|, |X(0)|). By applying these functors p* *ointwise to simplicial presheaves and presheaves of 2-groupoids, one obtains functors ___ G : SP re(C) Æ 2 - Gpd P re(C) : W and there is 3 PRESHEAVES OF 2-GROUPOIDS 12 Proposition_3.1.The functor G : SP re(C) ! 2 - Gpd P re(C) is left adjoint to t* *he func- tor W . A_map_f :_X_! Y_in_the category 2-GpdPre(C) is said to be a fibration if the* * induced map W (f) : W X ! W Y is a global fibration in the category SPre(C). A map g :_Z_! U in_the_category_2-GpdPre(C) is said to be a weak equivalence* * if the induced map W (g) : W Z ! W U is a weak equivalence in the category SPre(C). A cofibration in the category 2-GpdPre(C) is a map of presheaves of 2-groupo* *ids which has the left lifting property with respect to all fibations and weak equivalenc* *es. Say that a map of presheaves of 2-groupoids f is a trivial fibration if it i* *s both a fibration and a weak equivalence; a map g is a trivial cofibration if it is bot* *h a cofibration and a weak equivalence. Similarly, we can define the above concepts according to the category sGdPre* *(C). A map q : G ! H in the category 2-GpdPre(C) is a fibration if and only if it* * has the right lifting property with respect to all maps G(i) : GU ! GV induced by t* *he maps i : U ! V where i is a trivial cofibration in the category SPre(C) and V is ff-* *bounded, since there exist two adjoint diagrams similar to the diagrams D. For each S 2 C, Ob(GV (S)) = V (S)0, Mor(GV (S)) and 2-cell(GV (S)) are free* * gen- erated by V (S)1 and V (S)2, subject to some relations, respectively. So the ca* *rdinality of objects, morphisms and 2-cells of 2-groupoid GV (S) is smaller than fi = max(2f* *f, 1), where ff is a boundary of the simplicial presheaf V (S). We also call the presheaf of* * 2-groupoids GV is fi - bounded. For each 2-groupoid G and each object x of G, there are natural isomorphisms* * [8, Proposition 2.1(iii)]: ___ ß0(W G) ~=ß0(G), ___ ß1(W G, x) ~=ß1(G, x), ___ ß2(W G, x) ~=ß2(G, x), ___ ßi(W G, x) ~=0 (i > 2). According to the definition of topological weak equivalence of simplicial pr* *esheaves in [5] and the above relations, we can give an explicit description of weak equivalenc* *e of presheaves of 2-groupoids. For any presheaf of 2-groupoids X and any object U 2 C and x 2 Ob(X(U)), ass* *ociated to this presheaf of 2-groupoids X on C and x is a presheaf on the comma categor* *y C # U, the presheaf of homotopy groups of X|U, based at x, which is defined by (C # U)op! Grp ' : V ! U 7! ßi(X(V ), xV ), i = 1, 2 where xV is the image of x in X(V ) under the map X(U) ! X(V ) which is induced* * by V ! U. Let ßi(X|U, x), i = 1, 2 be the associated sheaves of the above presheaves. * * The sheaf ß0(X) of path components is defined similarly. 3 PRESHEAVES OF 2-GROUPOIDS 13 A map f : X ! Y of presheaves of 2-groupoids is said to be a weak equivalenc* *e if it induces isomorphisms of sheaves f* : ßi(X|U, x) ~=ßi(Y |U, fx), i = 1, 2; U 2 C, x 2 Ob(X(U)) f* : ß0(X) ~=ß0(Y ). In parallel with the corresponding arguments for presheaves of simplicial gr* *oupoids, we have Lemma 3.1. Given maps of presheaves of 2-groupoids f : X ! Y and g : Y ! Z, if* * any two of f, g, or g O f are weak equivalences, then so is the third. ___ Lemma 3.2. The functor X 7! W G(X) preserves weak equivalences of simplicial p* *resheaves. Proof.When T is a simplicial set, there are isomorphisms [8] ___ ß0(W GT ) ~=ß0(GT ) ~=ß0(T ), ___ ßi(W GT, t0) ~=ßi(GT, t0) ~=ßi(T, t0) (i = 1, 2), t0 2 T0. ___ ßi(W GT, t0) = 0 (i > 2). so there exist isomorphisms of sheaves ___ ß0(W GX) ~=ß0(GX) ~=ß0(X), ___ ßi(W GX|U, x) ~=ßi(GX|U, x) ~=ßi(X|U, x) (i = 1, 2) U 2 C x 2 X(U)0. ___ * * __ and ßi(W GX|U, x) = 0 (i > 2). * *|__| Lemma 3.3. The functor G : SP re(C) ! 2-Gpd P re(C) preserves cofibrations and* * weak equivalences. Lemma 3.4. The category 2-GpdPre(C) has all pushouts, and is hence cocomplete.* * The class of cofibrations in 2-GpdPre(C) is closed under pushout. Notice that there is a commutative diagram 2 "* 2 - GpdP re(C)_L__//_2 - GpdE___//_2 - GpdShv(B) __W| |__ __| | |W W| fflffl| fflffl| fflffl| SP re(C)__________//_SE____*____//_SShv(B) L2 " ___ W G is locally fibrant simplicial presheaf for any presheaf of 2-groupoids G, s* *o a map f : G ! H of presheaves of 2-groupoids on C is a weak equivalence if the induce* *d map "*L2 : "*L2G ! "*L2H is a pointwise weak equivalence. Proposition 3.2.Trivial cofibrations of presheaves of 2-groupoids are closed un* *der pushout. 3 PRESHEAVES OF 2-GROUPOIDS 14 Given a trivial cofibration i : A ! B in the category SPre(C), suppose that GA _____//C G(i)|| i0|| fflffl| fflffl| GB _____//D is a pushout in the category 2-GpdPre(C). Then the map i0is a trivial cofibrati* *on. Lemma 3.5. Every map f : X ! Y of presheaves of 2-groupoids may be factored f X ___________//@@Y??~ @@ ~~~ i@@__@~~p~ Z where i is a trivial cofibration and p is a fibration. A map p : G ! H of presheaves of 2-groupoids is a trivial fibration if and o* *nly if it has the right lifting property with respect to all inclusions GS G4nUof subob* *jects of the G4nU, U 2 C, n 0. A transfinite small object argument, as in Lemma 2.5, shows* * that Lemma 3.6. Every map g : Z ! W of presheaves of 2-groupoids may be factored g Z ____________//AAW==_ AAA ____ j AA__A__q_ M where j is a cofibration and q is a trivial fibration. Lemma 3.7. For the commutative diagram U ____//_X>>" "" i||s"" |p| fflffl|fflffl|"" V ____//_Y where i is a trivial cofibration and p is a fibration in the caregory 2-GpdPre(* *C), there exists a lifting s. Theorem 3.1. The category 2-GpdPre(C), with the classes of fibrations, weak eq* *uivalences and cofibrations as defined above, satisfies the axioms for a closed model cate* *gory. ___ Lemma 3.8. (1)The functor W : 2 - Gpd P re(C) ! SP re(C) preserves fibratio* *ns and weak equivalences. ___ (2) The functors G and W induce adjoint functors ___ G : Ho(SP re(C)) Æ Ho(2 - Gpd P re(C)) : W at the level of homotopy categories. REFERENCES 15 Proof.(1) This is implied_by the definitions of fibration and weak equivalence. (2) The functors W and G both preserve weak equivalences ((1) of this Lemma* * and Lemma 3.3), they localize to functors of homotopy categories. The triangular id* *entities for_ the unit and counit will still hold after localization. * * |__| ___ Corollary 3.1.The adjunction G : SP re(C) Æ 2 - Gpd P re(C) : W is a Quillen a* *djunc- tion. Proof.It's obvious from Theorem 3.1, Proposition 3.1, Lemma 3.3 and the definit* *ion_of Quillen adjunction. |* *__| Define the category 2 - typesSP re(C) of homotopy 2-types to be the full sub* *category of Ho(SPre(C)) given by those simplicial presheaves with sheaves ßi(X|U, x) = 0* * for any integer i > 2, any object U 2 C and any basepoint x 2 X(U)0. ___ Theorem 3.2. The functors G and W induce an equivalence of homotopy categories Ho(2 - Gpd P re(C)) ' 2-typesSP re(C) ___ Proof.For a simplicial presheaf X, the natural map j : X ! W G(X) is a weak equ* *ivalence if and_only if ßi(X|U, x) = 0, i > 2, U 2 C, x 2 X(U)0. For any presheaf_of_2-g* *roupoids K, ßi(W K|U, *) = 0, i > 2, U 2 C, * 2 Ob(K(U)), and the natural map ' : GW (K) !_* *K_is a weak equivalence. |_* *_| Acknowledgement I would like to thank my supervisor Dr. J.F. Jardine for suggesting the topic, * *and for his help and continuing encouragement. References [1]S.E. Crans, Quillen closed model structures for sheaves, J. Pure Appl. Algeb* *ra 101:35- 57, 1995. [2]W.G. Dwyer and D.M. Kan, Homotopy theory and simplicial groupoids, Indag. Ma* *th. 46(1984), 379-385. [3]P.G. Goerss and J.F. Jardine, Simplicial Homotopy T heory, Birkhäuser, PM174* *, 1999. [4]M. Hovey, Model Categories, Mathematical Surveys and Monographs, V. 63, Amer* *ican Mathematics Society, 1999. [5]J. F. Jardine, Simplicial presheaves, J. Pure and Appl. Algebra 47 (1987), 3* *5-87. [6]J. F. Jardine, Boolean Localization, In P ractice, Doc. Math 1, 1996, 245-27* *5. [7]A. Joyal, letter to A.Grothendieck, (1984). [8]I. Moerdijk and J. Svensson, Algebraic classification of equivariant homotop* *y 2 - types I, J. Pure Appl. Algebra 89:187-216, 1993. REFERENCES 16 [9]D. Quillen, Homotopical Algebra, Lecture Notes in Math., Vol.43, Springer, B* *erlin- Heidelberg-New York, 1967. [10]D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205-295. Department of Mathematics The University of Western Ontario London, Ontario Canada N6A 5B7 e-mail: zluo@uwo.ca