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