Homotopy classification of gerbes
J.F. Jardine
March 27, 2006
Abstract
Gerbes are locally connected presheaves of groupoids on a small Groth-
endieck site C. Gerbes are classified up to local weak equivalence by path
components of a cocycle category taking values in the diagram Grp (C)
of 2-groupoids consisting of all sheaves of groups, their isomorphisms and
homotopies. If F is a full subpresheaf of Grp (C) then the set [*, BF]
of morphisms in the homotopy category of simplicial presheaves classi-
fies gerbes locally equivalent to objects of F up to weak equivalence. If
St(ssF) is the stack completion of the fundamental groupoid ssF of F, if
L is a global section of St(ssF), and if FL is the homotopy fibre over L
of the canonical map BF ! B St(ssF), then [*, FL] is in bijective cor-
respondence with Giraud's non-abelian cohomology object H2(C, L) of
equivalence classes of gerbes with band L.
Introduction
Suppose that M is a closed model category, and that X and Y are objects of
M. A cocycle from X to Y is a picture
X- f Z g-!Y
of morphisms in M such that f is a weak equivalence. A morphism of cocycles
(f, g) ! (f0, g0) is a commutative diagram
fpp Z NNgN
wwpppp|| NNN''
X ffNNNN| pppY88
f0NN fflffl|g0ppp
Z0
and these cocycles and their morphisms together form the category H(X, Y ) of
cocycles from X to Y . The assignment (f, g) 7! gf-1 defines a function
OE : ss0H(X, Y ) ! [X, Y ]
from the path components of cocycle category H(X, Y ) to the set of morphisms
[X, Y ] from X to Y in the homotopy category Ho(M). Then it is a basic result
1
of [9] that this function OE is a bijection if the model category M is right pr*
*oper
and if its class of weak equivalences is closed under finite products.
The right properness condition is a serious restriction, but right proper mo*
*del
structures are fairly common in nature, and include the standard model struc-
tures for spaces, simplicial sets, and spectra, as well as more exotic structur*
*es
such as simplicial presheaves, simplicial sheaves and presheaves of spectra on
small Grothendieck sites.
The cocycle approach to constructing morphisms in the homotopy category
is proving to be very useful, particularly in connection with simplicial sheaves
and presheaves. Applications have so far appeared in new, short and conceptual
arguments for the homotopy classification of sheaf cohomology theories, both
abelian and non-abelian [9]. Cocycle categories are involved in the explicit
construction of the stack completion functor which is given in [8]. They have
been used to show [7], in a variety of settings, that morphisms [*, BI] in the
homotopy category can be identified with path components of a suitably defined
category of I-torsors for all small category objects I.
The present paper uses cocycles in presheaves of 2-groupoids, here called
2-cocycles, to give a homotopy classification of gerbes.
A gerbe is typically defined in the literature [2, p.129] to be a stack G wh*
*ich
is locally path connected. Stacks themselves have no conceptual mystery: they
are fibrant objects in model structures for sheaves of groupoids [11] or more
generally presheaves of groupoids [4], and one now can identify a stack with
the homotopy type that it represents in presheaves of groupoids. The model
structure for presheaves (or sheaves) of groupoids, over any small Grothendieck
site C, is easy to describe: a map f : G ! H of presheaves of groupoids is a lo*
*cal
weak equivalence (respectively global fibration) if the induced map BG ! BH
of classifying objects is a local weak equivalence (respectively global fibrati*
*on)
of simplicial presheaves. Local path connectedness is an invariant of homotopy
type in this sense, and we shall take the point of view that a gerbe is a presh*
*eaf
of groupoids G such that the classifying simplicial presheaf BG is locally path
connected. The local path connnectedness condition can be expressed this way:
given objects x, y 2 Ob (G)(U) in a section G(U), there is a covering family
OE : V ! U such that there is a morphism OE*(x) ! OE*(y) in G(V ) for any OE in
the cover.
Every gerbe G is locally equivalent to any of its sheaves of automorphism
groups. The category H(*, Grp (C)) of 2-cocycles taking values in the diagram
of all sheaves of groups, their isomorphisms and homotopies, is the vehicle by
which we classify gerbes up to local weak equivalence. Theorem 20 says that
the path components of this cocycle category are in one to one correspondence
with the path components of the category Gerbe (C) of gerbes and their local
weak equivalences, or that there is a bijection
ss0(Gerbe (C)) ~=ss0H(*, Grp (C)).
One has to interpret a statement like this correctly, because the categories in-
volved are not small. The path component functor ss0 means the class of equiv-
alence classes of objects, where two objects are equivalent if and only if ther*
*e is
2
a finite string of arrows connecting them in the ambient category, and we show
that there are functions
: ss0(Gerbe (C)) o ss0H(*, Grp (C)) :
which are inverse to each other. Here, is induced by a canonical cocycle con-
struction which is introduced in Example 12, and is defined by a generalized
Grothendieck construction, which is the subject of much of Section 2.
The 2-groupoid diagram Grp (C) has subobjects which are honest presheaves
of 2-groupoids. Examples include the sheaf of 2-groupoids G* associated to a
sheaf of groups of G on C: it has one object, the sheaf of 1-cells is the sheaf
of automorphisms of G, and its sheaf of 2-cells is the sheaf of homotopies (or
conjugations) of automorphisms. More generally, any presheaf of sheaves of
groups in Grp (C) determines a full subobject F Grp (C) which is a presheaf
of 2-groupoids, and one can discuss the homotopy type of F and its classifying
object BF in simplicial presheaves. It is shown in Theorem 23 that there is a
one to one correspondences
ss0H(*, F) ~=ss0(F - Gerbe )
between the set of path components of 2-cocycles taking values in the presheaf *
*of
2-groupoids F and path components of the category F -Gerbe of gerbes locally
equivalent to sheaves of groups appearing in F. By the result relating path com-
ponents of cocycle categories to morphisms in the homotopy category displayed
above, both of these objects are then in bijective correspondence with the set
[*, BF] of morphisms in the homotopy category of simplicial presheaves _ this
statement appears formally as Corollary 24. The bijection of path components
in the statement of Theorem 23 is a restriction of the bijection of Theorem 20.
In the special case where F = G* for some sheaf of groups G, Theorem 23
says that gerbes locally equivalent to G are classified up to weak equivalence
by morphisms [*, BG*] in the homotopy category of simplicial presheaves. This
result was originally proved, in a very different form, by Breen [1].
Finally, the presheaf of 2-groupoids has a fundamental groupoid ssF and a
canonical morphism F ! ssF. In the case where F = G*, the fundamental
groupoid ssG* is the sheaf of outer automorphisms of G. The fundamental
groupoid ssF has a functorial stack completion ssF ! St(ssF), and St(ssF) is
the stack of bands (liens) for F. Suppose that the band L is a fixed choice of
global section of St(ssF), and consider the homotopy fibre FL of the composite
BF ! BssF ! B St(ssF).
Theorem 27 (see also Corollary 28) identifies the set of morphisms [*, BFL] in *
*the
homotopy category with path components in a suitably defined category of L-
gerbes. In other words, Giraud's non-abelian invariant H2(C, L) is isomorphic
to [*, BFL]. Once again, the real thrust of the proof is to identify the set
of path components of the cocycle category H(*, FL) with path components
in L-gerbes, and then use the general result about cocycles to conclude that
[*, FL] ~=ss0H(*, FL).
3
Theorem 20, Theorem 23 and Theorm 27 are the main results of this paper.
The demonstration of these results appear in Sections 3 and 4, but depend on
some generalities about groupoids enriched in simplicial sets and presheaves of*
* 2-
groupoids which appear in Section 1, as well as the discussion of the generaliz*
*ed
Grothendieck construction of Section 2.
Contents
1 Simplicial groupoids 4
2 The Grothendieck construction 10
3 Cocycle classification of gerbes 14
4 Homotopy classification of gerbes 18
1 Simplicial groupoids
There are various equivalent ways to define a groupoid enriched in simplicial s*
*ets.
We shall initially take the point of view that such an object H is a simplicial
groupoid such that the simplicial set Ob(H) of objects is simplicially discrete*
*, or
just a set. The morphisms Mor(H) is a simplicial sets and the source, target s,*
* t :
Mor (H) ! Ob (H) and identity maps e : Ob (H) ! Mor (H) are all simplicial
set maps. The notation Hn will refer to the associated groupoid in simplicial
degree n.
For objects x, y of H the simplicial set of morphisms H(x, y) is defined by
the pullback diagram
H(x, y)________//_Mor(H)
| |(s,t)
| |
fflffl| fflffl|
*__(x,y)_//Ob(H) x Ob(H)
where * = 0 defines the one-point (terminal) simplicial set.
A 2-groupoid is a groupoid enriched in groupoids. Equivalently, a simplicial
groupoid G is a groupoid enriched in simplicial sets such that the simplicial s*
*et
Mor (G) is the nerve of a groupoid.
We shall routinely write BH for both the bisimplicial classifying space n 7!
BHn associated to H and its associated diagonal simplicial set dBH. The
vertical simplicial presheaf BHn in horizontal degree n is the iterated fibre
product
Mor (H) xt,sMor(H) xt,s. .x.t,sMor(H),
4
This simplicial set models composable strings of morphisms of length n, and is
the inverse limit for a diagram
Mor (H)D Mor (H)D Mor(H) . . .
DDDtDD szzzzz DDDtDD szzzzz
DD!! ""zzz D!!D ""zzz
Ob(H) Ob (H)
involving n copies of the morphism object Mor(H).
The vertices of BH are the objects of H, and two vertices of BH are in the
same path component if and only if they are in the same path component of
the space BH1, or in the same path component of the groupoid H1 in simplicial
degree 1. It is well known and easily seen that the degeneracy morphism H0 !
Hn induces a bijection ss0BH0 ~=ss0BHn for all n 0. It follows that there are
natural isomorphisms
ss0BH ~=ss0BH1 ~=ss0BHn (1)
for all n 0. We shall say that H is connected if BH is a path-connected
simplicial set. More generally, ss0H will often be written to denote ss0BH, and
will be called the set of path components of H.
There is a simplicial groupoid H1 whose objects in simplicial degree n are
morphisms h : x ! y of Hn and whose morphisms h ! h0are the commutative
squares
x __h__//_y
ff|| |fi|
fflffl|fflffl|
x0__h0_//y0
in Hn. There is a simplicial groupoid functor (s, t) : H1 ! H x H which is de-
fined in degree n by sending the square diagram above to the pair of morphisms
(x ff-!x0, y -fi!y0). The two projections s, t : H1 ! H are weak equivalences,
because they are weak equivalences in each simplicial degree.
Lemma 1. Suppose that H is a groupoid enriched in simplicial sets. Then the
pullback diagram
Mor (H) ___________//BH1
| |
| |
fflffl| fflffl|
Ob(H) x Ob(H) ____//_BH x BH
is homotopy cartesian.
Proof. Suppose first that H is an ordinary groupoid. There is a functor hom :
HxH ! Setdefined by (x, y) 7! H(x, y) and which sends the pair of morphisms
(ff : x ! x0, fi : y ! y0) to the function
H(x, y) ! H(x0, y0)
5
defined by sending f : x ! y to the composite
-1 f fi
x0-ff-!x -! y -! y0.
Observe that there is an isomorphism
holim---!HxHhom~=B(H1).
Suppose now that H is a groupoid enriched in simplicial sets. Applying the
construction of the previous paragraph in all simplicial degrees gives a simpli*
*cial
functor hom : H x H ! sSet and an isomorphism
holim---!HxHhom~=B(H1).
There is a homotopy cartesian diagram
X ______//_holim---!GX
| |
| |
fflffl| fflffl|
Ob (G) _______//_BG
for all simplicial set-valued diagrams X defined on groupoids G enriched in
simplicial sets [7, Lemma 2], [14], and this specializes to give the homotopy_
cartesian diagram required by the statement of the Lemma. |__|
Lemma 2. If f : G ! H is a morphism of groupoids enriched in simplicial sets
and if X : H ! sSet is a simplicial diagram defined on H, then the induced
diagram
holim---!GX_._f_//holim---!HX
| |
| |
fflffl| fflffl|
BG ___________//_BH
is homotopy cartesian.
Proof. Suppose given a factorization
holim---!HXj__//JZ
JJJ |p
JJJ |
JJ%%fflffl|
BH
of the canonical map holim---!HX ! BH such that j is a weak equivalence and p
is a fibration. The squares in the picture
X . f________//X______//holim---!HX
| | |
| | |
fflffl| fflffl| fflffl|
Ob (G) ____//_Ob(H)_______//_BH
6
are homotopy cartesian, so that the induced map
X . f__________//_Ob(G) xBH Z
?? tt
?? ttt
?? tt
?OO zztt
Ob(G)
is a weak equivalence of objects over Ob (G). It follows that the induced map
holim---!GX_._f____//BG xBH Z
FFF yyy
FFF yyy
F""F__yy
BG
induces a weak equivalence on all homotopy fibres. This map is also a homotopy
colimit of a comparison of diagrams made up of the homotopy fibres of the __
respective maps, and is therefore a weak equivalence. |__|
Corollary 3. Suppose that H is a groupoid enriched in simplicial sets. Then
the square
H(x, y)____//_B(H=y)
| |
| |
fflffl| fflffl|
* _____x___//_BH
is homotopy cartesian.
Proof. In the diagram of pullback squares
H(x, y)_____//B(H=y)______//_B(H1)
| | |
| | |
fflffl| fflffl| fflffl|
* ____x____//_BH_(1,y)//_BH x BH
the square on the right is homotopy cartesian, on account of Lemma 2 applied
to the composite functor
H (1,y)---!H x H hom---!sSet.
Lemma 1 implies that the composite square is homotopy cartesian, and the__
desired result follows. |__|
Let ssH denote the groupoid of path components of a groupoid H enriched
in simplicial sets. The object ssH will typically be called the path component
groupoid of H. It has the same objects as H, and the set of morphisms from x
to y is the set ssH(x, y) of path components of the simplicial set H(x, y). The*
*re
7
is a canonical map j : H ! ssH which is the identity on objects and is the
canonical map Mor (H) ! ss0Mor (H) on morphisms. The morphism j is one
of the canonical maps for an adjunction: the functor H 7! ssH is left adjoint to
the inclusion of groupoids in simplicial groupoids.
Corollary 4. The induced map j : BH ! BssH induces an isomorphism on
path components and all fundamental groups, so that ssH is naturally weakly
equivalent to the fundamental groupoid of BH.
Proof. The morphisms H(x, x) ! ssH(x, x) induce isomorphisms in path com-
ponents, and so the map BH ! BssH induces isomorphisms in path compo-
nents of all loop spaces, by Corollary 3. It follows that all homomorphisms
ss1(BH, x) ! ss1(BssH, x) are isomorphisms. The claim that ss0BH ! ss0BssH
is a bijection follows from (1) and the observation that the function ss0BH0_!
ss0BssH is a bijection. |__|
Suppose now that C is a small Grothendieck site.
If G is a presheaf of groupoids on C and x, y are objects of G(U), there is a
presheaf G(x, y) of homomorphisms from x to y on C=U. Write Gx = G(x, x)
for the presheaf of automorphisms of x in G, and let "Gxdenote the associated
sheaf of automorphisms on C=U.
Say that a presheaf of groupoids G is a ~Cech object if the canonical map
G ! ss0G is a local weak equivalence, where ss0G = ss0BG is the presheaf of
path components of G.
In particular, an ordinary groupoid H is a ~Cech groupoid if the groupoid
moprhism H ! ss0H is a weak equivalence. Equivalently, H is a ~Cech groupoid
if and only if there is at most one morphism between any two objects of H.
Example 5. The ~Cech groupoid C(p) for a function p : X ! Y has the objects
Ob (C(p)) = X, and there is a morphism x ! y in C(p) if and only if p(x) = p(y)
in Y . There is a canonical bijection ss0C(p) ~=p(X).
This construction is natural, and therefore applies to morphisms p : X ! Y
of presheaves on a site. If p is an epimorphism of sheaves, the simplicial pres*
*heaf
map BC(p) ! Y is the ~Cech resolution of Y corresponding to the epimorphism
p, and is a local weak equivalence. This constructionFspecializes to the standa*
*rd
C~ech resolution when applied to an epimorphism p : ffUff! Y arising from a
covering.
Lemma 6. Suppose that H is a presheaf of 2-groupoids, and let j : H ! ssH
be the canonical map to the presheaf of path component groupoids. Then j is
a local weak equivalence if and only if all presheaves of groupoids H(x, y) are
C~ech objects.
Proof. If the map j : H ! ssH is a local weak equivalence, then the map
Mor(H) ! ss0Mor (H)
is a local weak equivalence over Ob (H) x Ob(H), by Lemma 1. The object
Ob (H) x Ob(H) is a constant simplicial presheaf, so pullback along any map
8
Z ! Ob (H) x Ob(H) preserves local weak equivalences over Ob (H) x Ob(H)
[7]. In particular, for all choices x, y 2 Ob(U)(U), the induced map H(x, y) !
ss0H(x, y) is a local weak equivalence of presheaves of groupoids over C=U.
In general, one can show that a map
f
Z ______//2W
22 ffff
22 ffff
,,2ffff
A
of simplicial presheaves fibred over a presheaf A is a local weak equivalence if
and only if it induces weak equivalences Zx ! Wx of simplicial presheaves on
C=U for all x 2 A(U), U 2 C. One implication involved in this statement we
already know about, from [7]. For the other, if we know that all induced maps
on fibres are local weak equivalences, we can replace f by a sectionwise Kan
fibration, and check local lifting with respect to all inclusions @ n n, whi*
*ch
must take place in individual fibres.
Thus, if all morphism groupoids H(x, y) are ~Cech objects, then the map
Mor (H) ! ss0Mor (H) is a local weak equivalence of simplicial presheaves over
the presheaf Ob (H) x Ob(H). It follows that all maps
Mor (H)xt,sMor(H) xt,s. .x.t,sMor(H)
! ss0Mor (H) xt,sss0Mor (H) xt,s. .x.t,sss0Mor (H)
of iterated fibre products over Ob (H) are local weak equivalences. These are
the comparison maps of vertical simplicial presheaves making up the comparison
BH ! BssH of bisimplicial presheaves, and one concludes that this map is_a
local weak equivalence. |__|
Lemma 7. A presheaf of groupoids H is a ~Cech object if and only if for every
two morphisms f, g : x ! y in H(U) there is a covering sieve R hom ( , U)
such that OE*f = OE*g for all OE : V ! U in R.
Proof. Suppose that H ! "His the canonical map taking values in the associated
sheaf of groupoids "H. Then H is a ~Cech object if and only if all sheaves "H(x*
*, x)
of automorphisms of H" are trivial in the sense that the canonical sheaf map
H"(x, x) ! * are isomorphisms. This is equivalent to the assertion that all
presheaf maps H(x, x) ! * are local monomorphisms.
Thus, suppose that H is a ~Cech object, and suppose given f, g : x ! y
in H(U) the composite g-1f 2 H(x, x)(U), and there is a covering sieve R
hom ( , U) such that OE*(g-1f) = 1OE*xfor all OE : V ! U in R. But then
OE*(g) = OE*(f) for all OE 2 R.
The converse is clear: the local coincidence of all f, g : x ! y means that
all presheaf maps H(x, y) ! * are local monomorphisms, and so all sheaf maps
H"(x, x) ! * are isomorphisms. |___|
9
Lemma 8. Suppose that A is a presheaf of 2-groupoids, and that ss0A is its
presheaf of path components. Then the canonical map A ! ss0A is a local
weak equivalence if and only if all presheaves of groupoids A(x, y) and the path
component groupoid ssA are ~Cech objects.
Proof. Suppose that A ! ss0A is a local weak equivalence. Then all sheaves
of homotopy groups for BA are trivial, and so Lemma 1 implies that all maps
A(x, y) ! * are local weak equivalences. In particular, all A(x, y) are ~Cech
objects. But then j : A ! ssA is a local weak equivalence by Lemma 6, and so
the induced map ssA ! ss0(ssA) is a weak equivalence, so that the presheaf of
groupoids ssA is a ~Cech object.
Suppose conversely that all A(x, y) and ssA are ~Cech objects. Then Lemma
6 implies that A ! ssA is a local weak equivalence, and then the map
ssA ! ss0(ssA) ~=ss0A
is a local weak equivalence. It follows that the map A ! ss0A is a composite_of
two local weak equivalences. |__|
2 The Grothendieck construction
Let cat denote the 2-category whose 0-cells are the small categories, whose
1-cells are the functors between small categories, and whose 2-cells are the ho-
motopies of functors.
Suppose given a 2-category morphism F : A ! cat, such that A is a small
category enriched in groupoids. This morphism has an associated "Grothendieck
construction", which is a category EA F that is constructed as follows.
Consider the collection of pairs (x, i) where i is an object or 0-cell of A *
*and
x 2 F (i). Look at all pairs
(f, ff) : (x, i) ! (y, j)
where ff : i ! j is a 1-cell of A and f : ff*(x) ! y is a morphism of F (j). Say
that two such pairs
(f, ff), (f0, ff0) : (x, i) ! (y, j)
are equivalent if there is a 2-cell h : ff ! ff0in A such that the diagram
ff*(x) f
QQQ
F(h)| QQ((Q
| mm6y6m
fflffl|0mmm
ff0*(x)f
commutes, where F (h) is the homotopy associated to the 2-cell h by F . This
is an equivalence relation, since the homotopies h are isomorphisms in the
groupoids A(x, y). Write [(f, ff)] for the equivalence class containing the pa*
*ir
(f, ff).
10
Suppose given strings
(x, i) (f,ff)---!(y, j) (g,fi)---!(z, k)
and 0 0 0 0
(x, i) (f-,ff-)--!(y, j) (g-,fi-)--!(z, k)
and suppose that (f, ff) h1'(f0, ff0) and (g, fi) h2'(g0, fi0) via the displaye*
*d homo-
topies. Then there is a commutative diagram
fi*(f) g
fi*ff*(x)___//_fi*(y)__//z==_
__
F(h2)|| F(h2)||____ 0
fflffl|fi0*fflffl|(f)g__
fi0*ff*(x)__//_fi0*(y)::
uuu
fi0*(F(h1))||uuuu0 0
fflffl|fi*(fu)u
fi0*ff0*(x)
The composite homotopy fi0*(F (h1))(F (h2)ff*) is the image of the composite
2-cell h2 * h1 under the morphism F . It follows that the assignment
[(g, fi)] . [(f, ff)] = [(gfi*(f), fiff)]
gives a well defined law of composition. This composition law is associative,
and has 2-sided identities. Write EA F for the corresponding category.
Remark 9. This category EA F should seem familiar. Suppose that I is a small
category, and let x, y be objects of I. There is a small category Is(x, y) whose
objects are the functors ` : n ! I (strings of length n) such that `(0) = x and
`(n) = y. A morphism ` ! fl of Is(x, y) is a commutative diagram of functors
nNNN`
ff|| NN''N
| pp7I7p
fflffl|flppp
m
such that the ordinal number map ff is end-point preserving in the sense that
ff(0) = 0 and ff(n) = m. Concatenation of strings defines a composition law
Is(x, y)xIs(y, z) ! Is(x, z), and so there is a 2-category Is with the same obj*
*ects
as I and a canonical weak equivalence Is ! I (see also [3, IX.3.2]). Write GIs
for the category enriched in groupoids, having the same 0-cells as I, and such
that the groupoid GIs(x, y) is the free groupoid on the category Is(x, y). The
groupoid GIs(x, y) is a ~Cech groupoid with path components given by the set
I(x, y) of morphisms from x to y in I.
A pseudo-functor F defined on I and taking values in small categories can
be identified with a 2-category morphism F : GIs ! cat[3, IX.3.3], and one can
11
show that the Grothendieck construction EF GIs as defined above is isomorphic
to the standard Grothendieck construction for the pseudo-functor F .
The Grothendieck construction EA F given here for 2-catgory morphisms
defined on categories A enriched in groupoids generalizes the usual construction
for pseudo-functors, but there appears to be no corresponding construction for
lax functors.
We shall henceforth specialize to 2-category morphisms F : A ! catwhich
are defined on small 2-groupoids A. In this case, there is a canonical functor
p : EA F ! ssA, which is defined by the assignment (x, i) 7! i.
Lemma 10. Suppose that F : A ! cat is a 2-category morphism, where A is
a 2-groupoid. Suppose that [(f, ff)] : (x, i) ! (y, j) is a morphism of EA F su*
*ch
that f : ff*(x) ! y is an invertible morphism of F (j). Then [(f, ff)] is inver*
*tible
in EF A.
Proof. The inverse of [(f, ff)] is represented by [(ff-1*(f-1 ), ff-1)]. *
* |___|
Corollary 11. If the 2-category morphism F : A ! cat takes values in
groupoids, then EA F is a groupoid.
A diagram
B -ff'A F-!cat
such that ff : A ! B is a weak equivalence of 2-groupoids is a 2-cocycle taking
values in small categories. A morphism of 2-cocycles is a commutative diagram
of functors
ffppA OOOFO
xxp'ppp|| OO''
B ffN'NNN| oo7cat7
ff0NN fflffl|F0oooo
A0
and the corresponding 2-cocycle category is denoted by H(B, cat). There are
analogous definitions for 2-cocycles and 2-cocycle categories taking values in *
*of
groups and small groupoids. These 2-cocycle categories are typically not small.
In particular, write grp for the 2-groupoid whose objects are the groups,
whose 1-cells are the isomorphisms of groups G ! H, and whose 2-cells are the
homotopies of isomorphisms, and suppose now that there is a 2-cocycle
ssA-'jA K-!grp
taking values in groups. Then the associated Grothendieck construction EA K
can be identified with a category having as objects all i 2 Ob (A) and with
morphisms consisting of equivalence classes of pairs (f, ff) : i ! j, where ff :
i ! j is a 1-cell of A and f 2 K(j). In this case, there is a relation (f, ff) *
*~ (g, fi)
if the diagram
* PPfPP
h* || PP''*77
fflffl|gnnnnnn
*
12
commutes in the group K(j), where conjugation by h* defines the image of the
unique 2-cell ff ! fi.
The category EA K is a groupoid by Lemma 10.
Example 12. Suppose that G is a groupoid. The resolution 2-groupoid R(G)
has the same objects and 1-cells as G, and has a unique 2-cell f ! g between
any two morphisms f, g : x ! y of G. The path component groupoid ssR(G) of
R(G) is a ~Cech groupoid, and the natural maps
BR(G) ! BssR(G) ! ss0B(ssR(G))
weak equivalences. There are natural bijections
ss0G = ss0BG ~=ss0B(ssR(G)).
There is a canonical morphism F (G) : R(G) ! grp which takes the object
x 2 G(U) to the group Gx = G(x, x) on C=U, takes a 1-cell f : x ! y to the
isomorphism Gx ! Gy which is defined by conjugation by f, and takes the 2-
cell f ! g to the homotopy defined by conjugation by the element gf-1 2 Gy.
It follows that G determines a canonical 2-cocycle
ss0G -' R(G) F(G)---!grp.
~=
Lemma 13. There is a natural isomorphism of groupoids _ : ER(G)F (G) -! G
which is defined fibrewise over ssR(G) in the sense that there is a commutative
diagram
ER(G)F (G) _____~=__//_G
FFF
FFF
F""F
ssR(G)
Proof. The functor _ is the identity on objects. It is defined on morphisms by
sending the pair (f, ff) to the composite f . ff in G. If (f, ff) ~ (g, fi) and*
* ff ! fi
is the unique 2-cell in R(G), then the diagram
ffr9jLLLf9r
rrrr LLL
iLLL |fiff-1|&&j88r
fiL%fflffl|%LLgrrrrr
j
commutes in G, so that f . ff = g . fi and the assignment [(f, ff)] 7! f . ff i*
*s well
defined. The assignment is functorial, because the diagram
fiq8kMMfi*(f)MM8q
jqqqq MM&&
ffr99LLLfLrr firrk8MMMgM8r
irrr LL&&jrrr MM&&k
13
commutes in G.
The functor _ plainly induces surjective functions
_ : hom ER(G)F(G)(i, j) ! hom G(i, j)
Finally, if the diagram
r9jLL9f
ffrrr LL
irrL LL&&j88
LLL rrrr
fiL%%Ljrgr
commutes in the groupoid G then g . (fiff-1) = f, so that (f, ff) ~ (g,_fi),_an*
*d _
is injective on morphisms. |__|
3 Cocycle classification of gerbes
A gerbe G is a locally connected presheaf of groupoids. A morphism of gerbes
is a local weak equivalence G ! H of presheaves of groupoids. We shall write
Gerbe (C) for the category of gerbes and morphisms of gerbes.
Remark 14. If G is a gerbe and x is a global section of Ob (G), then the
inclusion map Gx ! G is a local weak equivalence. It follows that every gerbe
H is locally equivalent to a presheaf of groups, in the sense that there is a
covering U ! * by objects U 2 C and section xU 2 Ob (H)(U) such that the
morphisms HxU ! H|U are local weak equivalences over C=U for all U in the
covering.
Remark 15. Suppose that E is a presheaf, and identify E with a presheaf of
discrete groupoids. An E-gerbe is a morphism G ! E of presheaves of groupoids
such that the associated presheaf map ss0G ! E induces an isomorphism "ss0G ~=
E" of associated sheaves. A morphism of E-gerbes is a commutative diagram
f
G0_____//_H
00 flfl
00 flfl
0,,flfl
E
such that the morphism f : G ! H is a local weak equivalence of presheaves
of groupoids. Write Gerbe E(C) for the corresponding category. Categories of
E-gerbes do appear in applications _ see [12, p.22].
There is an equivalence of categories
Gerbe E (C) ' Gerbe (C=E)
between the category of E-gerbes on C and the category of gerbes for the fibred
site C=E. Equivalences of this sort are discussed at length in [6]. Classific*
*a-
tion results for E-gerbes can therefore be deduced from classification results *
*for
gerbes on the site C=E.
14
We shall write Grp (C) for the following monster: it is a contravariant di-
agram defined on C and taking values in 2-groupoids, such that the 0-cells of
Grp (C)(U) are the sheaves of groups on C=U, the 1-cells are the isomorphisms
of sheaves of groups on C=U, and the 2-cells are the (global) homotopies of she*
*af
isomorphisms. Grp (C) is not a presheaf of groupoids, because it does not take
values in small groupoids.
If G is a gerbe, then the corresponding resolution 2-groupoid R(G) (Example
12) is weakly equivalent to a point in the sense that the map R(G) ! * is a loc*
*al
weak equivalence of presheaves of 2-groupoids. There is a canonical morphism
F (G) : R(G) ! Grp (C) for which the 0-cell x 2 R(G)(U) is mapped to the
sheaf of groups "Gx, the 1-cell ff : x ! y is mapped to the sheaf isomorphism
cff: "Gx! "Gyon C=U which is defined by conjugation by the global section ff,
and each 2-cell h : ff ! fi of 1-cells x ! y maps to conjugation by the image of
h 2 "Gy(U) in global sections of "Gy. In this way, each gerbe G has a canonical*
*ly
associated 2-cocycle
* -' R(G) F(G)---!Grp(C).
Write H(*, Grp (C)) for the category of 2-cocycles taking values in the 2-
groupoid object Grp (C).
The assignment of the cocycle F (G) : R(G) ! Grp (C) to the gerbe G is not
functorial. It is true, however, that a map G ! H of gerbes induces~a 2-groupoid
morphisms f* : R(G) ! R(H). The sheaf isomorphisms fx : G"x =-!H"f(x)
induced by the local weak equivalence f determine a homotopy
R(G) x 1_! Grp (C)
from F (G) to F (H) . f*. It follows that F (G) and F (H) represent the same el*
*e-
ment of ss0H(*, Grp (C)), and so the assignment G 7! [F (G)] induces a function
: ss0Gerbe (C) ! ss0H(*, Grp (C)).
Suppose that
* -' A K-!Grp (C)
is a 2-cocycle with coefficients in Grp (C). Then K consists of 2-groupoid mor-
phisms K(U) : A(U) ! Grp (C)(U), and hence induces composite morphisms
A(U) K(U)---!Grp(C)(U) evU--!grp.
Here, evU : Grp (C)(U) ! grp is the 2-groupoid morphism which is defined by
U-sections.
Write EA K(U) for the Grothendieck construction corresponding to the com-
posite evU K(U). Then the assignment U 7! EA K(U) defines a presheaf of
groupoids EA K. From Section 2, we see that there is a canonical morphism
p : EA K ! ssA of presheaves of groupoids; it is defined in sections to be the
identity on objects, and it sends a class [(f, ff)] to the class [ff].
15
Lemma 16. Suppose that K : A ! Grp (C) is a 2-cocycle over the terminal
object * taking values in sheaves of groups. Then the presheaf of groupoids EA K
is a gerbe.
Proof. The map ss0EA K ! ss0(ssA) is an isomorphism of presheaves, since_each_
2-functor K(U) takes values in groups. |__|
Take i 2 A(U) and let K(i) be the corresponding sheaf of groups on C=U.
The functor OE : K(i) ! p=i of presheaves of groupoids on C=U is defined in
sections corresponding to an object _ : V ! U of C=U by sending the group
element f 2 K(i)(V ) = K(_*(i))(V ) to the class [(f, 1i)].
Lemma 17. Suppose that K : A ! Grp (C) is a 2-cocycle over * taking values
in sheaves of groups, and choose i 2 A(U). Then the homomorphism fli :
K(i) ! hom (i, i) EA K defined by f 7! [(f, 1i)] induces an isomorphism of
sheaves of groups on C=U.
Proof. Suppose that [(g, ff)] is an element of hom (i, i). By Lemma 7 there is a
covering sieve R hom ( , U) such that there is a 2-cell hOE: OE*(ff) ! 1ff*(i*
*)for
all OE 2 R. It follows that, locally, [(g, ff)] is in the image of fli.
Take group elements f, g 2 K(i)(U) and suppose that fli(f) = fli(g). Then
there is a 2-cell h : 1i! 1i in A(U) such that the diagram
* PPfPP
h* || PP''*77
fflffl|gnnnnnn
*
commutes in K(i)(U). The presheaf of groupoids A(i, i) is a ~Cech object by
Lemma 8 so that there is a covering OE : V ! U such that OE*(h) = 1 for all
members OE of the cover. But then OE*(h*) = 1 for all OE, and so h* = 1_since_
K(i) is a sheaf of groups. |__|
Corollary 18. Suppose that K : A ! Grp (C) is a 2-cocycle over * taking
values in sheaves of groups, and choose i 2 A(U). Then the corresponding map
fli: K(i) ! ss=i is a local equivalence of presheaves of groupoids on C=U.
Proof. The map fli: K(i) ! ss=i takes the group element f to the automorphism
[(f, 1i)] of the object [1i] : ss(i) ! i. The induced map K(i) ! hom ([1i], [1i*
*]) is
a surjection of presheaves of groups. The composite
K(i) ! hom ([1i], [1i]) ! hom (i, i)
is locally monic by Lemma 17. __
The map fli: K(i) ! ss=i is a sectionwise surjection on path components. |_*
*_|
16
Corollary 19. Suppose that the diagram
' q A RRKRR
qqqq | RR))
*xxqff`||MMM Grp5(C)5
'MMM|fflfflllllll
B G
is a morphisms of 2-cocycles. Then the induced map ` : EA K ! EB G is a local
weak equivalence of presheaves of groupoids.
Proof. The diagram
K(i) ___fli__//_hom(i, i)
= || |`*|
fflffl| fflffl|
G(`(i))_fl`(i)//_hom(`(i), `(i))
commutes, so that ` induces an isomorphism on all sheaves of fundamental
groups by Lemma 17. The map ` induces an isomorphism on sheaves of path_
components by Lemma 16. |__|
It follows that the assignment K 7! EA K defines a functor H(*, Grp (C)) !
Gerbe (C) and hence a function
: ss0H(*, Grp (C)) ! ss0(Gerbe (C)).
Theorem 20. The functions and are inverse to each other, and define a
bijection
ss0(Gerbe (C)) ~=ss0H(*, Grp (C)).
Proof. The relation = 1 is a consequence of Lemma 13.
Suppose that K : A ! Grp (C) is a 2-cocycle over *. There is a 2-groupoid
morphism ! : A ! R(EA K) which is the identity on objects, sends the 1-cell
ff : i ! j to the 1-cell [(e, ff)], and sends the 2-cell h : ff ! fi to the 2-c*
*ell
[(h*, 1)] : [(e, ff)] ! [(e, fi)].
The presheaf of groupoids ssR(EA K) has the same objects as A; it is a ~Cech
object (by Lemma 8), in which there is a morphism i ! j in R(EA K) if and
only if there is a 1-cell i ! j in A. It follows that the morphism ! induces
an isomorphism on presheaves of path components. It also follows that the
composite
A !-!R(EA K) F(EAK)-----!Grp(C)
defines a group-valued 2-cocycle on ss0A. This composite sends the object i 2 A
to the presheaf of groups EA K(i, i), sends a 1-cell ff : i ! j to the homomor-
phism cff: EA K(i, i) ! EA K(j, j) which is defined by conjugation with [(e, ff*
*)],
17
and sends a 2-cell h : ff ! fi to the homotopy defined by conjugation with the
element [(h*, 1)].
The assignments f 7! [(f, 1)] define homomorphisms
fli: K(i) ! EA K(i, i).
which induce isomorphisms of associated sheaves, by Lemma 17. The morphisms
fli further determine a homotopy
fl : A x 1_! Grp (C).
from the cocycle K to the cocycle F (EA K)!. It follows that there is a path
F (EA K) ~ F (EA K)! ~ fl ~ K
in the cocycle category, and so = 1 as required. |___|
4 Homotopy classification of gerbes
Suppose that G is a presheaf of groupoids on C, with automorphism sheaves
G"x, x 2 G(U). The presheaf of 2-groupoids G* has the same objects as G;
the 1-cells x ! y of G*(U) are the sheaf isomorphisms G"x ! G"y, and the
2-cells of G*(U) are the homotopies of isomorphisms. There is a 2-functor
G : G* ! Grp (C) which is defined by sending x to "Gx, and is the identity on
sheaf isomorphisms and homotopies. The canonical 2-cocycle F (G) : R(G) !
Grp (C) factors uniquely through a cocycle F (G)* : R(G) ! G* in the category
of presheaves of 2-groupoids.
Suppose that F Grp (C) is a subobject of Grp (C) such that
1)the imbedding is full: all simplicial presheaf maps
F(H, K) ! Grp (C)(H, K)
are isomorphisms,
2)F is a presheaf of groupoids, so that all classes Ob (F)(U) are sets,
We shall say that a subobject F of the diagram of 2-groupoids Grp (C) which
satisfies these conditions is a full subpresheaf of Grp (C).
The image of the presheaf of 2-groupoids G* in Grp (C) which arises from a
presheaf of groupoids G is an example of such an object F.
Lemma 21. Suppose that F F0 are full subpresheaves of Grp (C). Suppose
further that every automorphism group F0xof F0 is locally isomorphic to auto-
morphism groups of F. Then the inclusion F F0 is a local weak equivalence
of presheaves of 2-groupoids.
18
Proof. Write ff : F F0 for the inclusion morphism. Then ff is full, and
therefore induces a presheaf monomorphism ss0G* ! ss0F. Every sheaf of groups
F0x2 F0(U) is locally isomorphic to objects in the image of ff, by definition, *
*so
that ss0F ! ss0F0 is a local epimorphism.
The assertion that ff induces an isomorphism in all possible sheaves of_high*
*er
homotopy groups is a consequence of the fullness and Lemma 1. |__|
Say that two gerbes G and H are locally equivalent if there is a covering fa*
*mily
U ! *, U 2 Ob (C), such that the restricted gerbes G|U and H|U are locally
weakly equivalent on C=U for each object U in the covering of the terminal
object *. If there is a local weak equivalence G ! H then G and H are locally
equivalent in the sense just described, but the converse is not true.
Example 22. Suppose that the presheaf of 2-groupoids F is a full subpresheaf
of Grp (C), and that there is a 2-cocycle
* -' A F-!F Grp (C)
over the terminal sheaf *.
There is a covering family U ! *, U 2 C, such that A(U) 6= ;. In effect,
Ob (A) ! * is a local epimorphism, so there is a covering U ! * such that there
are liftings
Ob<(A)<
xUxxxx |
xx |
xx fflffl|
U _______//_*
where xU represents an object of A(U). The presheaf of groupoids EA F is
locally connected by Lemma 16, and the maps
F (xU ) ! hom EAF(xU , xU )
induce isomorphisms of associated sheaves of groups on C=U by Lemma 17. It
follows that the automorphism groups of the Grothendieck construction EA F
are locally equivalent to objects of F.
Write F - Gerbe for the full subcategory of the category of gerbes whose
automorphism groups are locally equivalent to sheaves of groups in F. The
assignment F 7! EA F for a cocycle F : A ! F takes values in F-gerbes, so
that there is a commutative diagram
ss0H(*, F)_______//ss0H(*, Grp (C))
| |~
| |=
fflffl| fflffl|
ss0(F - Gerbe )______//ss0(Gerbe )
Note that if f : G ! H is a local weak equivalence of gerbes, then G is an
F-gerbe if and only if H is an F-gerbe, and it follows that the induced map
ss0(F - Gerbe ) ! ss0(Gerbe )
19
is an injection.
Theorem 23. Suppose that F is a full subpresheaf of the Grp (C). Then the
Grothendieck construction defines a function
ss0H(*, F) ! ss0(F - Gerbe )
which is a bijection.
Proof. Suppose given cocycles F : A ! F and G : B ! F such that F and
G are in the same path component as cocycles taking values in Grp (C). Then
there is a string of maps of cocycles
F = F0 $ F1 $ . .$.Fn = G (2)
where Fi: Ai! Grp (C) are cocycles in Grp (C).
Suppose that F : A ! Grp (C) is a cocycle taking values in sheaves of groups
locally isomorphic to objects of F and that
A _____ff___//_::A0
::
F ::oo: F0
Grp (C)
is a morphism of H(*, Grp (C)). Take x 2 A0(U). Then there is a covering
family OE : V ! U with 1-cells OE*(x) ! ff(yV ) in A0(V ) for all OE. It follow*
*s that
the group F 0(x) is locally isomorphic to groups of the form F (yV ), and all of
these are locally isomorphic to objects of F. Thus, the cocycle F 0takes values
in sheaves of groups locally isomorphic to objects of F
It follows that all cocycles Fi in the list (2) take values in groups locally
isomorphic to objects of F. Write F0 for the presheaf of 2-groupoids which is
the full subobject of Grp (C) on the sheaves of groups appearing in the sets
F(U) and all Fi(Ob (Ai))(U). Then F F0, and Lemma 21 implies that this
map of presheaves of 2-groupoids is a weak equivalence. The string of cocycles
Fi in (2) all take values in F0 by construction, and the map
ss0H(*, F) ! ss0H(*, F0)
is a bijection. It follows that the original cocycles F and G are in the same p*
*ath
component of H(*, F). The function
ss0H(*, F) ! ss0H(*, Grp (C))
is therefore a monomorphism, as is the function
ss0H(*, F) ! ss0(F - Gerbe ).
Suppose that the every automorphism presheaf of the gerbe H is locally
equivalent to an object of F. Then all automorphism sheaves of H are locally
20
isomorphic to automorphism sheaves of G. Choose a full subpresheaf F0
Grp (C) whose 0-cells are sheaves of groups locally equivalent to objects of F
and which contains both F and H*. Then the canonical cocycle
F (H) : R(H) ! Grp (C)
takes values in F0. The map F ! F0 is a local weak equivalence, and so F (H)
can be represented by a cocycle taking values in F. It follows that the functio*
*n_
ss0H(*, F) ! ss0(F - Gerbe ) is surjective. |__|
Corollary 24. Suppose that F is a full subpresheaf of 2-groupoids in Grp (C).
Then there is a bijection
[*, dBF] ~=ss0(F - Gerbe ).
Suppose that G is a gerbe, and write G - Gerbe for the category of gerbes
which are locally equivalent to G. This category coincides with the category
G*-Gerbe arising from the full subpresheaf of 2-groupoids G*, and so we have
the following:
Corollary 25. Suppose that G is a gerbe on a site C with associated 2-groupoid
object G* of isomorphisms and homotopies of automorphism sheaves of G. Then
there is a bijection
[*, dBG*] ~=ss0(G - Gerbe ).
A special case of Corollary 25, corresponding to the case of a sheaf of grou*
*ps
G, was proved by Breen in [1].
Remark 26. Recall that gerbes on C can be identified with gerbes on the
fibred site C=E up to natural equivalence. Given an gerbe G, write GE for the
corresponding gerbe on C=E. Then Corollary 25 gives a homotopy classification
[*, dBGE*] ~=ss0(GE - Gerbe ).
for gerbes, up to local equivalence defined on the site C=E.
Suppose that F0 is a full subpresheaf of Grp (C), and write Gerbe (F0) for
the full subcategory of gerbes G such that G* F0 _ say that the objects
of Gerbe (F0) are the gerbes in F0. The category F - Gerbe of gerbes with
automorphism sheaves locally isomorphic to objects of F is a filtered colimit
of subcategories Gerbe (F0), indexed over all inclusions F F0 of full sub-
presheaves of Grp (C) such that every object of F0 is locally isomorphic to
objects of F. It follows that there is an isomorphism
ss0(F - Gerbe ) ~= lim-!ss0(Gerbe (F0)).
F' F0
Write St(F) and St(ssF) for the stack completions (really, fibrant models)
for the presheaf of 2-groupoids F and its path component object ssF. The path
21
component object is a groupoid of outer automorphisms and its stack completion
St(ssF) is the stack of bands (liens) for F. These stack completion constructio*
*ns
are functorial, since the underlying model structures are cofibrantly generated
[13].
A (global) band L is a global section of the presheaf of groupoids St(ssF), *
*or
equivalently [8] a torsor for the presheaf of outer automorphism groupoids ssF.
Write pF for the composite
F ! ssF ! St(ssF).
The homotopy fibre over a global band L of the induced map BF ! B St(ssF)
is the classifying object B(pF =L) of the simplicial groupoid pF =L [7].
The objects of 2-cocycle category H(*, B(pF =L)) can be identified with the
collection of pairs ( , OE) consisting of a 2-cocycle
* -' A OE-!F
and a natural (iso)morphism : OE* ! L in St(ssF), where OE* : ss(A) ! St(ssF)
is the unique induced morphism in the diagram
OE
A ________//_F (3)
| |
| |
fflffl| fflffl|
ss(A)__OE*//_St(ssF)
and L has been identified with the composite
ss(A) ! * L-!St(ssF).
The morphisms f : (OE, ) ! (OE0, 0) are cocycle morphisms
A NN OE
| NNN''N
f || p8F8
fflffl|OE0ppppp
A0
such that 0. f* = : OE* ! L.
Suppose that G is a gerbe in F. Consider the diagram
F(G)
G ______//EER(G)___//_F_____//St(F) (4)
EE | | |
EEE | | |
EE""fflffl| fflffl| fflffl|
ssR(G)F(G)*//_ssFj__//St(ssF)
and write !G = jF (G)* : ssR(G) ! St(ssF).
22
An L-gerbe (G, G ) in F is a gerbe G in F together with a natural isomor-
phism G : !G ! L, where L has been identified with the composite
G ! * L-!St(ssF).
Observe that there is a canonical natural isomorphism
~=
hf* : F (G)* -! F (H)*f*
for any morphism f : G ! H of gerbes. A morphism f : (G, G ) ! (H, H ) of
L-gerbes is a morphism f : G ! H of gerbes such that the diagram of natural
isomorphisms
!G j(hf*)//_!H f* (5)
G || ||H f*
fflffl| fflffl|
L ___1___//L
commutes. The natural isomomorphisms hf* arising from gerbe morphisms f
are coherent; this gives the law of composition for a category of L-gerbes in F,
which will be denoted by Gerbe (F)=L.
Every L-gerbe (G, G ) in F determines an object (F (G), G ) in the cocycle
category H(*, B(pF =L)).
Suppose that f : (G, G ) ! (H, H ) is a morphism of L-gerbes in F. Then
the homotopy of cocycles hf : R(G) x 1_! F from F (G) to F (H)f* determines
a diagram of path component groupoid morphisms
ssR(G)TTT
| TTTTTF(G)*TT
| TTTTT
fflffl| hf* TTTTT** j
ssR(G)OxO1_________________//ssF____//St(ssF);;
ww
| ww
| www
| ww F(H)*
ssR(G)___f*__//ssR(H)
The natural isomorphism G : jF (G)* ! L extends uniquely to a natural
isomorphism h : jhf* ! L, and h restricts to H f* : jF (H)* ! L on
ssR(G) x {1} on account of the commutativity of the diagram (5).
It follows that every morphism f : (G, G ) ! (H, H ) of L-gerbes determines
a path between the associated objects (F (G), G ), (F (H), H ) in the cocycle
category, and that there is a function
F : ss0(Gerbe (F)=L) ! ss0H(*, B(pF =L))
which is defined by ([(G, G )]) = [(F (G), G )].
23
Suppose that the 2-cocycle
*oo_'___A ____OE__//_F
| |
| |
fflffl| fflffl|
ss(A)_OE*//_St(ssF)
and the natural isomorphism : OE* ! L in St(ssF) define an object ( , OE) of
the cocycle category H(*, B(pF =L)). Then the associated presheaf of groupoids
EA OE is a gerbe which has automorphism sheaves locally isomorphic to objects
of F, and then from the proof of Theorem 20 we know that there is a homotopy
A TTTTT
| TTTTTOET
| TTTTT
fflffl| fl TTTTT**
A xO1___________________//F0O;;w
| wwww
| wwF(EAOE)w
| w
A ___!___//R(EA OE)
where F0is a full subpresheaf of Grp (C) containing F such that the map F F0
is a local weak equivalence. We also know that the induced map !* : ssA !
ssR(EA OE) is an isomorphism. It follows that the induced natural isomorphism
(or homotopy)
~=
fl* : OE* -! F (EA OE)*!*
of functors ssA ! St(F0) induces a unique natural isomorphism
jF (EA OE)* "-!L
which restricts to the isomorphism : OE* ! L along the homotopy
ss(A) x 1_fl*-!ssF0 ! St(ssF0).
In other words, (EA OE, ") is an L-gerbe in F0.
Suppose that f : (OE, ) ! (OE0, 0) is a morphism of the 2-cocycle category
H(*, B(pF =L)). Then there is a full subpresheaf F00 Grp (C) containing F,
such that F F00is a weak equivalence and such that the associated gerbes
EA OE and EA0OE0are gerbes in F00. There is a diagram of homotopies
OE*(i)__=____//OE0*(f(i))
fli|| flf(i)||
fflffl| fflffl|
j Aut(i)__f*//_j Aut(f(i))
24
where Aut(i) is the sheaf of automorphisms of i in EA OE and Aut(f(i)) is the
sheaf of automorphisms of f(i) in EA0OE0. Then the morphisms i : OE(i) ! L
and 0f(i): OE0(f(i)) ! L coincide on OE(i) = OE0(f(i)) since f is a morphism of
the cocycle category H(*, B(pF =L)). Furthermore, the vertical isomorphisms
uniquely determine the natural isomorphisms " : j Aut(i) ! L and " 0f* :
j Aut(f(i)) ! L, respectively. It follows that the map f* : EA OE ! EA0OE0defin*
*es
a morphism of L-gerbes in F00. We therefore have a well defined function
: ss0H(*, B(pF =L)) ! lim-!ss0(Gerbe (F0)=L).
F' F0
Theorem 27. Suppose that L is a band. Then the function
is a bijection.
Proof. Suppose that F F00 is a weak equivalence of full subpresheaves of
Grp (C). Then the diagram
ss0(Gerbe (F)=L)__F__//_ss0H(*, B(pF =L))
| |~
| |=
fflffl| fflffl|
ss0(Gerbe (F0)=L)____//ss0H(*, B(pF0=L))
F0
commutes, where the indicated vertical map is a bijection since the comparison
map B(pF =L)) ! B(pF0=L) is a local weak equivalence. It follows that the
maps F0 together induce a function
: lim-!ss0(Gerbe (F0)=L) ! ss0H(*, B(pF =L)).
F' F0
The function is the inverse of . |___|
Corollary 28. Suppose that L is a band. Then there are bijections
[*, B(pF =L)] ~=ss0H(*, B(pF =L)) ~= lim-!ss0(Gerbe (F0)=L).
F' F0
25
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