ButterAEies II: Torsors for 2-group stacks
Ettore Aldrovandi
Department of Mathematics, Florida State University
1017 Academic Way, Tallahassee, FL 32306-4510, USA
aldrovandi@math.fsu.edu
Behrang Noohi
Department of Mathematics, King's College London
Strand, London WC2R 2LS, UK
behrang.noohi@kcl.ac.uk
Abstract
We study torsors over 2-groups and their morphisms. In particular,
we study the orst non-abelian cohomology group with values in a 2-group.
ButterAEy diagrams encode morphisms of 2-groups and we employ them to
examine the functorial behavior of non-abelian cohomology under change
of coeOEcients. We re-interpret the orst non-abelian cohomology with co-
eOEcients in a 2-group in terms of gerbes bound by a crossed module. Our
main result is to provide a geometric version of the change of coeOEcients
map by lifting a gerbe along the ifractionj (weak morphism) determined
by a butterAEy. As a practical byproduct, we show how butterAEies can
be used to obtain explicit maps at the cocycle level. In addition, we dis-
cuss various commutativity conditions on cohomology induced by various
degrees of commutativity on the coeOEcient 2-groups, as well as specioc
features pertaining to group extensions.
Contents
1 Introduction 2
1.1 Content of the paper . . . . . . . . . . . . . . . . . . . . . . . . .*
* 3
1.2 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . *
* 7
1.3 Conventions and notations . . . . . . . . . . . . . . . . . . . . . . 8
2 Recollection of results from [Part I] 8
2.1 Crossed modules and gr-stacks . . . . . . . . . . . . . . . . . . . 8
2.2 ButterAEies and weak morphisms . . . . . . . . . . . . . . . . . . . 9
2.3 Composition of butterAEies and the bicategory of crossed modules 11
1
3 Torsors and non-abelian cohomology 12
3.1 Non-abelian cohomology . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 1-Cocycles with values in crossed modules . . . . . . . . . . . . . 13
3.3 Bitorsor cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. 15
3.4 Torsors for gr-stacks . . . . . . . . . . . . . . . . . . . . . . . . *
*. 17
4 Pushing cohomology classes along butterAEies 19
4.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* 20
4.2 Lift of a 1-cocycle along a butterAEy . . . . . . . . . . . . . . . . *
*. 20
4.3 Computing the map F* . . . . . . . . . . . . . . . . . . . . . . . 21
5 Gerbes bound by a crossed module 23
5.1 Recollections on gerbes . . . . . . . . . . . . . . . . . . . . . . . .*
* 23
5.2 Gerbes bound by a crossed module . . . . . . . . . . . . . . . . . 24
5.3 Local description . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. 26
5.4 The class of a gerbe bound by a crossed module . . . . . . . . . . 27
5.5 Gerbes vs. torsors . . . . . . . . . . . . . . . . . . . . . . . . . *
*. 30
6 Extension of gerbes along a butterAEy 35
6.1 Extension of torsors . . . . . . . . . . . . . . . . . . . . . . . . *
*. 35
6.2 Debremaeker's extension along strict morphisms . . . . . . . . . 36
6.3 Extension along a butterAEy . . . . . . . . . . . . . . . . . . . . . *
*38
6.4 Induced map on non-abelian cohomology . . . . . . . . . . . . . . 41
7 Commutativity conditions 43
7.1 Commutativity conditions and butterAEies . . . . . . . . . . . . . 43
7.2 The monoidal 2-stack of G -torsors . . . . . . . . . . . . . . . . . *
*44
7.3 Group structures on cohomology and butterAEies . . . . . . . . . . 44
7.4 Explicit cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. 45
8 ButterAEies and extensions 47
8.1 The Schreier-Grothendieck-Breen theory of extensions . . . . . . 47
8.2 Remarks on extensions by commutative crossed modules . . . . . 49
8.3 ButterAEies, extensions, and simplicial morphisms . . . . . . . . . 51
1 Introduction
This paper is the second part of a series aimed at a systematic study of n-group
stacks and their torsors. The orst part, [AN09 ], is dedicated to the case n = *
*2 of
2-group stacks, or gr-stacks, in a slightly older terminology, and especially t*
*heir
morphisms. The most important result is that if 2-group stacks are made strict
by replacing them with (sheaves of) crossed modules, the groupoid of morphisms
between 2-group stacks is equivalent to that of certain special diagrams called
butterAEies between corresponding crossed modules. This allows to overcome
the longstanding problem, even present in the non sheaf-theoretic setting, that
replacing a monoidal category with a strict one is not a functorial constructio*
*n.
2
Moving up one step in the cohomological ladder, the present paper, which
is a direct sequel to [AN09 ], is concerned with the torsors for 2-group stacks.
In a very general sense, torsors are the global geometric objects from which
1-cocycles with values in a 2-group stack arise, once suitable local trivializi*
*ng
data have been chosen. In eoeect, after a rigidiocation has been performed by
replacing a 2-group stack by a crossed module, such cocycles will take values
in a complex of sheaves (of length 2). This is the categorioed version of the
familiar process which associates to a principal G-bundle (or ordinary G-torsor)
with local sections a 1-cocycle with values in G. Indeed the case n = 1 is the
one of ordinary group objects. (In general a similar situation holds in the case
of n-group stacks, as we shall see in later installments of this series.)
Our aim is to the study morphisms of torsors by harnessing the power of
butterAEies developed in the orst part of this series, and to illustrate a few *
*appli-
cations.
1.1 Content of the paper
It is useful to describe the context of our work in its more general terms. Thu*
*s,
if F :H ! G is a morphism of 2-group stacks over a certain site S, we want an
appropriate morphism
(1.1.1) F*: TORS (H ) -! TORS (G ),
where TORS (G ) denotes the 2-stack of G -torsors. One obtains in this way a
geometric deonition of degree-one non-abelian cohomology sets, with built-in
functoriality. Namely, if by TORS (G )(*) we denote the 2-groupoid of global
torsors, we can deone H1(G ) simply as ss0 TORS (G )(*) , the connected com-
ponents of that 2-groupoid; once F* is deoned, the functoriality of the orst
cohomology follows automatically.
A viable general mechanism by which torsors are extended ialongj a mor-
phism of n-group stacks is in fact well-known: given an H -torsor X , one deones
F* via the icontracted productj
H
(1.1.2) F*(X ) = X ^ G ,
see [Bre90, #6], and section 6.1 below for all the details. The construction on
the right-hand side above is the icategoriocationj of the standard one in the
case of ordinary torsors, that is n = 1. The above deonition of F* provides a
conceptual answer to onding a morphism (1.1.1), and therefore, by the above
geometric deonition of cohomology, an induced morphism
(1.1.3) H1(H ) -! H1(G ).
On the other hand, the recently introduced butterAEy diagrams aoeord a rather
one-grained picture of morphisms of 2-group stacks, to be recalled below, so one
asks for a similar description of (1.1.1)and the induced map (1.1.3).
3
To discuss this, let us recall from the orst part that a butterAEy allows to
decompose a morphism F :H ! G into a ifractionj
H oo_Q_E__P__//G,
where Q is an equivalence of 2-group stacks. Actually, if we introduce crossed
modules H1 ! H0 and G1 ! G0 for H and G , respectively, the fraction above
is determined by a butterAEy diagram of group objects:
H-1________ G-1________
| ___________________________________________________*
*_______________________________________________________________________|
| _________________________________________________*
*_________________________________________________________________________|
| _%%____________________________________yy______*
*______________________________|
(1.1.4) | _E________|____________________
| _____________|___________________________________*
*______________________________________________________________________________
| _______________|___________________________________*
*____________________________________________________________________
fflffl|__________fflffl|______________________""_____*
*_________________________
H0 G0
The NW-SE sequence is a complex, and the NE-SW sequence is a group exten-
sion. One onds the resulting map H1 x G1 ! E is a crossed module in its own
right, which is quasi-isomorphic to H1 ! H0, and determines the stack E . In
sum, the butterAEy allows to split F :H ! G into a fraction of morphisms corre-
sponding to morphisms of crossed modules. In fact, diagram (1.1.4)corresponds
to a fraction in the derived category of crossed modules
(1.1.5) Hooo_q_Eo __p_//_Go
where now p and q are genuine morphisms of crossed modules (the latter being
a quasi-isomorphism) inducing the corresponding ones denoted by upper-case
letters between corresponding 2-group stacks.
We have alluded to the fact that classes in, say, H1(G ) can be represented
by 1-cocycles with values in the crossed-module G1 ! G0. Let us remind the
reader, following [Bre90], that such cocycles can equivalently be described as
simplicial maps from hypercovers of objects of S to a reasonable_model of the
classifying space of G , as provided for instance by the W construction applied
to the simplicial group G_determined by the crossed module. It is possible to
prove using (1.1.2)that a cocycle with values in Ho determines one with values
in Go. The argument mostly rests on the construction of a morphism
___ ___
(1.1.6) W H_-! W G_
between classifying objects. (Note, in passing, that this is the very deonition
of weak morphism of crossed modules in the set-theoretic case, see [Noo05 ].)
Unfortunately, starting from the morphism F as a whole is not very explicit or
constructive, not only because it requires a chosen rigidiocation of .the other*
*wise
weak group laws of H and G , but chieAEy because F does not determine a direct
morphism Ho ! Go between crossed modules.
Our orst result is to exploit the butterAEy technology to provide a much more
direct approach to computing the morphism (1.1.3). As explained in section 4
4
below, the morphism (1.1.3)can be computed by, oguratively_speaking, lifting
a 1-cocycle, or equivalently a simplicial map j :Uo ! W H_along the trian-__
gle (1.1.5). More concretely, one constructs a new simplicial map j0:Uo ! W E_
such that its projection via q is j (possibly after passing to a oner hypercover
which will not be notationally distinguished); in eoeect j0 represents the same
class as j, since Ho and Eo are quasi-isomorphic. Then the sought-after mor-
phism is simply obtained by projecting j0 along p. Diagrammatically, we have:
j oo_q_j0_p__//,
where , denotes the resulting simplicial map or 1-cocycle with values in Go.
This same method, in simpler form, works for 0-cocycles as well, such as
those dealt with in the orst part, and it is expected to do so for higher degree
classes in the case the involved 2-group stacks are symmetric or Picard.
The construction just outlined embodies the general idea that informs our
main result, a novel geometric construction of the morphism (1.1.1). Starting
from the butterAEy decomposition of F we want to decompose F* as
TORS (H )ooQ*_TORS (E )_P*_//TORS (G ),
where P* and Q* are expected to be simpler than F*, since P and Q each
arise from a strict morphism. Moreover, this decomposition should be such that
passing to cohomology classes, or better yet to representative cocycles, provid*
*es
a calculation of the map (1.1.3)of cohomology set outlined above.
Now, in practice, we do not implement our program within the context of of
torsors over a 2-group stack, essentially due to the fact that the direction of*
* P
is at odds with the natural notion of extension of torsors along a morphism (i.*
*e.
P goes in the wrong direction). One can of course make the choice of a quasi-
inverse P *to it, but that defeats the purpose, so to speak; we want something
more canonical.
It turns out the concept of gerbes iboundj by a crossed module is the ap-
propriate notion. In very broad terms, the general idea, originally due to Debr*
*e-
maeker (see [Deb77 ]), is that a gerbe P bound by a crossed module G1 ! G0
is a gerbe equipped with a morphism
~: P -! TORS (G0)
subject to certain additional conditions, recalled in section 5, which in parti*
*cular
make P into a G1-gerbe. These gerbes give rise to non-abelian cohomology
classes with values in the crossed module (or in fact in a 2-group stack) too.
Torsors do the same of course, and indeed we prove there in an equivalence
(1.1.7) TORS (G ) -! GERBES (G1, G0),
which generalizes a similar result of Breen (for the 2-group stack of G-bitorso*
*rs
for a group object G and G-gerbes) put forward in [Bre90]. While the equiva-
lence and the statement have pretty much identical forms, the proof is however
quite dioeerent, and we have included it here.
5
Thus the actual version of the decomposition we provide is is to deone a
morphism
F+ :GERBES (H1, H0) -! GERBES (G1, G0)
by means of the following diagram
Q0+ P0+
(1.1.8) GERBES (H1, H0)oo__GERBES (E1, E0)____//_GERBES (G0, G1)
where the deonition of P+0and Q0+is direct (available in [Deb77 ]), since p and*
* q
are strict morphisms of crossed modules. The quasi-inverse to the arrow pointing
to the left, the one that would be diOEcult in the itorsorj version of (1.1.8),*
* is
surprisingly simple in the gerbe context: from a gerbe Q bound by the crossed
module Ho, the gerbe bound by Eo we need is simply the stack obered product:
Q0= Q xTORS(H0)TORS (E).
The image of Q0 by Q+ is equivalent to Q, and by ipushingj along P , that
is, considering the image under P+ , we obtain a gerbe bound by Go. We then
prove, essentially by comparing cohomology classes, that F+ is equivalent to F*,
modulo the equivalence (1.1.7), so in other words we obtain a 2-commutative
square
TORS (H ) ____F*_____//_TORS(G )
| |
| |
fflffl| F+ fflffl|
GERBES (H1, H0)_____//GERBES (G1, G0)
After having gone through these general result, we move on to consider some
applications, mainly to the abelian structures on cohomology resulting when
braided, symmetric, or Picard structures are imposed on the coeOEcients, and
speciocally when group extensions in the sense of Grothendieck ([Gro72 ]) and
Breen ([Bre90, #8]) are concerned. In the end we make contact with the deo-
nition of weak morphism between crossed modules as simplicial maps between
classifying spaces. Since several results are already known, our discussion as-
sumes a more informal character compared to the previous sections, and many
arguments are just sketched.
Let us conclude with a comment about the use of gerbes bound by crossed
modules. The original intent behind the introduction of the concept of gerbe
bound by a crossed module was to correct the perceived lack of functoriality
inherent in Giraud's deonition of higher non-abelian cohomology using liens
(see [Gir71]). Functoriality was addressed in Debremaeker's paper [Deb77 ] by
considering only morphisms of crossed modules, that is what we now call strict
morphisms. This restriction to strict morphisms is not the natural thing to
do, and since non abelian cohomology depends on the associated 2-group stack,
rather than on the coeOEcient crossed module itself, introducing torsors led to*
* a
better conceptual understanding of the functoriality of non abelian cohomology.
Thus the notion has not been developed or used until recently, when it became
useful in dioeerent contexts (see for instance [Ald08, Mil03]).
6
This state of aoeairs has been changed by the better control of morphisms
aoeorded by the use of butterAEies, since they allow a description of all morph*
*isms
by way of crossed modules. Thus now the use of gerbes bound by crossed
modules plus the use of butterAEies aoeords a geometrization of the non abelian
derived category equivalent to the one obtained by using the torsor picture.
1.2 Organization of the paper
Here is a brief synopsis of this paper's content. Since this is a direct contin*
*uation
of [AN09 ], the reader will unavoidably be constantly referred to that paper. In
order to make this process a little less burdensome, we recall in section 2 some
of the results of that orst part we shall most often need here. In section 3 we
have collected results and deonitions concerning torsors over gr-stacks and non-
abelian cohomology. Our purpose was of course to make a moderate attempt
at being self-contained and at a uniformity of conventions.
By design the material in these sections is not new, except maybe in the
presentation. New results begin in earnest in section 4, where we explicitly
describe in terms of butterAEies the morphism of non-abelian orst cohomology
sets induced by a morphism of gr-stacks.
In section 5 we present the idea of a gerbe bound by a crossed module, orig-
inally due to Debremaeker. In addition to re-introduce the main deonitions, we
analyze the local structure and prove the cohomology class determined by such
an object takes values in the gr-stack associated to the crossed module. Since
this is almost the same idea as that of a torsor for said gr-stack, we determine
the precise relation between the two. In this way we obtain a generalization of
an analogous result due to [Bre90, Proposition 7.3]. The sort of rigidiocation
that the passage from G ! Aut(G) to a general crossed module G ! entails
makes the proof very dioeerent, so we discuss it in detail.
The morphism of orst non-abelian cohomology sets induced by a morphism
of gr-stacks discussed in purely algebraic terms in section 4 has a well-known
geometric realization in terms of extension of torsors along that morphism (this
is the categoriocation of the well-known extension of structural groups for pri*
*n-
cipal bundles). The analogous procedure in terms of gerbes bound by crossed
modules is described in section 6. It generalizes Debremaeker's notion of mor-
phism of gerbes bound by crossed modules, which only uses what we call strict
morphisms of crossed modules. The general case is treated in section 6.3. We
prove that the morphism so obtained is equivalent, modulo the equivalence be-
tween torsors and gerbes, to the morphism given by the extension of torsors,
and in section 6.4 we show that the induced cohomology class is precisely the
one computed by the procedure described in section 4.
Sections 7 and 8 are devoted to some applications. In section 7 we brieAEy
analyze the commutativity conditions on cohomology ensuing from the assump-
tion that the coeOEcient crossed module (or gr-stack) be at least braided. It is
well-known that in this case the orst a priori non-abelian cohomology acquires
a group structure which becomes abelian if the coeOEcient gr-stack is symmetric.
Our approach is to analyze these structures in terms of specioc butterAEy dia-
7
grams associated to braided crossed modules which express the fact that for a
braided gr-stack the monoidal structure is a weak morphism. This is discussed
in detail in [AN09 , #7]. Using these special butterAEies, we are in position *
*to
apply the general theory of section 6 to obtain a novel description of the group
structures on cohomology, for which we can write explicit product formulas
at the cocycle level. Section 8 contains some remarks about group extensions.
First about how the classical Schreier theory of extensions, from the geometric
perspective of Grothendieck and Breen, ots in the butterAEy framework. We
then discuss again commutative structures, and to some extent abelianization
maps. Some onal informal paragraphs are devoted to making contact with the
simplicial deonition of weak morphism of crossed modules.
1.3 Conventions and notations
In the sequel we shall refer to [AN09 ] simply as iPart I.j We keep its standing
assumptions, notations, and typographical conventions: in particular, S denotes
quite generally a site with subcanonical topology, and T = S~ denotes the topos
of Set-valued sheaves over S. Again as in Part I we break our convention usage
in the introduction by reverting to the older term igr-stackj in place of the
more recent 2-group (stack). Concerning the numbering scheme, references to
the orst part are made using that paper's numbering sequence. For this one, we
have chosen to cut the numbering ooe by one level, due to the its reduced length
(compared to [Part I]).
2 Recollection of results from [Part I]
2.1 Crossed modules and gr-stacks
Let G be a gr-stack (or 2-group stack), that is a stack over S endowed with a
group-like monoidal structure
: G x G -! G ,
see, for example, [Bre90, Bre92, Bre94a], and [S#75, JS93] for the point-wise
case. Many of the results from the previous references which are required in
this text are summarized in [Part I], to which the reader is referred for more
details. Here we limit ourselves to recall that starting from G we can always
construct a homotopy obration
G1__@__//G0ssG//_G,
where @ :G1 ! G0 has the structure of a crossed module, so that in fact G
can be recovered as its associated gr-stack. More precisely, the crossed module
G1 ! G0 provides us with a concrete model for the associated gr-stack, namely
there in an equivalence
G -~! TORS (G1, G0).
8
Following Deligne [Del79], the right-hand side denotes the stack of those G1-
torsors which become trivial after extension P _ P ^G1 G0. Thus, G is realized
as the homotopy ober
G ____//_TORS(G1)__@*_//TORS(G0),
that is, an object of G is a pair (P, s), comprising a right G1-torsor P and
a trivialization s: P ^G1 G0 ~! G0. When combined with the crossed module
structure, this picture allows to realize G as a sub-gr-stack of BITORS (G1) by
observing that the underlying G1-torsor in the pair (P, s) acquires a G1-bitors*
*or
structure by deoning a left G1-action through s as:
g . p := p gs(p),
where p 2 P , g 2 G1, and s is viewed as a G1-equivariant morphism s: P ! G0.
A morphism ': (P, s) ! (Q, t) in G is therefore a commutative diagram
'
P___________//___________Q.
______________________________________________
_______________________________________________
s _''____________________________tww______________*
*_______________
G0
It follows that the monoidal structure of G can be expressed through standard
contraction of bitorsors: for two objects (P, s) and (Q, t) of G we set
G1
(P, s) (Q, t) = (P ^ Q, s ^ t),
where s ^ t is the G1-equivariant map given by (p, q) 7! s(p)t(q), where (p, q)
represents a point of P ^G1 Q. It results from the compound trivialization:
G1 G1 G1 G1 1^t G1 s
P ^ Q ^ G0 ' P ^ Q ^ G0 --! P ^ G0 -! G0.
In dealing with gr-stacks and crossed modules we will always~often tacitly~
make use of the interplay outlined in the previous paragraphs, and therefore
move freely between gr-stacks and crossed modules.
2.2 ButterAEies and weak morphisms
Let Ho and Go be crossed modules of T, and let H and G denote their associated
gr-stacks, respectively.
A morphism F :H ! G , that is, an additive functor, is by deonition a weak
morphism from Ho to Go. All weak morphisms from Ho to Go form a groupoid,
denoted WM (Ho, Go).
A butterAEy from Ho to Go is by deonition a commutative diagram of group
9
objects of T:
H1 __ G1__
| ___~_______________________________________________*
*_________________-___________________________________________________________*
*________|
| _________________________________________________*
*_________________________________________________________________________|
| _$$________________________________zz__________*
*_______________________|
(2.2.1) @| _E_______|@________________
| ss___________|____________________________________*
*_____________________________________________________________________________*
*___
| _____________|____________________________________*
*___________________________________________________________________
fflffl|""_______fflffl|_______________________!!_____*
*_________________________
H0 G0
such that the NW-SE sequence is a complex, and the NE-SW sequence is a
group extension. The various maps satisfy the equivariance conditions written
set-theoretically as:
(2.2.2) -(g_(e)) = e-1-(g)e, ~(hss(e)) = e-1~(h)e
where g 2 G1, h 2 H1, e 2 E. An easy consequence of (2.2.2)is that the images
of _ and ~ commute in E.
The short-hand notation [Ho, E, Go] will be used for a butterAEy from Ho to
Go.
A morphism of butterAEies ': [Ho, E, Go] ! [Ho, E0, Go] is given by a group
isomorphism ': E ~!E0 such that the diagram:
H1 _______//_______________E0G1oo_
_____________________________________________________*
*_____________________________________________________________________________*
*_OO
| ___________________________________________________*
*________________________________________________|____________________________*
*___________________|
| __________________________________________________*
*_|__________________________________________________________|
|| ______%%____________________________________________*
*________________________________________________________________||yy_________*
*_____________________________|
| _____________________________________________________*
*_________|E__________________________________________________________________*
*____________________________________________________
|________________________________________________|_____*
*_____________________________________________________________________________*
*______________________
|______________________________________________|_______*
*_____________________________________________________________________________*
*______________
fflffl|____________"~~____________fflffl|"_____________*
*___________!!________________________
H0 G0
commutes and is compatible with all the conditions involved in diagram (2.2.1).
Two morphisms are composed in the obvious way. In this way butterAEies from
Ho to Go form a groupoid, denoted B(Ho, Go).
One of the main results of [Part I, Theorem 4.3.1] reads, in part:
2.2.1 Theorem. There is an equivalence of groupoids
B(Ho, Go) -~! WM (Ho, Go).
A pair of quasi-inverse functors
: B(Ho, Go) -! WM (Ho, Go)
and
: WM (Ho, Go) -! B(Ho, Go).
is explicitly described in Part I.
10
Strict morphisms of crossed modules (described in detail in Part I, section
3.2) correspond to butterAEy diagrams whose NE-SW diagonal is split~with a
deonite choice of the splitting morphism, see Part I, section 4.5. Conversely, a
splittable butterAEy, namely one whose NE-SW diagonal is in the same isomor-
phism class of a semi-direct product, by deonition corresponds to a morphism
equivalent to a strict one.
A butterAEy diagram is called AEippable or reversible if both diagonal are e*
*x-
tensions. The corresponding weak morphism is an equivalence.
It easy to verify that from the butterAEy diagram (2.2.1)the homomorphism
(2.2.3) @E :H1 x G1 -! E,
where @E (h, g) = ~(h)-(g), is a crossed module with the obvious action of E on
H1 x G1 through that of H0 and G0 on the respective factors. Let us denote
this crossed module by
Eo: E1 ! E0,
with E0 = E and E1 = H1 x G1.
From Part I we have that the weak morphism given by the butterAEy (2.2.2)
factorizes as a ifractionj
(2.2.4) Hooo~__Eo ____//_Go
of strict morphisms of crossed modules. The one to the left is a quasi-isomorph*
*isms,
that is, it induces isomorphisms on the corresponding homotopy sheaves:
ssi(Eo) ' ssi(Ho), i = 0, 1.
2.3 Composition of butterAEies and the bicategory of crossed
modules
Composition of butterAEies is by juxtaposition: Given two butterAEies
K1 __ _H1_ H1 __ G1__
| ________________________________________________________________*
*___-0___________________________________________________________________||~__*
*_____________________________________________________________________________*
*_______________________________________________________|
| ______________________________________________________________*
*_____________________________________________________________________________*
*_____||______________________________________________________________________*
*______________________________________________|
________zz____________________________________||$$__________*
*__________________________zz____________________________________|
@K || _$$F______|@H_____________@H|E______|@G______________
| ____________|_________________________________________________*
*_____0_________________________________________________________________||ss__*
*_____________________________________________________________________________*
*___________________________________________________
| _____________|_________________________________________________*
*______________________________________________________||_____________________*
*_____________________________________________________________________________*
*__________________
fflffl|""_______fflffl|_______________________!!__________________*
*____________fflffl|fflffl|""______________________________!!_________________*
*_____________
K0 H0 H0 G0
their composition is the butterAEy (deoned set-theoretically in [Noo05 ]):
K1 ______ _G1_____
| ______________________________________________________*
*__________________________________________________________|
| ___________________________________________________*
*_____________________________________________|
| _$$_______________________________________zz_____*
*___________________________________|
(2.3.1) @K|| F xH1H0E |@G|
| _______________|_________________________________*
*________________
| _________________|_________________________________*
*______________________________________________
| ___________________|_________________________________*
*________________________________________
fflffl|_______________fflffl|___________""______________*
*__________
K0 G0
11
The center is given by a kind of pull-back/push-out construction: we take the
ober product F xH0 E and mod out the image of H1 (see also [Part I, #5.1], for
details).
This composition is not associative: if [Lo, M, Ko] is a third butterAEy , t*
*hen
there only is an isomorphism
K H ~ K H
M xK10F xH10E -! M xK10 F xH10E .
An almost immediate consequence is
2.3.1 Theorem (Part I, Theorem 5.1.4). When equipped with the morphism
groupoids B(-, -), crossed modules in T form a bicategory, denoted XMod__(S).
There are obered analogs of the various entities we have introduced so far:
so, for instance, one deones a obered category B(Ho, Go), which is deoned as
usual by assigning to U 2 Ob S the groupoid
B(Ho|U , Go|U ),
and to every arrow V ! U of S the functor
B(Ho|U , Go|U ) -! B(Ho|V , Go|V ).
Starting from WM (Ho, Go) instead, an identical procedure leads to a obered
category WM (Ho, Go) over S. It is proved in [Part I, 4.6.1, 4.6.2] that both
are stacks (in groupoids) over S. In a more general, but similar, fashion, the
bicategory XMod__(S) has a obered analog, denoted XMod (S). Thanks to the fact
that B(Ho, Go) is itself a stack, XMod (S) is a pre-bistack over S. All gr-stac*
*ks
comprise a 2-stack denoted Gr-STACKS (S). Hence the obvious morphism
XMod (S) ! Gr-STACKS (S) is 2-faithful. Moreover, it is shown that every gr-
stack G is equivalent to the gr-stack associated to a crossed module~see [Part
I, Proposition 5.3.7]. Hence the morphism above is essentially surjective, and *
*it
follows that XMod (S) is a bistack.
3 Torsors and non-abelian cohomology
In this section we recall some facts about G -torsors, where G is a gr-stack. T*
*his
is necessary in order to compare them with one of the main objects of study in
this text, the gerbes bound by the crossed module G1 ! G0, whose associated
gr-stack is G . Those gerbes will be introduced below. Since we shall also be
concerned with classes of equivalence of such objects, as well as functoriality
properties, it is useful to go through a quick review of the some deonitions in
non-abelian cohomology.
3.1 Non-abelian cohomology
Let us recall the main deonitions, following [Bre90] and [Ill71, Jar89, Jar86].*
* Let
G_ be a simplicial group-object of T. The non-abelian cohomology with values
12
in G_can be deoned as
( -i
Hi(*, G_) = Hom D(T)(*, G_),i 0,
Hom D(T)(*, BG_), i = 1.
Here * denotes the terminal object of T, denotes the loop construction,
whereas B G_is_some (in fact any) form for the classifying space construction,
for example W G_. D(T) denotes the derived category of simplicial objects of T
in the same sense as [Ill71, Bre90], that is, by localizing at the morphisms of
simplicial objects that induce isomorphisms of homotopy sheaves.
Note that the simplicial group structure is only relevant in order to deone
H 1, whereas for all other degrees i 0 the deonition only uses the underlying
simplicial set structure. But also note that the former will only be a pointed
set, as opposed to the others which carry group structures (abelian for i < 0).
If we use the convention that B-1 def= , the various Hi(*, G_) are computed as a
colimit:
Hi(*, G_) = lim-!*, BiG_,
V !*
where the colimit runs over homotopy classes of hypercovers of * and [-, -]
denotes (simplicial) homotopy classes.
Our main focus will be the pointed set H 1(*, G_) when the coeOEcient sim-
plicial group arises from a crossed module G1 ! G0, which we denote by
H 1(*, G1 ! G0). In view of the fact that any gr-stack G can be realized as
the gr-stack associated to a crossed module G1 ! G0, as explained in Part I,
we can write the same cohomologies by emphasizing the stack, rather than the
crossed module, as coeOEcients, as Hi(*, G ), i 1. In fact more stress will be
put on the cocycles representing cohomology classes, rather than on the classes
themselves. After all, the former naturally arise from any appropriate decom-
position (i.e. local description) of geometric objects, such as torsors and ger*
*bes,
as it will be clear below.
Following [Bre90], it will be convenient to recall the simplicial deonition *
*of 1-
cocycles, as well as the more geometric one that simply categorioes the standard
deonition by replacing a group with a gr-stack.
3.2 1-Cocycles with values in crossed modules
If G_ois a simplicial_group object of T, there_is a model for its classifying s*
*pace
provided by the W -construction. Namely, W G_o is the simplicial object of T
given by:
___ ___
W G_0= *, W G_n= G_0x G_1x . .x.G_n-1, n 1.
The face and degeneracy maps are:
8
><(d1g_1, . .,.dn-1g_n-1) i = 0
di(g_0, . .,.g_n-1)= > (g_0, . .,.g_i-1d0g_i, g_i+1, . .,.dn-i-1g_n-1)0 < i *
*< n
: (g
__0, . .,.g_n-2) i = n
13
and
8
><(1_, s0g_0, . .,.sn-1g_n-1) i = 0
si(g_0, . .,.g_n-1)= > (g_0, . .,.g_i-1, 1_, s0g_i, . .,.sn-i-1g_n-1)0 < i <*
* n
: (g
__0, . .,.g_n-1, 1_) i = n
We have slightly changed the formulas of ref. [May92 , #21] in order to better *
*ot
with our iaction on the rightj convention.
If G is a group object of T, identioed with the constant simplicial group,
then the previous construction reduces the standard classifying simplicial space
B G.
3.2.1 Deonition. Let Vo_!_U be a hypercover. A 1-cocycle over U is a
simplicial map , :Vo ! W G_o. Two such_cocycles_,, ,0 are equivalent if there is
a simplicial homotopy ff: , ) ,0:Vo ! W G_o.
Let G_o be the nerve of the_groupoid_G_determined_by a crossed_module_
G1 ! G0. In this case we have W G_1= G0, W G_2= G0 x (G0 x G1),_W_G_3=
G0 x (G0 x G1) x (G0 x G1 x G1), etc. A simplicial map , :Vo ! W G_owill be
determined by its 3-truncation ([Bre90]).
A rather tedious, but otherwise straightforward calculation shows that the
simplicial map , determines, and is determined by, a pair (x, g) where x: V1 !
G0 and g :V2 ! G1 satisfying the condition
(3.2.1a) d*1x= d*2x d*0x @g
*x*
(3.2.1b) d*0g d*2g= (d*3g)(d0d1) d1g
and the normalizations s*0x = 1, s*0g = s*1g = 1. The explicit expressions of
the maps ,i, i = 0, . .,.3 are as follows: ,0 = *, ,1 = x: V1 ! G0, whereas
,2: V2 ! G0 x (G0 x G1) and ,3: V3 ! G0 x (G0 x G1) x (G0 x G1 x G1) are
given by
,2 = (d*2x, (d*0x, g))
,3 = ((d2d3)*x, ((d0d3)*x, d*3g), ((d0d1)*x, d*0g, (d*0g)-1d*1g)
A simplicial homotopy ff: , ! ,0 is uniquely determined by y :V0 ! G0
and a0, a1: V1 ! G1 such that:
(d*1y) x0= x (d*0y) @(a1a-10)
(3.2.2) * 0 *
d*0(a1a-10) d*2(a1a-10)d0xg0= g(d0d1) yd*1(a1a-10)
Note that the change a0 ! a0a, a1 ! a1a gives another homotopy between ,
and ,0.
The simplicial homotopy itself (again as in [May92 , #5]) in this case is gi*
*ven
by maps ff00:V0 ! G0, ff1i:V1 ! G0 x (G0 x G1) for i = 0, 1, and ff2i:V2 !
14
G0 x (G0 x G1) x (G0 x G1 x G1), i = 0, 1, 2, given by
ff00= y
ff10= (d*1y, (x0, a0))
ff11= (x, (d*0y, a1))
*x 0 *
ff20= ((d1d2)*y, (d*2x0, d*2a1), (d*0x0, g0, g0-1(d*2a-10)d0 g d1a0))
*x00 *
ff21= (d*2x, ((d0d2)*y, ), (d*0x0, d*0a0, d*0a-10(d*2a-10)d0 g d1a0))
ff22= (d*2x, (d*0x, g), ((d0d1)*y, d*0a1, d*0a-11d*1a1)
These results are essentially the same (barring a dioeerent set of conventions)*
* as
those of [Bre90, #6.4~6.5] for the crossed module ': G ! Aut(G).
3.3 Bitorsor cocycles
Let G be a gr-stack. Let Uo be a hypercover, for example the #ech complex
~CU of a generalized cover U ! *.
3.3.1 Deonition. A 1-cocycle with values in G consists of a pair (g, fl), where
g is an object of G over U1, and fl a morphism of G over U2, satisfying the
cocycle conditions
(3.3.1a) fl :d*1g -~! d*2g . d*0g
over U2, and the coherence condition
* * * * * *
(3.3.1b) (d2d3) g . d0fl O d2fl = a O d3fl . (d0d1) g O d1fl,
over U3, where a is the associator isomorphism for the group law in G . Two
cocycles (g, fl) and (g0, fl0) (assumed for simplicity to be deoned over the sa*
*me
Uo) are equivalent if there is a pair (h, j), where h 2 Ob GU0 and j 2 Mor GU1,
such that:
(3.3.2a) j :g . (d*0h) -~! (d*1h) . g0
and the diagram
d*1j
d*1g . (d0d1)*h_________/(d1d2)*h/. d*1g0
|fl| fl0||
fflffl| fflffl|
(d*2g . d*0g) . (d0d1)*h(d1d2)*h .O(d*2g0.Od*0g0)
(3.3.2b) |a| a||
fflffl| |
d*2g . (d*0gO.O(d0d1)*h)((d1d2)*h .Od*2g0)O. d*0g0
|d*0j| d*2j||
| |
d*2g . ((d0d2)*h . ooa___(d*2gd.*(d0d2)*h)0.gd*0g00)
15
commutes.
In view of the discussion on the relationship between G and the crossed
module reviewed in sect. 2.1, whereby the monoidal structure of G is described
in terms of contracted products of G1-bitorsors, a 1-cocycle such as (g, fl) in
Deonition 3.3.1 will be referred to, albeit imprecisely, as bitorsor cocycle.
It is easy_to pass from a 1-cocycle with values in G to a 1-cocycle with
values in W G_o. Indeed, recall from [Bre90] or from the remarks in sect. 2.1
that G ' TORS (G1, G0), the gr-stack of G1-torsors equipped with a chosen
trivialization of their extensions to G0. Thus g 2 Ob GU1 can be thought of as
such an object. In other words, we may write g as the pair g = (E, s), where E *
*is
the underlying G1-torsor and s: E ! G0 is the equivariant morphism providing
the trivialization as a G0-torsor. So we have:
3.3.2 Lemma. There exists a reonement_Vo_of Uo such that the bitorsor cocycle
(g, fl) determines a 1-cocycle Vo ! W G_o.
Proof. Let V ! U1 be a generalized cover such that the restriction of the
underlying G1-torsor E of g = (E, s) becomes trivial. Then by [SGA72 , V,
Th#or#me 7.3.2] there exists a hypercover Vo and a map Vo ! Uo which for
degree n = 1 factorizes through the chosen cover:
V1 ! V ! U1.
Over V1 we have E|V1 ' G1|V1, and s is determined by its value s(1) 2 G0. Thus
g may simply be identioed with this element of G0(V1). In turn, the morphism
fl is identioed with an element of G1 over V2, since the underlying map of G1-
torsors is a morphism of trivial torsors. That is, the required element is simp*
*ly
fl(1) 2 G1(V2). Notice that from the identiocation of g with s(1) 2 G0 it follo*
*ws
that d*2g . d*0g is identioed with the product d*2s(1)d*0s(1). Since fl is a mo*
*rphism
of (G1, G0)-torsors, we must have that
d*1s(1) = d*2s(fl(1))d*0s(fl(1)) = d*2s(1)d*0s(1) @fl(1).
Furthermore, it is not diOEcult to realize that the coherence condition for fl *
*on
V3 becomes *
d*0fl(1)d*2fl(1) = d*3fl(1)(d0d1) s(1)d*1fl(1).
These are precisely the cocycle relations (3.2.1)(modulo exchanging x $ g_and
g $ fl in the notation). |__|
The procedure in the proof of Lemma 3.3.2 will repeatedly be used in the
sequel.
3.3.3 Remark. There is a converse procedure,_namely one that allows to ob-
tain, starting from a simplicial map Uo ! W G_o, where again G_ois the simplici*
*al
group determined by a crossed module, to a 1-cocycle with values in the associ-
ated gr-stack G relative to the #ech nerve ~C(U0), where U0 is the degree n = 0
objects in Uo. The (long) proof can be extracted from [Bre90, #6.5]. No explicit
use will be made of such procedure in the rest of this paper.
16
3.4 Torsors for gr-stacks
The deonition of torsor under a gr-stack has been given in full generality
in [Bre90, 6.1], so here we will conone ourselves to only recalling the main
points. Let G be a gr-stack over S. In modern parlance, a G -torsor is the
categoriocation of the standard notion of torsor, as follows.
A right-action of G on a stack in groupoids X is given by a morphism of
stacks
m: X x G -! X
plus a natural transformation
(m,idG)
X x G x G _______//_X x G
(3.4.1) (idX, G)|| ~_____m||_______________________________*
*_____________________________________________________________________________*
*_________________________
fflffl| "_______fflffl|___________________________*
*____________________
X x G ____m______//_X
which amounts, for objects x, g0, g1, to a functorial isomorphism
~x,g0,g1:(x . g0) . g1 -~!x . (g0 . g1) ,
where x . g stands for m(x, g). We require that:
1.the pair (m, ~) satisfy the standard pentagon diagram;
2.the composite
X __~__//X x 1____//X x G_m__//X
where 1 ! G sends the unique object to the identity object of G , be
isomorphic to the identity functor of idX. Moreover, this morphism must
be compatible with m and ~, in the sense that the two diagrams [Bre90,
(6.1.4)], resulting from combining it with (3.4.1), must be commutative.
Most importantly, we require that the morphism
"m= (pr1, m): X x G -! X x X
be an equivalence. Having so far deoned what ought to be called a pseudo-torsor,
we need to complete the deonition by adding the condition that there exist a
(generalized) cover U ! * such that the ober category XU be non-empty.
A morphism of G -torsors X ! X 0consists of a stack morphism F :X !
X 0together with a natural transformation
(F,IdG)
X x G _______//_X x G
m || '____|m0|__________________________________*
*_____________________________________________________________________________*
*____________________________
fflffl| "_____fflffl|_______________________________*
*___________
X _____F____//_X 0
17
compatible with the transformations ~ and ~0. (That is, with the diagrams (3.4.*
*1).)
A 2-morphism of G -torsors is a 2-morphism ff: F ) F 0such that the dia-
grams
(F,IdG)
____________________________________*
*_____________________________________________________________________________*
*____________________________________________________OOOO
(F,IdG) __________________________________##f*
*f'ff
X x G _______//_X x G X x G (F0,IdG)//_X x G
m || '_____m0||______________________________________________*
*_____________________________________________________________________________*
*_______________m||m0||'0_____________________________________________________*
*_____________________________________________________________________________*
*_____________
fflffl| "______fflffl|_________________________________________ff*
*lffl|fflffl|"_______________________________________________
X ___F_______//___________________;;OOOOXX0//_X 0 *
* 0
______________________________________________________________*
*____________________________________________ff'ffF
___________________________________________________________*
*__________________
F0
deone a commutative diagram of 2-morphisms.
3.4.1 Remark. We have deoned the notion of right torsors. That of left torsor
is deoned in the same way. It is actually the one adopted in [Bre90].
With the notions of morphism and 2-morphism outlined above, G -torsors
comprise a 2-category. In fact, all together they form a neutral 2-gerbe over S
denoted TORS (G ). The ober above U 2 Ob(S) is the 2-category of G |U -torsors
(cf. [Bre92, Bre94a]).
3.4.1 Contracted product
We will need to consider the notion of contracted product of torsors over a gr-
stack in some detail. It is introduced in [Bre90, #6.7] (credited to J. B#nabou*
*).
(We use a slightly dioeerent convention for some of the diagrams.)
If X (resp. Y ) is a right (resp. left) G -torsor, or more generally stacks *
*with
G -action, their contracted product X ^G Y is deoned as follows. The objects
are pairs (x, y) 2 Ob X x Y . A morphism (x, y) ! (x0, y0) is an equivalence
classes of triples (a, g, b), where g 2 Ob G , and a: x ! x0. g and b: g . y ! *
*y0
are morphisms of X and Y , respectively. Two triples (a, g, b) and (a0, g0, b0)*
* are
equivalent if there is a morphism fl :g ! g0 in G such that the diagrams
a _5x0.5g______________________________________gb._y______*
*___________________________________
__________________________________________________________*
*______________________________________________________________________|
________|___________________________________________________*
*__""____________________________
x_ |idx0.fl fl.idy| y0
________|___________________________________________________*
*_______||==________________________________________
___((_fflffl|_____________________________________________*
*_____________________________________________________________________________*
*________________
a0 x0. g0 0fflffl|b0_____________________________*
*_____
g . y
commute. The composition of two morphisms (x1, y1) ! (x2, y2) and (x2, y2) !
(x3, y3) represented by triples (a, g, b) and (a0, g0, b0), respectively, is re*
*presented
by the triple given by the expected compositions
0.g0 ~
x1 -a!x2 . g a-! (x3 . g0) . g -! x3 . (g0. g)
0.b b0
(g0. g) . y1 -~!g0. (g . y1) g-! g0. y2 -! y3
18
and, of course, g0. g.
It should be observed that the foregoing procedure produces a obered cate-
gory over S with group law. We denote by X ^G Y the associated stack. One
may also characterize X ^G Y as the i2-Limitj of the diagram
X x Y x G _____////_X x Y
where one arrow is the projection and the other is the (right) action (x, y, g)*
* !
(x . g, g* . y), where x, y, g are objects and g* is a choice for the inverse o*
*f g.
Properties analogous to the familiar ones for ordinary torsors hold. For
example, one has the isomorphism
(x . g, y) -~! (x, g . y) ,
represented by the triple (idx.g, g, idg.y).
3.4.2 Cohomology classes and classiocation of torsors
3.4.2 Proposition ([Bre90, Proposition 6.2]). Let G1 ! G0 be a crossed
module of T. The elements of the pointed set H 1(*, G1 ! G0) are in bijec-
tive correspondence~with equivalence classes of right G -torsors over S, where
G = G1 ! G0 .
General idea of the proof.The central argument goes through the standard com-
putation with 1-cocycles subordinated to hypercovers Uo. Suppose X is a right
G -torsor over S, as described above. The choice of an object x of X over U0
leads to establishing the existence of an object g of G over U1 such that
d*0x -~! d*1x . g.
After pulling back to U2, the local equivalence of X and G allows to conclude
that there must exist a morphism (3.3.1a)over U2, with fl satisfying (3.3.1b)
over U3. The choice of another object x0of X , still over U0 say, leads to anot*
*her
1-cocycle (g0, fl0) equivalent to (g, fl), in the sense of Deonition 3.3.1, tha*
*t is there
is a pair (h, j) where h is an object of G over U0 and a j morphism over U1
satisfying equations (3.3.2).
From a 1-cocycle (g, fl) one can extract a 1-cocycle with values in the cros*
*sed
module G1 ! G0 as explained at the end of sect. 3.3.
Conversely, as mentioned in Remark 3.3.3, the procedure from the proof
of [Bre90, Proposition 6.2], in particular #6.5, allows to reconstruct a bitors*
*or_
cocycle, and ultimately a G -torsor, from a 1-cocycle with values in G1 ! G0. *
*|__|
4 Pushing cohomology classes along butterAEies
Changing the coeOEcients results in a morphism in non-abelian cohomology.
From the point of view of the general deonition recalled in sect. 3.1, this is
done by means of a morphism of simplicial groups H_o! G_o, which in our case
19
is the one induced by a morphism of crossed modules, and ultimately by a mor-
phism F :H ! G of gr-stacks. We are also speciocally interested in the case
i = 1, and we want to provide a short account of how the morphism
F*: H1(*, H ) -! H1(*, G )
can be prootably described in terms of butterAEies. This is a necessary stepping
stone in the more geometric description of the orst non-abelian cohomology
group with values in a gr-stack to be presented further down in the paper. After
some general observations, we begin with an elementary approach to the above
morphism in terms of explicit 1-cocycles with values in crossed modules. We
then show how the more conceptual formulation in terms of bitorsor cocycles
can be reduced to these explicit calculations.
4.1 General remarks
Let (F, ~): H ! G be a morphism of gr-stacks over S, where we have explicitly
marked the natural isomorphism ~ providing the additivity:
~y1,y2:F (y1y2) -~! F (y1)F (y2),
for any two objects y1, y2 of H . The following is an easy claim whose proof is
left to the reader.
4.1.1 Lemma. Let (F, ~) be as above, and let (y, h) be a 1-cocycle with values *
*in
H relative to a hypercover Uo ! * as in Deonition 3.3.1. Then (F (y), ~OF (h))
is a 1-cocycle with values in G (relative to the same hypercover). If (y, h) and
(y0, h0) are two equivalent 1-cocycles with values in H , then so are their ima*
*ges
(F (y), ~ O F (h)) and (F (y0), ~ O F (h0)).
Our goal is to explicitly calculate (F (y), ~ O F (h)) by means of a butterA*
*Ey
representing F .
4.2 Lift of a 1-cocycle along a butterAEy
Since a butterAEy [Ho, E, Go] corresponds to a morphism F :H ! G , it is
expected_that it will be possible_to iliftj a 1-cocycle j = (y, h) with values *
*in
W H_oto one with values in W G_o. Note that, after having observed that_the
butterAEy_E or equivalently the morphism F lead to a simplicial map W H_o !
W G_o, the lift is only a matter of composing j with said map. We prefer to
present a direct approach, which will be useful here and elsewhere_in this text.
Let Vo be a hypercover as above, and let j = (y, h): Vo ! W H_obe a
1-cocycle, with y :V1 ! H0 and h: V2 ! H1. Since ss :E ! H0 is a sheaf
epimorphism, there will be a local lift of y to E, namely a (generalized) cover
p1: U ! V1 and e: U ! E such that
U __e__//_E
p1|| ss||
fflffl| fflffl|
V1 __y__//H0
20
commutes. Using [SGA72 , V, Th#or#me 7.3.2], there is a hypercover Vo0dom-
inating Vo, with a factorization V10! U ! V1. All objects will be considered
relative to Vo0by pull-back along the latter map. In particular,_j = (y, h) can
now be considered as a 1-cocycle relative to Vo0via Vo0! Vo ! W H_o.
The explicit form of the cocycle condition on (y, h), the relation @H = ss O*
* ~,
and the injectivity of -: G1 ! E show that there must exist g :V20! G1 such
that
(4.2.1) d*1e = d*2e d*0e ~(h) -(g).
Set x = __O_e: V10! G0. We show that the pair (x, g) determines a 1-cocycle
, :Vo0! W G_o.
Applying _ to the previous relation gives the orst cocycle condition (3.2.1a*
*).
After a pull-back to V30, and using (4.2.1)to reduce (d2d3)*e(d0d3)*e (d0d1)*e
in both possible ways, by a routine calculation we obtain the equality
*b* * (d d )*x*
(4.2.2) ~(d*2h d*0h) -(d*2g d*0g) = ~((d*3h)(d0d1) d1h) -((d3g) 0 1 d1g),
so that the second cocycle condition (3.2.1b)for (x, g) also holds. (This uses
the fact that - is injective and that its image commutes with that of ~.)
4.2.1 Remark. From (4.2.1)and (4.2.2), it follows that "j= (e, (h, g)) deones
a 1-cocycle with values in the crossed module (~, -): H1 x G1 ! E.
4.2.2 Remark. The technique adopted in this section can also_be used to
describe the explicit lift of a 0-cocycle with values in W H_oof the type discu*
*ssed
in [Part I]. It is an exercise to show that the geometric view in terms of tors*
*ors
given there reduces to this one when trivializations are chosen. This view is
implicit in the proof of Theorem 2.2.1 given in [Part I, Theorem 4.3.1].
4.3 Computing the map F*
When H ~![H1 ! H0]~ , G !~ [G1 ! G0]~ , and (F, ~) is expressed through
the butterAEy [Ho, E, Go], the image of a 1-cocycle (y, h) with values in H
can be explicitly computed. Most of the necessary calculations follow in a
straightforward way from the explicit treatment of the equivalence between the
morphism F and the butterAEy provided in [Part I, Theorem 4.3.1] (recalled here
as Theorem 2.2.1).
Recall that we have the equivalence H ' TORS (H1, H0), and therefore,
if the object y corresponds to the (H1, H0)-torsor (Q, t), then F (y) can be
computed as
F (Q, t) = Hom__H1(Q, E)t,
as shown in Part I. The right-hand side is the G1-torsor of local H1-equivariant
lifts of t: Q ! H0 to E. In fact it is a (G1, G0)-torsor: the section
s: Hom__H1(Q, E)t- ! G0
21
is simply the map sending a local lift e of t to _ O e. The morphism h is the
isomorphism of torsors
H1 * * *
h: d*1(Q, t) -~! (d*2Q ^ d0Q, d2td0t),
so that the composite ~ O F (h) arises, again as explained in Part I, from the
isomorphism of G1-torsors
H1 * ~ * G1 *
Hom__H1(d*2Q ^ d0Q, E)t- ! Hom__H1(d2Q, E)t ^ Hom__H1(d0Q, E)t.
Assume the hypercover Uo with respect to which (y, h) is deoned is such that
the underlying H1-torsor Q is trivial, and the whole cocycle can be expressed
via a 1-cocycle with values in the crossed module H1 ! H0. Let us keep the
notation (y, h) for the latter, so that now y 2 H0(U1) and h 2 H1(U2).
Recalling that y 2 H0(U1) corresponds to the object (H1, y) of H (U1), its
image under F is given by:
Hom__H1(H1, E)y~-!Ey
(4.3.1)
e7-! e(1)
where the G1-torsor on the right-hand side is the ioberj of E ! H0 above y. It
follows that the resulting cocycle with values in G is given by the datum of Ey
plus the morphism
G1
(4.3.2) fl :Ed*1y~-!Ed*2y^Ed*0y.
arising from the application of (F, ~) to the orst relation in the 1-cocycle co*
*ndi-
tion, i.e.
d*1y = d*2y d*0y @h,
which really is the morphism
h: (H1, d*1y) -! (H1, d*2y d*0y).
So (4.3.2)is the result of the composition
G1
(4.3.3) Ed*1y-! Ed*2y d*0y-! Ed*2y^Ed*0y.
A trivialization of the G1-torsor Ey will produce a 1-cocycle with values in the
crossed module G1 ! G0. More precisely, we have:
4.3.1 Proposition. The choice_of a trivialization e 2 Ey amounts to a lift of t*
*he
1-cocycle j = (y, h): Uo ! W H_oalong the butterAEy [Ho, E, Go], as described in
section 4.2.
22
Proof. One needs to show that the choice of a trivialization e 2 Ey leads to
formulas (4.2.1)and (4.2.2). Indeed, after pullback the choice of e 2 Ey yields
d*1e, d*2e, and d*0e.
The orst morphism of (4.3.3)sends d*1e to (d*1e) ~(h)-1. This is a conse-
quence of the following observation: suppose we have y = y0@h, for y, y0 2 H0
and h 2 H1. Consider the diagram
Hom__H1(H1, E)y____//_Hom_H1(H1, E)y0
| |
| |
fflffl| fflffl|
Ey ________________//_Ey0
where the top horizontal arrow sends a local lift e to eOh-1. Then, using (4.3.*
*1)
for the vertical arrows, we can calculate the bottom horizontal arrow and ond
that a section e is sent by to e ~(h)-1.
Returning to the problem at hand, since the product d*2e d*0e provides a
trivialization of Ed*2y^G1Ed*0y, there must exist a g 2 G1 such that
(d*1e) ~(h)-1 = d*2e d*0e -(g),
which clearly is the same as (4.2.1), as wanted.
Relation (4.2.2)follows from this last one by direct calculation. Alterna-
tively, one can show that it follows from the cocycle condition (3.3.1b)applied
to the morphism (4.3.2), by pulling back to U3 and moving from (d1d2)*e to
the product (d2d3)*e (d0d3)*e (d0d1)*e in the two possible ways. The second
approach subsumes the second. In any event, both are straightforward and_left
to the reader. |__|
5 Gerbes bound by a crossed module
5.1 Recollections on gerbes
For gerbes, our main references will be [Gir71, Bre94a]. Recall that a gerbe P
over S is by deonition a stack in groupoids over S which is ilocally non-emptyj
and ilocally connected.j Following [LMB00 ], this can be expressed as follows.
Let X be a ispace,j i.e. a sheaf of sets, over S. A gerbe over X is a stack in
groupoids P over S equipped with a morphism p: P ! X such that both p and
the diagonal : P ! P xX P are (stack) epimorphisms. The usual deonition
of gerbe over S without reference to another space is recovered by setting X = *
**.
Any stack X is equipped with a canonical morphism
X - ! ss0(X )
which makes X into a gerbe over ss0(X ) ([LMB00 , #3.19] and [Bre94a, #7.1]).
This construction and its analog for 2-stacks were applied at dioeerent points *
*in
Part I.
23
If U ! * is a generalized cover and G is a sheaf of groups over S=U, then
P is a G-gerbe if there exists an object x 2 Ob(PU ) and an isomorphism
G -! Aut_U(x) .
(The choice of the isomorphism is called a labeling of P in [Bre94a]). It is we*
*ll
known from loc. cit. that a G-gerbe gives rise to a non-abelian cohomology
class with values in the crossed module [': G ! Aut(G)]. Essentially identical
cohomology classes are shown in [Bre90] to arise from G -torsors, where G =
[G ! Aut(G)]~ is the associated gr-stack. In fact, it is also shown in loc. cit.
that there is an equivalence (of 2-gerbes) between G -torsors and G-gerbes. This
section is devoted to tie together these strands for a general crossed module
G1 ! G0 of T.
5.2 Gerbes bound by a crossed module
Let Go: G1 @!G0 be a crossed module of T. The concept of gerbe bound by Go
is a sort of rigidiocation, due to Debremaeker [Deb77 ], of the idea of G-gerbe
recalled above.
5.2.1 Deonition. A gerbe P bound by Go, or equivalently, a (G1, G0)-gerbe,
is a gerbe P over S equipped with the following data:
1. a functor ~: P ! TORS (G0);
2. for each object x of P an isomorphism _x: Aut_(x) ~! ~(x) ^G0G1 such
that the diagram
Aut_(x)__________//_Aut_(~(x))
(5.2.1) _x|| '||
fflffl| id^@ fflffl|
~(x) ^G0G1 _______//_~(x) ^G0G0
commutes and it is functorial with respect to morphisms f :x ! y in
P. The right vertical morphism is the standard one identifying the
automorphism group of a G-torsor P with the twisted adjoint group
Ad P = P ^G G.
Let us explicitly remark that the functoriality of diagram (5.2.1)means we
must have, for each morphism f :x ! y in P, over (say) U, a commutative
diagram:
Aut_(x)_____f*______//Aut_(y)
(5.2.2) _x|| _y||
fflffl| fflffl|
~(x) ^G0G1__~(f)^id_//~(y) ^G0G1
f* is deoned, as usual, by sending a section fl of Aut_(x) to f O fl O f-1 .
24
5.2.2 Example. TORS (G1) is evidently (G1, G0)-gerbe with ~ = @* and _
given by
G1 ~ G0
_P :P ^ G1 -! @*(P ) ^ G1
for a G1-torsor P . TORS (G1) will be called the trivial (G1, G0)-gerbe when
equipped with the structure just described. We shall see shortly, in sect. 5.3,
that all (G1, G0)-gerbes are locally of this type.
We will denote a gerbe bound by Go synthetically as (P, ~, _). We have
morphisms and 2-morphisms of gerbes bound by Go, as follows:
5.2.3 Deonition. A morphism (F, '): (P, ~, _) ! (P0, ~0, _0) of gerbes bound
by Go is given by a morphism F :P ! P0 of gerbes plus a 2-morphism
P _________________F//_P0
_____________________________________
________________________________________~0||'(0__*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____________________
~__''___________________________________________*
*_____fflffl|
TORS (G0)
such that for every object x 2 Ob(P) the following diagram commutes:
Aut_(x)_____F*__//_Aut_(F (x))
(5.2.3) _x|| |_0F(x)|
fflffl| fflffl|
~(x) ^G0G1 __'x_//~0(F (x)) ^G0G1
A 2-morphism ` :(E, ") ) (F, ') is a 2-morphism of gerbes ` :E ) F such that
~0* ` O " = '.
In ref. [Deb77 ] the deonition of morphism is given in greater generality th*
*an
in Deonition 5.2.3 above, by allowing a strict morphism of crossed modules.
Recall that a strict morphism fo: Ho ! Go is a commutative diagram of group
objects
f1
H1 _____//G1
@|| |@|
fflffl| |fflffl
H0 __f0_//G0
where f1 is an f0-equivariant map.
5.2.4 Deonition. Let (P, _, ~) be a (G1, G0)-gerbe and (Q, ~, ) an (H1, H0)-
gerbe. An fo-morphism (F, '): Q ! P is the datum of a morphism F :Q ! P
of gerbes plus a 2-morphism
Q ______F______//P
|| '____|~|.6________________________________*
*__________________________________________________________
fflffl| _______fflffl|______________________________*
*___________________________________________________________________
TORS (H0) _(f0)*//_TORS(G0)____
25
such that for each object y of Q there is a functorial diagram
Aut(F)
Aut_(y)________//_Aut_(F (y))
~y || |_F(y)|
fflffl| fflffl|
(y) ^H0H1___!_//~(F (y)) ^G0G1
where ! is the composite
H0 H0 ~ H0 G0 G0
(y) ^ H1 -! (y) ^ G1 -! ( (y) ^ G0) ^ G1 -! ~(F (y)) ^ G1.
There is an obvious generalization of the notion of 2-morphism too. The
reader can formulate the appropriate diagram.
5.2.5 Remark. An abelian crossed module is simply a homomorphism of
abelian groups of S. Gerbes bound by crossed modules in this sense have ap-
peared in refs. [Mil03] and [Ald08]. As it is shown in the latter, the notion
encompasses several well-known examples such that of connective structure due
to Brylinski and McLaughlin ([Bry93]) and hermitian structure due to one of
the authors ([Ald05]).
5.3 Local description
We want to explicitly show that a (G1, G0)-gerbe (P, _, ~) is always locally
equivalent to TORS (G1) with the structure described in Example 5.2.2.
First, it will be useful to carry out a few local calculations to translate *
*the
global structure aoeorded by the (G1, G0)-structure on the gerbe P into the ope*
*r-
ations of the crossed module @ :G1 ! G0. To this end, consider diagram (5.2.2),
and assume two trivializations u, v of the G0-torsors ~(x) and ~(y) are given.
It follows that f determines an element af 2 G0(U) by
u 7-! ~(f)(u) = v af.
Then fl in Aut_(x) determines, via the trivialization u, an element g 2 G1(U):
jx(fl) = u ^ g.
From diagram (5.2.2)we have that the action of f* amounts to:
-1
u ^ g 7-! v a ^ g = v ^ gaf .
Thus, if the trivializations are oxed, the action of f* can be identioed with t*
*he
automorphism of G1 given by:
-1
g 7-! gaf .
26
If in particular y = x, so that f 2 Aut_(x) too, then _x(f) = u^hf, and by (5.2*
*.1)
we must have af = @hf. Since _x(f O fl O f-1 ) = u ^ (hfghf-1 ), it immediately
follows that
g@hf = hf-1g hf .
Returning to the question of the local structure of P, let x be the choice
of an object of PU , for a suitable U ! *. We can assume that there exists a
trivialization s of the G0-torsor ~(x), reoning U if necessary.
5.3.1 Lemma. The pair (x, s) determines an equivalence of (G1, G0)-gerbes
(Lx,s, ~x,s): P|U -~! TORS (G1).
Proof. The underlying functor Lx,s:P|U ! TORS (G1|U) is the standard one
deoned by the assignment
y _ Hom__P(x, y)
(see [Bre94a, Bre94b]). It is the choice of s that allows to conclude that Hom_*
*_P(x, y)
is a G1|U -torsor.
Let f :x ! y be a morphism of PU (over some V ! U) and let a be an
element of G0 over V . The claim is that the required isomorphism of G0|U -
torsors G
~-1x,s:Hom_P(x, y) ^1G0 -! ~(y)
is deoned by the assignment
(5.3.1) (f, a) 7-! ~(f)(s) a.
Indeed, let f be replaced by f Ofl, where fl is an automorphism of x. Then there
is an element g of G1 such that _x(fl) = s ^ g, and by deonition we have
~(fl)(s) = s @(g),
so that
~(f O fl)(s) a = ~(f)(s) @(g) a.
Thus the pairs (f O fl, a) and (f, @(g) a) map to the same point of ~(y),_hence
the claim. |__|
5.3.2 Remark. For a (G1, G0)-gerbe P choosing an object x and an appropri-
ate trivialization of the resulting G0-torsor ~(x) shows that P is in particular
a G1-gerbe.
5.4 The class of a gerbe bound by a crossed module
For gerbes bound by G1 ! G0 there is an analogous statement to Proposi-
tion 3.4.2.
5.4.1 Proposition. The elements of the pointed set H 1(*, G1 ! G0) are in
bijective correspondence with equivalence classes of (G1, G0)-gerbes over S.
27
Proof. Let Vo ! * be a hypercover such that we can choose an object x 2
Ob PV0 and a morphism f :d*0x ! d*1x in Mor PV1. The choice of the pair
(x, f) is a labeling of P relative to Vo. Let us temporarily put G = Aut_(x).
The computations in [Bre94b, #5.2], show that there exists an element fl of
Aut_ (d0d1)*x) ' (d0d1)*G over V2, deoned by the diagram
d*2f
(d0d2)*x_______//_(d1d2)*xOOOO
| |
(5.4.1) d*0f|| |d*1f|
| |
| fl |
(d0d1)*xoo_____(d0d1)*x_
such that the non-abelian cocycle condition holds:
(d*1f)*= (d*2f)* O (d*0f)* O ('fl)
(5.4.2) (d0d1)*f
d*0fl O d*2fl= d*3fl O d*1fl,
where 'fldenotes the image of fl 2 G in Aut(G) and flf is a short-hand for
(f-1 )*(fl). The orst equation holds over V2, whereas the second over V3.
We can assume the G0|V0-torsor P def=~(x) is trivial over some W ! V0, via
some choice of s: W ! P . Using [SGA72 , V, Th#or#me 7.3.2], we can work
with a new hypercover Vo0equipped with a map Vo0! Vo such that for n = 0
we have a factorization V00! W ! V0. Let us from now on relabel Vo0to Vo.
Given the foregoing assumptions, it now follows that G = Aut_(x) ' G1|V0
and f determines an element a of G0 over V1, whereas fl corresponds to an
element g of G1 over V2. The local calculations of section 5.3 show that (5.4.2)
becomes
(d*1a)= d*2a d*0a @g
(5.4.3) (d0d1)*a
d*0g d*2g= d*3g d*1g,
where this time ga denotes the action of G0 on G1 in the crossed module. This
is a 1-cocycle in the same sense as put forward in sect. 3.2, equations. (3.2.1*
*).
The choice of a dioeerent labeling (y, f0), which for simplicity we assume t*
*o be
relative to the same hypercover Vo, will determine another pair (a0, g0) satisf*
*ying
the same non-abelian cocycle condition (5.4.3). (To obtain it, we must assume
as well that the G0-torsor ~(y) is trivialized by an appropriate choice, possib*
*ly
changing the cover again in the process.) Following [Bre94b, #5.3] we may also
assume, up to further reoning Vo, that we have chosen a morphism
O: y -! x
over V0. Such choices determine an element jO of Aut_(d*0y) via
d*1O-1 O f O d*0O = f0O jO.
28
Again, the calculations of section 5.3 show that the pair (O, jO) determines,
via the chosen trivializations, a pair (u, h), with u 2 G0(V0) and h 2 G1(V1).
Combining the latter relation with the primed and unprimed versions of (5.4.2),
and using (5.4.3), we arrive at the relation
a d*0u= d*1u a0@h
(5.4.4) * 0 *
g0(d*2h)d0ad*0h= d*1h g(d0d1) u.
By comparison with (3.2.2), the pair (O, jO) (or equivalently (u, h)) determines
a homotopy between the two 1-cocycles corresponding to the two dioeerent la-
belings of P.
The quickest way to reverse the process and to reconstruct a (G1, G0)-gerbe
starting from the datum of (a, g) satisfying (5.4.3), relative to Vo, is to fol*
*low
the procedure outlined at the end of [Bre94b, #5.2]. BrieAEy, from a we can deo*
*ne
a trivial (G1, G0)-torsor E over V1. Now, as observed in [Bre90] and [Part I], a
(G1, G0)-torsor is in particular a G1-bitorsor, hence refs. [Bre94b, Bre90] may
be followed to descend E (if necessary) to V0 x V0 and then to use (5.4.3)to
conclude that E deones a ibitorsor cocyclej relative to the #ech cover cosk0Vo,
analogously to the cocycle that appeared in the proof of Proposition 3.4.2. From
there, we can construct a (G1, G0)-gerbe by gluing local copies of TORS (G1|V0*
*),
considered as (G1, G0)-gerbes, according to Example 5.2.2. (For the gluing we __
must invoke the eoeectiveness of 2-descent data for (G1, G0)-gerbes.) |_*
*_|
5.4.2 Remark. Embedded in the proof of the previous proposition is the fact
that, given two objects x, y 2 Ob PU above U 2 Ob S, with chosen trivializa-
tions of the G0-torsors ~(x) and ~(y), the (Aut_(x), Aut_(y))-bitorsor
Ex,ydef=Hom_P(y, x)
is in fact a (G1, G0)-torsor. This follows at once from the calculations of se*
*c-
tion 5.3. From this point of view an arrow f :y ! x deoned over a (generalized)
cover V ! U is to be considered as a local section of such torsor. In particula*
*r,
the assignment deoned in section 5.3 of af 2 G0(V ) to f ought to be seen as
the G1-equivariant map
s: E -! G0
which is part of the deonition of (G1, G0)-torsor. Indeed, if f is replaced by
f O fl, where fl 2 Aut_(y)(V ) and fl is then identioed with an element g 2 G1(*
*V ),
then we have
afOfl= af @g.
5.4.1 Bitorsor cocycle associated to a labeling
According to ref. [Bre94b] and to the last remark, the proof of Proposition 5.4*
*.1
can be reformulated in terms of the bitorsor cocycles introduced in section 3.3.
Indeed, the local equivalence of (G1, G0)-gerbes provided by a labeling, analyz*
*ed
29
in section 5.3, in particular in Lemma 5.3.1, determines a bitorsor cocycle as
follows. Let
'U :TORS (G1|U ) -! P|U
be such an equivalence, where P is a (G1, G0)-gerbe. Now, let U be the degree
zero stage of a (generalized) cover Uo, and consider the two possible pull-backs
d*0' and d*1' to U1. We obtain in this way a 2-commutative diagram
TORS (G1|U1)_____j_____//______TORS(G1|U1)
_______________________________________________________*
*____
______________________________________________________*
*_______________________
____________________________________________________*
*__________________________________
d*1'__))______________________________________________*
*______d*0'uu______________________________________________________
P|U1
of (G1, G0)-gerbes. By Morita's theory (see [Bre94a, BM05 ]) j is induced by a
G1-bitorsor E. It is relatively easy to see that E is in fact an object of GU1,*
* that
is a (G1, G0)-torsor over U1. The formal argument will constitute the proof of
Lemma 5.5.2 below. The pull back to U2 determines a 2-morphism
fl :d*1j ) d*2j O d*0j :TORS (G1|U2) -! TORS (G1|U2),
which results in the morphism of bitorsors
G1 *
fl :d*1E -! d*2E ^ d0E,
with fl to satisfy the appropriate coherence conditions over U3. From Lemma 3.3*
*.2,
or rather its proof, we can once again extract from (g, fl) a cocycle with valu*
*es
in the crossed module G1 ! G0.
5.5 Gerbes vs. torsors
Let G be the gr-stack TORS (G1, G0). Propositions 3.4.2 and 5.4.1 hold that
G -torsors and (G1, G0)-gerbes give rise to the same equivalence classes of ob-
jects, in other words they are both classioed by the non-abelian cohomology set
H 1(*, G1 ! G0). The following is the analog of [Bre90, Proposition 7.3] and the
non-abelian counterpart of [Ald08, Theorem 5.4.4]. For the statement, recall
that Eq denotes the stack of equivalences, as deoned in[Gir71, IV Proposition
5.2.5].
5.5.1 Proposition. There is a pair of quasi-inverse Cartesian 2-functors
G o
(5.5.1) : TORS (G ) -! GERBES (G1, G0), X 7-! TORS (G1) ^X
and
(5.5.2) : GERBES (G1, G0) -! TORS (G ), P 7-! Eq(TORS (G1), P)
where for a right-G -torsor X the symbol X odenotes the opposite (left) torsor,
which deone a 2-equivalence between the 2-stacks TORS (G ) and GERBES (G1, G*
*0)
over S.
30
In fact the pair deones a 2-equivalence between neutral 2-gerbes over S. For
the proof the following lemma, which is also of independent interest, is needed:
5.5.2 Lemma. There is an equivalence of gr-stacks
G -~! Eq(TORS (G1), TORS (G1))
where TORS (G1) is considered as a (G1, G0)-gerbe in the manner described by
Example 5.2.2.
Proof. The functor in the statement is the one sending the (G1, G0)-torsor (E, *
*s)
to the equivalence
G1
P 7-! P ^ E
where, according to [Bre90], recalled in [Part I, #3.4.8], E is a G1-bitorsor u*
*sing
the left G1-action deoned as g . e = egs(e). The functor is clearly fully faith*
*ful.
Let (F, '): TORS (G1) ! TORS (G1) be an equivalence of (G1, G0)-gerbes
(see Deonition 5.2.3). Recall that by standard arguments of Morita theory, the
underlying functor F determines and is determined, up to equivalence, by a
G1-bitorsor E so that for any right G1-torsor P there is an isomorphism
G1
F (P ) ' P ^ E.
E is simply the image under F of the trivial torsor G1. By Deonition 5.2.3, this
must be compatible with @*: TORS (G1) ! TORS (G0), so there must exist
an isomorphism
G1 ~ G1 G1
'P :P ^ G0 -! P ^ E ^ G0
for all torsors P . If in particular P = G1, it reduces to
G1
'G1 :G0 -~!E ^ G0,
that is E must be equipped, as a right G1-torsor, with a trivialization of its
extension to a G0-torsor. Thus E is a (G1, G0)-torsor, and it is relatively easy
to verify that the resulting left G1-torsor structure recalled above is_the_same
as the original one. |__|
Main lines of the proof of Proposition 5.5.1.The proof closely mirrors the one
in [Bre90, Proposition 7.3], except for the details pertaining to the (G1, G0)-
gerbe structure.
By Lemma 5.5.2, G acts on the right on (P). As observed in loc. cit.,
for any two equivalences F, F 0we have F 0' F O (F -1O F 0), for a choice F -1
of the quasi-inverse to F , and F -1O F is an auto-equivalence of TORS (G1).
Furthermore, (P) is locally non void, since from 5.3 the choice of an object
x of P and of a trivialization s of ~(x) over some U 2 Ob (S) determines an
equivalence TORS (G1) ~!P of (G1, G0)-gerbes over U.
As for (X ), it is a gerbe since, as already noted in loc. cit., the very f*
*act
that X is itself locally equivalent to G shows that (X ) is locally equivalent
to TORS (G1).
31
It is to be shown that (X ) actually is a (G1, G0)-gerbe. To this end, let
~: (X ) ! TORS (G0) be deoned by
G1
(5.5.3) ~(P, X) def=@*(P ) = P ^ G0.
If the triple (ff, g, fi), where g = (E, s) denotes a (G1, G0)-torsor, represen*
*ts a
morphism
(P1, X1) -! (P2, X2)
in (X ) as described in 3.4.1, then @*([ff, g, fi]) is deoned to be the compos*
*ition
G1 ff^idG0 G1 G1 ~ G1 G1 idP2^s G1
(5.5.4)P1 ^ G0 -----! (P2 ^ E) ^ G0 -! P2 ^(E ^ G0) -----! P2 ^ G0.
It is immediately checked that it does not depend on the specioc choice of the
triple representing the morphism.
For two morphisms (P1, X1) ! (P2, X2) and (P2, X2) -! (P3, X3) com-
posed as in 3.4.1, a diagram chase, using Mac Lane's pentagon, reveals that the
composition of the corresponding images (5.5.4)equals (as expected) the image
of the composition under ~.
Having deoned ~, it must be proved that there is a functorial isomorphism
G0
(5.5.5) _P,X: Aut_(P, X) -~! ~(P, X) ^ G1,
as per Deonition 5.2.1. Note that from (5.5.3)it follows that:
G0 G1
~(P, X) ^ G1 ' P ^ G1 ' Aut_(P ),
so that (5.5.5)amounts to showing that:
Aut_(P, X) ' Aut_(P ).
This actually follows from the fact that the choice of the object X of X oes-
tablishes a local equivalence with G , and hence one of (X ) with TORS (G1).
Explicitly, and somewhat more precisely, an automorphism of (P, X) is given
by a triple (ff, g, fi) such that
G1
ff: P -! P ^ E fi :g . X -! X , g = (E, s).
Since X is a torsor, it follows there must be an arrow
fl :g -! IG ,
in G , that is the (G1, G0)-torsor (E, s) is isomorphic to the trivial (G1, G0)*
*-torsor
(G1, 1). It follows that the triple (ff, g, fi) is equivalent in the sense of 3*
*.4.1 to
(ff0, IG , lX ), where lX is the structural functorial isomorphism
lX :IG . X -~! X
32
which is part of the deonition of G -torsor. On the other hand, ff0 is the com-
position (idP .fl) O ff: P ! P ^G1 G1 ' P , which is the sought-after element of
Aut_(P ). It is clear the requirements of Deonition 5.2.1 and in 5.3 are met.
As a last point, since TORS (G1) ^G X ois actually deoned by a process of
stackiocation, it should also be checked that ~ as deoned glues along descent
data. If (P, X) is an object deoned over V with a morphism
': d*0(P, X) -! d*1(P, X)
over, say, V xU V such that the cocycle condition
d*1' = d*2' O d*0'
holds, the deonition (5.5.3)should give rise to a well-deoned G0-torsor over U
(via descent in TORS (G0)). Writing ' as being represented by a triple (ff, g,*
* fi),
the descent datum above gives rise to two diagrams
d*2g . (d*0g . (d0d1)*X)//_d*2g . (d0d2)*X__//(d1d2)*XOO
|o| ||
fflffl| |
(d*2g . d*0g) . (d0d1)*X__________________//d*1g . (d0d1)*X
and
(d0d1)*P _____//(d0d2)*P . d*0g_//((d1d2)*P . d*2g) . d*0g
| |
| |o
fflffl| fflffl|
(d1d2)*P oo_____________________(d1d2)*P . (d*2g . d*0g)
Applying ~ produces an object P ^G1 G0 over V , a morphism d*0P ^G1 G0 !
d*1P ^G1 G0 of type (5.5.4)over V xU V , and another long but totally straight-
forward diagram chase applying ~ to the second diagram above allows to obtain
a corresponding cocycle condition. Hence P ^G1 G0 can be descended to a_G0-_
torsor over U, as wanted. |__|
Passing to classes of equivalences, we have the identiocations
1
TORS (G ) ' GERBES (G1, G0) ' H (*, G1 ! G0),
where [.] denotes taking classes of equivalences of objects over *. The orst id*
*en-
tiocation is of course induced by (and its inverse by ). It follows at once
from Proposition 5.5.1 and from Propositions 3.4.2 and 5.4.1 that the above
identiocations constitute a commutative diagram, namely the isomorphism in-
duced by is compatible with taking cohomology classes, so that the induced
map on H1 is the identity. We record this as a lemma.
5.5.3 Lemma. The maps induced by and preserve equivalence classes.
33
For future use, it is nevertheless convenient to have a computational verio-
cation.
Proof of the lemma.If X is a G -torsor, then the choice of an object x in the
ober XU over U establishes an equivalence
X |U -~! G |U
which gives (see [Bre90] and the proof of Proposition 5.5.1)
G|U ~ G|U
(X |U ) = TORS (G1|U ) ^ X o|U-!TORS (G1|U ) ^ G o|U
-~! TORS (G
1|U ).
Explicitly, an inverse equivalence is given by:
G|U
'U :TORS (G1|U )-~! TORS (G1|U ) ^ X o|U
P 7-! (P, x).
According to section 5.4.1, this equivalence will determine a bitorsor cocycle
for the gerbe (X ), which we want to identify with the one determined by
the choice of the object x of X . Indeed, let the latter be given by the pair
(g, fl), with g = (E, s) is a (G1, G0)-torsor over U = U0, as in the proof of
Proposition 3.4.2. From the morphism
, :d*0x -~! d*1x . g
in XU1 consider the morphism (g* is a choice of the inverse for g):
* * ~ * * ~ *
d*0x . g* -~! d1x . g . g -! d1x . g . g -! d1x,
which by deonition corresponds to a morphism ,o in X o:
,0: g . d*0x -! d*1x.
By the deonition of contracted product given in sect. 3.4.1, we have that the
triple (id, g, ,o) determines a morphism
G1 G1 * * ~ * *
d*0' P ^ E = (P ^ E, d0x) (P . g, d0x) -! (P, d1x) = d1' P .
By comparison with the results of section 5.4.1, we see that resulting self-
equivalence of TORS (G1|U1) is indeed given by g = (E, s), as wanted.
In the opposite direction, let P be a (G1, G0)-gerbe. If x is an object of P*
*U ,
this choice will determine as in section 5.4.1 a bitorsor cocycle (g, fl), rela*
*tive to
some cover of U, where we write again g = (E, s). In view of Lemma 5.5.2, and
the deonition of , it is immediate that the bitorsor cocycle for the G -torsor*
* __
Eq (TORS (G1), P) (relative to the trivialization induced by x) is still (g, f*
*l). |__|
5.5.4 Remark. The preceding proof in fact shows that both and act as
identities on bitorsor cocycles, thereby implying the statement of the lemma.
34
6 Extension of gerbes along a butterAEy
Functoriality of cohomology under a change of coeOEcients is one of the most
important properties which are required to hold in the realm of non-abelian
cohomology. In the case of groups it is well known that the map H1(*, H) !
H 1(*, G) induced by a homomorphism ffi :H ! G is realized by the standard
extension of torsors ffi*: TORS (H) ! TORS (G), which sends an H-torsor P
to its extension ffi*P = P ^H G. (In fact there is a ffi-morphism P ! ffi*P , s*
*ee
[Gir71].)
In the case of a morphism F :H ! G of gr-stacks, the categoriocation of the
above extension of torsors yields the required map H1(*, H ) ! H1(*, G ), see r*
*ef.
[Bre90]. These matters are brieAEy recalled, mostly for convenience, in sect. 6*
*.1
below. Just note that the categoriocation entails considering the morphism of
2-gerbes F*: TORS (H ) ! TORS (G ) given by sending the H -torsor Y to
F*Y = Y ^H G . In view of the equivalence between torsors and gerbes stated
in Proposition 5.5.1, this picture could be reinterpreted in terms of gerbes bo*
*und
by crossed modules, albeit not in an immediately explicit form.
Our purpose is to remedy this by putting forward a better and more ex-
plicit picture which leverages on the equivalence (cf. Theorem 2.2.1) between
morphisms of gr-stacks and butterAEies between crossed modules, and on the
interpretation of the orst non-abelian cohomology group with values in a gr-
stack as equivalence classes of gerbes. The procedure to be expounded below
starts with a gerbe bound by the crossed module Ho and uses the butterAEy
representing F :H ! G to construct in a fairly explicit way a gerbe bound by
Go, compatibly with the categoriocation above. It builds upon and improves
an earlier notion of Debremaeker [Deb77 ].
6.1 Extension of torsors
A morphism F :H ! G of gr-stacks induces a morphism
F*: TORS (H ) -! TORS (G )
between the corresponding 2-gerbes of torsors. The deonition of F* is the cat-
egoriocation of the standard iextension of the structural groupj for torsors,
namely if Y is an H -torsor, then we deone
H
F*(Y ) = Y ^ G .
This was extensively used~without deonition, but referring instead to [Bre90]~
in Part I. Passing to cohomology, that is, to isomorphism classes of objects, it
is clear that there results a corresponding maps of pointed sets:
H1(*, H ) -! H1(*, G ).
Indeed, still according to [Bre90], this is the enabling framework to interpret*
* the
functoriality of non-abelian cohomology with values in a crossed-module. Insofar
35
as cohomology only depends on the quasi-isomorphism class of the coeOEcient,
and every gr-stack is equivalent to one associated to a crossed module, this
covers the general case.
Let H and G be associated to crossed modules H1 ! H0 and G1 ! G0,
respectively. In view of the equivalence stated in Proposition 5.5.1, there is *
*an
abstract description of F* in terms of gerbes. Following ref. [Bre90], let us u*
*se
the notation F** for the morphism GERBES (H1, H0) ! GERBES (G1, G0)
resulting from F* via the following 2-commutative diagram:
TORS (H ) ____F*_____//_TORS(G )
|| ||
fflffl| fflffl|
GERBES (H1, H0)_F**_//GERBES (G1, G0)
The deonition is F** = O F* O .
It is clear that modulo the obvious isomorphism above the statement of
Lemma 5.5.3, F* and F** induce the same map H1(*, H ) ! H1(*, G ).
Unfortunately, without additional input, F** cannot be easily characterized.
If Y is again an H -torsor, a simple manipulation gives that the gerbe (F*(Y ))
is equivalent to TORS (G1) ^H Y 0, where TORS (G1) carries an H -action via
H -F! G -~! Eq(TORS (G1), TORS (G1)).
Thus, if Q is an (H1, H0)-gerbe, the previous observation suggests that its ima*
*ge
under F** is
G o H o
F**(Q) = TORS (G1) ^ (Q) = TORS (G1) ^ Eq(TORS (H1), Q) .
To improve on this picture, we propose to provide an explicit characterization
of F** by employing the butterAEy construction of the morphism F :H ! G .
6.2 Debremaeker's extension along strict morphisms
Let fo: Ho ! Go be a strict morphism of crossed modules, as in Deonition 5.2.4.
Let (P, _, ~) be an (H1, H0)-gerbe. In [Deb77 ], Debremaeker proved that there
exists a (G1, G0)-gerbe (P0, _0, ~0) and an fo-morphism P ! P0.
The gerbe (P0, _0, ~0) is constructed in two steps. First, a obered category
P* is deoned with the same objects as P and morphisms given by the extension
of torsors
~(y)^^H0H1 H
(6.2.1) Hom__*(y, x) def=Hom_P(y, x) ~(y) ^0G1 ,
for any two objects x, y of P. Note that in the above formula, to deone
~(y) ^H0G1, G1 is considered as an H0 object via the homomorphism f0: H0 !
G0, and that the homomorphism id~(y)^f1: ~(y) ^H0H1 ! ~(y) ^H0G1 is used
36
for the extension. Then, the second step is to deone P0 as the stack associated
to P*. The fo-morphism from P ! P0 is induced by the corresponding
P ! P* simply given by the identity on objects and the map f 7! (f, 1) on
morphisms.
To see that P0 is a (G1, G0)-gerbe, one can argue that a choice of trivializ*
*a-
tions of ~(y) and ~(x) above makes Hom__P(y, x) into an (H1, H0)-torsor. Conse-
quently, Hom__*(y, x) ' Hom__P(y, x) ^H1G1 is a (G1, G0)-torsor. The conclusion
follows from the application of this argument to the class of P constructed in
Proposition 5.4.1. Still according to the proposition, the modioed cohomology
class according to (6.2.1)is therefore the class of a (G1, G0)-gerbe.
To elaborate further, according to [Deb77 ], there is a composition
Hom__*(y, x) x Hom__*(z, y) -! Hom__*(z, x)
deoned as follows. If fly is an element of Aut_(y) ' ~(y) ^H0G1, and similarly
for flz, then the composition law is deoned as:
((f, fly), (g, flz)) 7-! (f O g, ~(g)-1(fly)flz),
where ~(g)-1 is a short-hand for the homomorphism of group objects
H0 H0
~(y) ^ G1 -! ~(z) ^ G1
induced by ~(g)-1 :~(y) ! ~(z). Note that the functor ~0:P0 ! TORS (G0)
is simply induced by the composition of ~ with
(f0)*: TORS (H0) -! TORS (G0),
in other words to any object x we assign ~(x) ^H0G0. Moreover, from (6.2.1)
it immediately follows that if y = x then
H0 H0 G0
Aut_*(x) ' ~(x) ^ G1 ' ~(x) ^ G0 ^ G1,
which gives the required isomorphism _0x. All the necessary requirements can be
easily checked by the reader as an exercise.
It is also not hard to realize that Debremaeker's construction is actually
functorial with respect to morphisms (and 2-morphisms) of (H1, H0)-gerbes (see
[Deb77 ] for details). This provides us with a 2-functor
(6.2.2) F+0:GERBES (H1, H0) -! GERBES (G1, G0)
which we seek to generalize in section 6.3, to a morphism which is not necessar*
*ily
assumed to be strict.
6.2.1 Remark. The object Ex,y= Hom__P(y, x) is a (~(x) ^H0H1, ~(y) ^H0H1)-
bitorsor. It must be characterized (see again [Bre90]) by a ~(y) ^H0H1-equivari*
*ant
morphism
H0 H0
Ex,y-! Isom_(~(x) ^ H1, ~(y) ^ H1) ' Hom__H0(~(y), ~(x))
37
from Ex,yconsidered as a right torsor. This map is simply given by
(6.2.3) f 7-! ~(f)-1
where we use the same short-hand notation as above. Consequently, E*x,y=
Hom__*(y, x) given by (6.2.1)has the structure of (~(x) ^H0G1, ~(y) ^H0G1)-
bitorsor, since by (6.2.3)above we get an obvious map
H0 H0 H0 H0
Isom_(~(x) ^ H1, ~(y) ^ H1) -! Isom_(~(x) ^ G1, ~(y) ^ G1),
which is equivariant with respect to
H0 H0
id^f1: ~(x) ^ H1 -! ~(x) ^ G1.
According to [Bre90, Proposition 2.11], this is what is required to obtain an
extension of bitorsors. Thus an alternative way to construct the gerbe P0 is to
start from the bitorsor cocycle E* as described in [Bre94b].
6.3 Extension along a butterAEy
Let now F :H ! G be a general morphism of gr-stacks, and let [Ho, E, Go]
be the corresponding butterAEy (2.2.1), under the equivalence theorem 2.2.1 (we
assume equivalences H ' [H1 ! H0] and G ' [G1 ! G0] have been chosen).
Let also Eo: H1 x G1 ! E be the intermediate crossed module (2.2.3), quasi-
isomorphic to Ho. Recall that there is a ifraction,j (2.2.4), which, denoting by
E the gr-stack associated to Eo, factors the morphism F into
(6.3.1) H - E -! G ,
where the left-pointing arrow is an equivalence. Also, let (Q, k, ) be a gerbe
bound by Ho. The following theorem generalizes the analogous statement of
[Deb77 , Theorem, #2, p. 66].
6.3.1 Theorem. For a butterAEy [Ho, E, Go] as above, and a gerbe Q bound by
Ho, there exists a gerbe P bound by Go. The construction of P is purely in
terms of the butterAEy [Ho, E, Go].
Proof.The construction of the gerbe P is carried out in two steps:
# orst, construct an intermediate gerbe bound by Eo;
# second, apply the construction of sect. 6.2 to the strict morphism
pr2
H1 x G1 ____//_G1
(6.3.2) || |@|
fflffl| fflffl|
E _______//_G0
to obtain the required (G1, G0)-gerbe P.
38
To realize the orst step, let us consider the gerbe:
(6.3.3) Q0def=Q xTORS(H0)TORS (E),
where the ober product is of course taken in the sense of stacks: an object of
Q0is a triple (x, f, P ), where x is an object of Q, P is an E-torsor, and f is*
* an
isomorphism
E
f : (x) -~! ss*(P ) = P ^H0.
There is an obvious morphism Q0 -! Q given by the projection to the orst
factor. The proof is completed by showing that Q0 is bound by Eo, which we_
state in the following lemma, below. |__|
6.3.2 Lemma. The gerbe Q0 is bound by Eo: H1 x G1 ! E.
Proof. Indeed, orst of all there is a morphism
0:Q0- ! TORS (E)
given by the projection to the second factor, and, second, there is a functorial
isomorphism
E E E
(6.3.4) k0: Aut_(x, f, P ) -~! P ^(H1 x G1) ' (P ^H1) x (P ^G1)
satisfying the requirements in Deonition 5.2.1. To see this, observe that by the
very deonition of stack ober product an automorphism of (x, f, P ) is given by
a pair
': x -! x ff: P -! P
such that
(x)___f_//P ^E H0
(')|| |ff^id|
fflffl| fflffl|
(x)___f_//P ^E H0
commutes. In other words, f determines an isomorphism
E
f*: Aut_( (x)) -~! Aut_(P ^H0)
so that f*( (')) = ff ^ idH0. Note that it coincides with
H0 E H0 E
f ^ idH0: (x) ^ H0 -! P ^H0 ^ H0 ' P ^H0
modulo the canonical isomorphism which identioes, for any G-torsor R, Aut_(R)
with R ^G G. Thus, the following diagram
'
Aut_(x)__kx_//_ (x) ^H0H1f^id//_P ^E H0 ^ H0H1____//P ^E H1
|| id^@|| id^@|| |id^@|
|fflffl ' fflffl|f^id fflffl| ' fflffl|
Aut_( (x))____// (x) ^H0H0____//_P ^E H0 ^ H0H0_____//P ^E H0
39
commutes. It shows that there is an isomorphism
E E E
(6.3.5) Aut_(x, f, P ) -~! P ^H1 x(P ^EH0)P ^E ' P ^ H1 xH0 E ,
and moreover, everything is clearly functorial. From the butterAEy (2.2.1)it
readily follows that
H1 xH0 E ' H1 x G1,
so that (6.3.5)is the promised isomorphism (6.3.4), and this concludes_the proof
of the lemma. |__|
6.3.3 Remark. Since the strict morphism (6.3.2)involves just the projection
from H1 x G1 to G1, the eoeect of (6.2.1)is to just kill ooe the H1-part of the
automorphisms. More precisely, given two objects (x, f, P ) and (y, g, Q) of Q0,
the torsor
Hom__Q0(y, g, Q), (x, f, P )
is isomorphic, via (6.3.4), to a product. In this simpler situation, the net eo*
*eect
of (6.2.1)is that of killing the factor relative to P ^E H1.
6.3.4 Remark. The construction of the gerbe P provided by Theorem 6.3.1
can be described by the diagram
__Q0___________________________________
_________________________________________________*
*__________
____________________AEAE___________________
Q P
which resembles the fraction (6.3.1).
Both steps in the construction of the gerbe P in the proof of Theorem 6.3.1
are (2-)functorial: this is clear for the orst step involving the ober product
construction of the gerbe
Q0= Q xTORS(H0)TORS (E)
bound by Eo, and for the second step it follows from the functoriality of Debre-
maeker's construction itself, recalled in sect. 6.2.
Let F :H ! G be the morphism of gr-stacks corresponding to the butterAEy
[Ho, E, Go]. By the above, we have another 2-functor. We state it as follows:
6.3.5 Deonition. Let
(6.3.6) F+ :GERBES (H1, H0) ! GERBES (G1, G0)
be the 2-functor given by sending the (H1, H0)-gerbe Q to its extension along
the butterAEy [Ho, E, Go].
F+ generalizes the functor F+0(see (6.2.2)), and reduces to it when F arises
from a strict morphism of crossed modules. However, note that while for a strict
morphism fo: Ho ! Go the resulting functor F+0reviewed in section 6.2 is such
that there always is an fo-morphism Q ! F+0(Q), it is not so in the current
more general situation, unless one reverts to a torsor picture.
40
6.4 Induced map on non-abelian cohomology
We now consider the eoeect of F+ on cohomology. To this end, consider the
cohomology class determined by the (H1, H0)-gerbe Q, and let (y, h) be a rep-
resentative 1-cocycle with values in Ho, relative to a hypercover Uo ! *. The
class of P = F+ (Q) is obtained by applying the procedure of section 4 to the
class of Q. More precisely, we have:
6.4.1 Proposition. The lift of (y, h) along the butterAEy, as described in sect*
*. 4.2,
provides a representative for the cohomology class of the (G1, G0)-gerbe P con-
structed in Theorem 6.3.1.
Proof.The cocycle (y, h) is determined by the choice of an object z 2 Ob QU0, a
trivialization s of the H0-torsor (z), and the choice of an appropriate morphi*
*sm
a: d*0z ! d*1z over U1, see the proof of Proposition 5.4.1.
To prove the proposition, we show the lift of (y, h) along the butterAEy com*
*es
from a labeling of the (H1 x G1, E)-gerbe Q0 provided by a pair (z0, a0), where
z0 is an object, and a0:d*0z0! d*1z0 a morphism, respectively mapping to z and
a under the projection Q0 ! Q. (The pair (z0, a0) determines a non-abelian
1-cocycle with values in H1 x G1 ! E for the gerbe Q0.)
Only the construction of z0 and a0 will be carried out, leaving the details
of the calculation that this indeed yields the lift of (y, h) along the butterA*
*Ey to
the reader. In the process, the hypercover Uo will need replacing with a oner
one, say U0o, by a process we have already met several times, now, and it will
be silently done without further mentioning. The need for some construction to
hold ilocallyj will signify the need for said replacement.
The object z0 can be found as follows: if :Q ! TORS (H0) is the functor
which is part of the (H1, H0)-gerbe structure of Q, choose a (local) lift of the
H0-torsor (z) to an E-torsor P , so that there is a ss-morphism of torsors
(6.4.1) oe :P -! (z),
where ss :E ! H0. Then set z0 = (z, f, P ), where f is the inverse of the
morphism induced by oe:
E
~oe:P ^H0-! (z)
(p, y)7-! oe(p) y.
A morphism a0:d*0z0! d*1z0 mapping to a: d*0z ! d*1z under the projection
Q0 ! Q is of the form a0 = (a, ff), where ff: d*0P -! d*1P . In fact ff can be
constructed as a (local) lift of (a) with respect to the ss-morphism (6.4.1), *
*so
that we have a commutative diagram
d*0P___ff__//d*1P
| |*
(6.4.2) d*0oe| d1oe|
fflffl| fflffl|
d*0 (z)_(a)//_d*1 (z)
41
as follows. Choose "sof P such that oe("s) = s, again changing Uo if necessary.
Indeed, note that the ioberj Ps = oe-1(s) is a G1-torsor, so onding "samounts
to a trivialization of Ps. Let e 2 E(U1) be a local lift of y 2 H0(U1) and deone
ff as:
ff(d*0"s) = (d*1"s) e.
Since y is determined by the relation (a)(d*0s) = (d*1)y, it is clear that ff *
*so
deoned satisoes (6.4.2).
Now, a further pull-back to U2 determines an automorphism j0 of (d0d1)*z0
such that
(6.4.3) d*1a0= d*2a0O d*0a0O j0
via the analog of diagram (5.4.1)in the proof of Proposition 5.4.1. By con-
struction, the projection Q0 ! Q maps j0 to the automorphism j of (d0d1)*z
obtained in the same way from a: d*0z ! d*1z. It follows that j0= (j, "), where
" is an automorphism of (d0d1)*P covering (j). By using (6.3.5)we have that
E
Aut_((d0d1)*z0!~ (d0d1)*P ^(H1 xH0 E),
so that, relative to the chosen a trivialization "sof P (suitably pulled back to
U2), j0 is identioed with an element of H1 xH0 E. In particular, " is identioed
with the E-factor, call this particular element e02 E(U2), whereas the H1 factor
is h 2 H1(U2), which corresponds to j via the chosen trivialization s of (z).
So, explicitly, the pair (h, e0) satisoes @(h) = ss(e0). Finally, the isomorph*
*ism
H1xH0 E ' H1x G1, identioes (h, e0) with (h, g), for a suitable g 2 G1(U2), or
put it dioeerently, e = ~(h) -(g).
Calculating the relation (6.4.3)with respect to the chosen trivializations s
and "s, we ond that e, h, and g satisfy
d*1e = d*2e d*0e ~(h) -(g),
which is the same as (4.2.1). Moreover, from the second relation of (5.4.2)
applied to the pair (a0, j0), or alternatively performing the calculation sugge*
*sted
at the end of 4.2, it follows that e, h, and g also satisfy (4.2.2), and so the*
* 1-
cocycle (x, g), where x = _(e), is the lift of (y, h) along the butterAEy, as w*
*anted.
To complete the proof, we must make sure (x, g) indeed is the 1-cocycle
arising from a labeling of the gerbe P, obtained from Q0via the strict morphism
Eo ! Go. This is clear, since from section 6.2 we have that P has locally the
same objects as Q0, the functor ~: P ! TORS (G0) is locally the composition
of 0 with _*: TORS (E) ! TORS (G0), and the automorphism group of an __
object is locally isomorphic to G1 via |__|
H1 xH0 E ' H1 x G1 -! G1.
It follows from the previous proposition and from the arguments in section 4
that the class gerbe P is therefore the image of that of Q under F . The
following is an immediate consequence of the previous results.
6.4.2 Theorem. The gerbe P constructed in Theorem 6.3.1 is equivalent to
F**(Q). The two 2-functors F** and F+ are equivalent.
42
7 Commutativity conditions
The group law of a gr-stack may be equipped with commutativity constraints.
Cohomology with values in such a gr-stack will inherit corresponding structures,
actually in a more rigid form due to the process of modding out by the relation
generated by (functorial) equivalence. ButterAEies help to obtain explicit forms
for these structures. (Commutativity conditions for gr-stacks are thoroughly
discussed [Bre94a, Bre99], see also the discussion in [Part I, #7].)
7.1 Commutativity conditions and butterAEies
The very orst commutativity condition one may impose on a gr-stack is that
the group law1
(7.1.1) m: G x G -! G
be braided, that is that there be a functorial isomorphism
sx,y:x y -! y x
for each pair of objects x, y of G . Following the convention adopted in [Part
I] (which is not the same as refs. [Bre94a, Bre99]) we say that the braiding is
symmetric if for all pairs of objects x, y of G the additional condition
sy,xO sx,y= idx y
holds. In addition the symmetric braiding is Picard if it satisoes
sx,x= idx x
for each object x. A braiding is equivalent to the group law being a morphism
of gr-stacks, rather than just a morphism of the underlying stacks, which is the
categorical analogue of the very well-known fact that a group is abelian if and
only if its multiplication map is a group homomorphism. Therefore there is a
butterAEy
G1 x G1__ G1__
| ___ff______________________________________________*
*_________________fi__________________________________________________________*
*_________|
| ________________________________________________*
*__________________________________________________________________________|
| __&&_________________________________________z*
*z_________________________________|
(7.1.2) @x@ | __P_______|@_________________________
| ae____________|__________________________________*
*_________________oe__________________________________________________________*
*_____
| _______________|__________________________________*
*___________________________________________________________________
fflffl|zz_________fflffl|____________________!!______*
*________________________
G0 x G0 G0
representing the morphism (7.1.1), see [Part I, 7.1.3], once an equivalence G '
[G1 ! G0]~ has been chosen. This particular butterAEy has certain additional
____________________________1
We are going to use a plain symbol m to denote the monoidal structure of G ,*
* in place of
the forbidding G used in [Part I].
43
properties, in particular it is always strong, namely it always possesses a glo*
*bal
set-theoretic section o of the epimorphism ae: P ! G0 x G0, so that a classical
braiding map ([JS93])
(7.1.3) c: G0 x G0 -! G1
can be obtained, see [Part I, #7.1]. The group law of P can then be described e*
*x-
plicitly in terms of the set-theoretic isomorphism P ~! G0xG0xG1 determined
by o and the braiding.
Depending on whether the braiding is symmetric or Picard, the butter-
AEy (7.1.2)satisoes extra symmetry conditions, described in detail in [Part I,
#7]. BrieAEy, if G is braided symmetric the corresponding butterAEy (7.1.2)has
the property that its pull-back under the map that swaps the two factors in
Go x Go is isomorphic to P . If in addition G is Picard, then the pull-back of
this isomorphism to the diagonal is the identity.
7.2 The monoidal 2-stack of G -torsors
Let G be at least braided. Since the monoidal structure of G is a morphism of
gr-stacks, we obtain a 2-functor:
(7.2.1) m*: TORS (G ) x TORS (G ) -! TORS (G )
where we have used the identiocation TORS (G x G ) ' TORS (G ) x TORS (G ).
Thus, m* assigns to the G x G -torsor (X , X 0) the G -torsor (X , X 0) ^GxG G .
By the theory of section 6.3 the gerbe counterpart of (7.2.1)is the 2-functor
(7.2.2)m+ :GERBES (G1, G0) x GERBES (G1, G0) -! GERBES (G1, G0)
given by the lift of the gerbe (P, P0) along the butterAEy (7.1.2).
A full investigation of the monoidal structure (7.2.1)or (7.2.2)is beyond the
scope of the present work, but it is necessary to at least point out that it is*
* the
entire collection (in this case: 2-gerbe) of geometric objects itself that acqu*
*ires a
(weak) group structure. The one on cohomology is then obtained by considering
equivalence classes, and it is examined in the next section.
7.3 Group structures on cohomology and butterAEies
If G is at least braided, its monoidal structure (7.1.1)induces morphisms
(7.3.1) m*: Hi(*, G ) x Hi(*, G ) -! Hi(*, G ),
by the mechanisms expounded both in [Part I] (for degree i 0) and in the
present work (for degree i = 0, 1). The morphism (7.3.1)is obtained starting
from either (7.2.1)or (7.2.2)and using functoriality.
At the level of representing cocycles, the group laws (7.3.1)can be computed
by applying the lifting along the butterAEy (7.1.2)described in section 4.2 (By
remark 4.2.2, it applies equally well to 0-cocycles, i.e. descent data for obje*
*cts
44
of gr-stacks). The weak form of the group law for G translates into a standard
rigid one for the m*, including the case i = 1.
We collect the main facts in the following
7.3.1 Proposition. Let G be a braided gr-stack.
1. H 0(*, G ) is an abelian group;
2. H 1(*, G ) is a group;
3. If in addition G is symmetric, H1(*, G ) is an abelian group.
Sketch of the proof.The result is quite well-known, so we only sketch the main
ideas.
For 1, given that H0(*, G ) ' ss0(G (*)), the result is obvious (it follows *
*im-
mediately from the weak group law of G ). As noted, for case 2, that is H1(*, G*
* ),
it follows from either morphism in section 7.2 and functoriality.
More interesting is the case of a symmetric gr-stack. It was proved in [Part
I, Propositions 7.2.2 and 7.2.3] that the symmetry condition is equivalent to t*
*he
braiding being a 2-morphism
s: m =) m O T :G x G -! G
of gr-stacks, where T is the swap functor. Passing to cohomology and us-
ing (7.3.1)yields the commutative structure
H1(*, G ) x H1(*,_G~)_//H1(*, G x G_)m_//H1(*, G )
T*|| T*|| ||||
fflffl| ~ fflffl| m ||
H1(*, G ) x H1(*,_G_)_//H1(*, G x G_)__//H1(*, G )
|___|
7.4 Explicit cocycles
Besides iexplainingj how the orst non-abelian cohomology group with values in
a crossed module acquires a group structure, the butterAEy allows to calculate
explicit formulas for the product. The computations involved are tedious and
straightforward overall, so we will not dwell on the details and only report the
main formulas.
As already observed the butterAEy (7.1.2)is strong, so the group law of P
can be explicitly described in terms of the set-theoretic isomorphism P ' G0 x
G0 x G1 and the braiding (7.1.3)as
(7.4.1) (x0, y0, g0) (x1, y1, g1) = (x0x1, y0y1, c(x1, y0)y1gy0y10g1),
with x0, x1, y0, y1 2 G0, and g0, g1 2 G1. In the foregoing the strong set-
theoretic section o :G0 x G0 ! P is obviously of the form
(7.4.2) o(x, y) = (x, y, 1),
45
with x, y 2 G0. In fact, all the maps in (7.1.2)have explicit descriptions in t*
*hese
coordinates, and their form will be left as an exercise to the interested reade*
*r;
here we only mention that oe :P ! G0 has the form
(7.4.3) oe(x, y, g) = x y @g.
Note that the composition with o gives the multiplication map of G0, which is
of course not a homomorphism.2 The two main computations are as follows.
7.4.1 degree zero
Assume two global objects X, X0 2 Ob G (*) are represented by zero-cocycles
(descent data) (x, g) and (x0, g0) relative to some common (hyper)cover Uo !
*. Here x, x0 2 G0(U0) and g, g0 2 G1(U1). The object (X, X0) of G x G is
represented by the direct product of the corresponding cocycles. Applying the
procedure of section 4.2 (adapted to 0-cocycles, as per Remark 4.2.2) one onds
that the image of (X, X0) under the multiplication map (7.3.1)is represented
by the cocycle *
(xx0, gd1xg0).
This formula coincides with the one for the group law of the gr-stack G express*
*ed
in terms of descent data found in [Part I, 3.4.3]. So the lift along the butter-
AEy computes exactly the same (abelian) group law as induced by the braided
structure on G .
Remark. A priori there appear to be two group laws on H0(*, G ). One inherited
from the monoidal structure of G , while the second is m* in (7.3.1). One is a
homomorphism of the other, so by the classical argument they coincide, and the
resulting structure is abelian.
7.4.2 Degree one
Assume now P, P0 are two gerbes bound by the crossed module Go. Recycling
symbols, assume they are represented by 1-cocycles (x, g) and (x0, g0) relative
to some common (hyper)cover Uo ! *. This time x, x0 2 G0(U1) and g, g0 2
G1(U2). The product gerbe P x P0 is represented by the direct product of the
corresponding cocycles. Applying again the procedure of section 4.2 the gerbe
m+ (P x P0) of section 6.3 (see in particular Deonition 6.3.5) is represented by
a 1-cocycle relative to Uo given by the expression:
0 * * 0 -d*x0 d*x0d*x00
(7.4.4) x x , c(d0x, d2x ) 0 g 2 0 g .
We could have used G -torsors X and X 0to arrive at the same conclusion. In
particular, if (x, g) and (x0, g0) are assumed to be 1-cocycles corresponding to
X and X 0, then the 1-cocycle of expression (7.4.4)represents the G -torsor
(X x X 0) ^GxG G .
____________________________2
In this way one arrives at the standard interpretation of the braiding map a*
*s the isomor-
phism relating the multiplication map and its swapped version.
46
In summary, modulo the appropriate notion of equivalence, expression (7.4.4)
gives an explicit form to the group law (7.3.1)when i = 1.
If G is braided symmetric, the geometric condition on the butterAEy (7.1.2)
translates into the standard notion that the braiding map satisoes the symmetry
condition c(x, y) = c(y, x)-1. In this instance it is possible to explicitly ve*
*rify
that H1(*, G ) becomes an abelian group; exchanging the role of (x, g) and (x0,*
* g0)
in expression (7.4.4)leads to a 1-cocycle which can be seen to be equivalent to
the original one. We omit the details.
8 ButterAEies and extensions
Group extensions and non-abelian cohomology in degree one have a close re-
lationship, which one can trace from Dedeker's classical approach based on
cocycle calculations, to Grothendieck's and Breen's more geometric one, where
the category of extensions
1 -! G -! E -! -! 1
of the topos T is given geometric meaning by showing its equivalence to that a
morphism of gr-stacks
-! BITORS (G).
BITORS (G) is the gr-stack associated to the crossed module G ! Aut(G), and
is considered as a gr-stack in the obvious way. These ideas ot very well with*
*in
the butterAEy framework.
8.1 The Schreier-Grothendieck-Breen theory of extensions
Following ref. [Bre90, #8.11], consider an extension of by the crossed module
G1 ! G0, a notion due to Dedecker and deoned by the following commutative
diagram:
1_____//G1__-_//_E_ss_//__________________//_1
_____________________________
(8.1.1) @ || ________________________________________________*
*__________________
fflffl|_ww__________________________________________*
*___________________________
G0
where the map _: E ! G0 is subject to the additional condition
(8.1.2) e-1-(g)e = -(g_(e)).
We recognize (8.1.2)as the orst relation in (2.2.2), as well as Bre90, equation
(8.11.2), after the obvious changes due to the dioeerent conventions adopted in
this paper.
The trivial extension corresponds to E = n G1, where acts on G1 via
a homomorphism , : ! G0 and the action of G0 on G1, whereas _ is given
set-theoretically as
_(x, g) = ,(x) @g,
47
for x 2 and g 2 G1.
A comparison with diagram (2.2.1)suggests diagram (8.1.1)ought t be con-
sidered as a ione-winged butterAEy,j namely a butterAEy diagram from the crossed
module [1 ! ] to [G1 ! G0]. Therefore, by the results in [Part I, #4 and #5],
recalled in section 2.2, the extension (8.1.1)corresponds to a morphism of gr-
stacks
(8.1.3) FE : -! G
where G ' [G1 ! G0]~ . The form of this morphism is as follows. If x: U !
is a point, it follows from [Part I, #4.3] (see also section 4.3 for a quick re*
*view),
that it maps to the (G1|U , G0|U )-torsor
Hom__1(1, E)x ' x*E Ex.
This retrieves the expression [Bre90, 8.2.2]. Observe also that (8.1.2)is none
other than the expression of the left G1-action on x*E in terms of the right
one (cf. section 2.1). In this language a trivial extension corresponds to a sp*
*lit
butterAEy. Note also that for a split extension the (G1|U , G0|U )-torsor x*E *
*is
isomorphic to (G1|U , x).
The obvious notion of morphism of extensions of the form (8.1.1)is clearly
the same as that of morphism of one-winged butterAEies, in other words an iso-
morphism ': E ! E0 of group objects compatible with (8.1.1). With reference
to the notation used elsewhere in this series (see, e.g. section 2.2) we have
Ext( , G1 ! G0) B( , Go),
where the left-hand side denotes the category (in fact, the groupoid) of exten-
sions of the form (8.1.1), and the right-hand side the one of butterAEies. It
immediately follows from Theorem 2.2.1 that there is an equivalence of cate-
gories
(8.1.4) Ext( , G1 ! G0) -~! Hom( , G ).
There is also the obered analog of the preceding construction. Again from [Part
I, #4 and #5] (see also the summary in section 2.3), and using the same notatio*
*n,
we obtain the following analog of [Bre90, Lemme 8.3]:
8.1.1 Lemma. There is an equivalence
Ext( , G1 ! G0) -~! Hom ( , G ),
where the left-hand side is the stack whose ober over U is Ext( |U , Go|U ).
The cohomological classiocation of the extensions is obtained by applying
ss0 to (8.1.4),
Ext( , G1 ! G0) -~! Hom ( , G ),
48
and rephrasing the right-hand side in terms of the non-abelian cohomology of the
classifying object B . BrieAEy, the group structure of is encoded by diagram
8.1.2 of [Bre90], which we write in the form
G1 *
(8.1.5) fl :d*1E -~! d*2E ^ d0E,
subject to the coherence condition for fl expressing the associativity of the g*
*roup
law. Pulling back by x: U ! , and then d*0x, d*1x, d*1x, we can see (8.1.5)plus
the coherence condition for fl deone a 1-cocycle on B with values in G . By
a reasoning entirely analogous to the one of section 4.3, we can compute the
class with values in the crossed module Go, thereby obtaining the sought-after
element in H1(B , G ). Thus we have:
8.1.2 Proposition (Bre90, Proposition 8.2). There is a functorial isomorphism
of sets
Ext( , G1 ! G0) -~! H1(B , G ).
Functoriality is built-in the butterAEy representation of morphisms of gr-
stacks.
8.2 Remarks on extensions by commutative crossed mod-
ules
We can combine the idea of extension by a crossed module (8.1.1)with the con-
ditions studied in section 7. In this situation the orst non-abelian cohomology
set H1(B , G ) acquires a group structure, possibly abelian if Go is symmetric
or Picard.
8.2.1 Baer sums
The explicit cocycle multiplication formula of section 7.4.2 could be easily tr*
*ans-
lated in terms of group cohomology. This is easier in the case of a strong but-
terAEy, that is for an extension (8.1.1)possessing a global set-theoretic secti*
*on
s: ! E, and it is left as an exercise to the reader.
There is a more interesting ibutterAEy explanationj of the existence of the
product; while the basic mechanism is the one already explained in section 7,
the translation in terms of group cohomology gives it a slightly dioeerent AEav*
*or
that further underscores the role of butterAEy diagrams. The procedure outlined
below is the analog in the context of non-abelian cohomology of the standard
Baer sum of extensions in ordinary homological algebra (see ML95 ).
From two extensions of type (8.1.1), we can form the direct product (drawn
49
with a dioeerent orientation) one-winged butterAEy:
G1_x_G1__
(-,-)_____________________________________*
*__________________________________________________|
ww_________________________________________*
*______________||
(8.2.1) E x E0 |(@,@)
(ss,ss)______________|_________________________________*
*_____________(_,_)____________________________________________________________
____________________|_________________________________*
*_____________________________________________________________
yy___________________fflffl|____________&&_____________*
*________________
x G0 x G0
which then can be composed with (7.1.2), which encodes the monoidal structure,
to yield
G1_x_G1_____ G1___
(-,-)__________________________________________*
*_____________________________________________|ff_____________________________*
*___________________________________________________________fi________________*
*________________________________________________________________________|
ww______________________________________________*
*_________||&&________________________________________________________________*
*__________________zz_________________________________________________________*
*________________||
(8.2.2) E x E0 |(@,@) __P________@|__________________*
*_________
(ss,ss)______________|__________________|___________________*
*_________(_,_)____________________________________________________________ae_*
*________________________________________________________________oe___________*
*_____________________________________________________
____________________|__________________|___________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_________________
yy___________________fflffl|____________fflffl|&&___________*
*__________________zz_________________________________________________________*
*__
x G0 x G0 G0
that is, according to [Part I, #5.1],
___G1____________________________
___________________________________*
*_____________|
___________________________________|
vv____________________________________|
E x E0 xG1xG1G0xG0P |@|
________________________|___________________________*
*__________________________________________________________
____________________________|___________________________*
*__________________________________________________
vv____________________________fflffl|_''__________________*
*_________________________
x G0
which is then pulled back to via the diagonal homomorphism : ! x .
The overall picture for the product is as follows:
1_____________//1________________________________G1__________________
| | ___________________________________________________*
*_______________________________________________________________|
| | _______________________________________________*
*_____________________________________________________________|
| | __))___________________vv___________________*
*__|
| | E x E0 xG1xG1 P @|
| | _____G0xG0____ |
| | _________________________|____________________*
*__________________________________________________________________
| | _____________________________|____________________*
*_______________________________________________________
fflffl| fflffl|vv________________________fflffl|''____________*
*__________________________
___________// x G0
The composition expressed by the above diagram is the full butterAEy diagram
expressing the product structure on the orst cohomology with coeOEcients in G .
Thus we obtain a monoidal structure on the category Ext( , Go).
8.2.2 Abelian structure on H1
If G (or equivalently Go) is symmetric, the butterAEy (7.1.2)is isomorphic to
itself under pull-back by the morphism T that switches the factors. By [Part I,
50
#7.2.4] this means there exists _ :P ~! P such that:
_
P ___________//P
| |
| |
fflffl|T fflffl|
G0 x G0 _____//G0 x G0
compatible with all the morphisms in (7.1.2). The same kind of swap of course
exchanges the factors in the butterAEy (8.2.1). Therefore there is a diagram of
juxtaposed butterAEies
____//_ xoo___ E x E0_____//Go x Gooo___P _____//Go
|| | | | | ||
|| |T T| |T _| ||
|| fflffl| fflffl| fflffl| fflffl|||
____//_ xoo___ E0x E _____//Go x Gooo___P _____//Go
which leads to a morphism of one-winged butterAEies
G xG
* (E x E0) xG10x1G0P
________________________________________________________*
*____________________________________
_____________________________________|_____________________*
*_____________________________________________
vv___________________________________||((____________________*
*_______G
hh____________________________________66_____________________*
*_______ o
______________|____________________________________________*
*________________________________________________________________
__________fflffl|______________________________________*
*_______________________
* (E0x E) xG1xG1G0xG0P
from to Go. This provides a purely diagrammatic proof that the group struc-
ture of H1(B , G ) is abelian when G is symmetric. At the level of diagrams, it
is a braiding on the category Ext( , Go).
8.3 ButterAEies, extensions, and simplicial morphisms
Consider again a generic morphism F :H ! G of gr-stacks and the correspond-
ing butterAEy (2.2.1). Using a sheaooed nerve construction, F corresponds to a
simplicial map
___ ___
(8.3.1) W H_o -! W G_o,
via the map H_o ! G_o in the sense of A1 -spaces, thanks to considerations
analogous to those of [Bre90, #8.5]. In the set-theoretic case this simplicial *
*map
is the starting point for the deonition of weak-morphism of crossed module,
which is then computed by a butterAEy diagram. In the sheaf-theoretic context
the starting point for the deonition of weak morphism is dioeerent (See the
discussion in AN09 , #4.2). Thus, it is of some interest to re-obtain the simpl*
*icial
map in the present context.
Rather than appealing to A1 -geometry, we sketch a dioeerent way to arrive
at the same conclusion, as follows. If in the butterAEy (2.2.1)we isolate the
51
ione-wingedj one,
_G1_______
-_________________________________________*
*________________________________________________________||
zz_________________________________________*
*_________________|
(8.3.2) _E________|@_______________
ss___________|____________________________________*
*_____________________________________________________________________________*
*___
_____________|____________________________________*
*___________________________________________________________________
""____________fflffl|__________________!!__________*
*____________________
H0 G0
analogous to (8.1.1), we obtain a class in H1(B H0, G ), corresponding to a wel*
*l-
deoned morphism
H0 -! G ,
in the sense of gr-stacks. Thus, the underlying geometric object to the exten-
sion (8.3.2)is a G -torsor, or equivalently, a gerbe bound by Go, over BH0.
Next, the standard pull-back (see ML95 ) of the extension (8.3.2)to H1 via
@ :H1 ! H0 is trivial, due to the existence of the homomorphism ~: H1 ! E
in the full butterAEy (2.2.1). It follows that the class of the extension (8.3.*
*2)dies
under the pull-back map
(8.3.3) (B @)*: H1(B H0, G ) -! H1(B H1, G ).
The condition that the pullback of the cocycle corresponding to the exten-
sion (8.3.2)vanish leads to an explicit simplicial map (8.3.1). The actual com-
putation via cocycles is uneventful and quite laborious, so we omit it.
More interesting is the geometric reason, which we record in the following
informal assertions~not all veriocation having being carried out. Essentially,_
the G -torsor over B_H0_deoned by the extension (8.3.2)idescendsj to W H_o
along the map BH0 ! W H_o.
8.3.1 Assertion. The vanishing of the image of the class of the extension (8.3.*
*2)
under the map (8.3.3)determines 2-descent data for_the_G -torsor determined
by the extension (8.3.2)relative to the map BH0 ! W H_o.
Sketch of the proof.Consider the augmented (bi)simplicial object
___ _____// _____// ___
Uoo = cosk0B H0 ! W H_o : . ._.__////_BH0 x__WH_oBH0_//BH0____//WH_o
where the orst index is the iexternalj one, whose face maps are explicitly drawn
above. We compute BH0 x__WH_oBH0 ' B(H0 n H1), and so on, therefore Uoo
is equivalent to B applied degree-wise to H_o:
_____//. _____// _____// ___
. ...._//B(H0 n (H0 n H1))___////_B(H0 n_H1)_//BH0____//WH_o
The face maps are actually induced by those of H_o._Note that the diagonal of
the above bisimplicial object is equivalent to W H_o.
The extension (8.3.2)determines a bitorsor cocycle of the type (8.1.5)which
we write as: G
flx,y:Exy -~!Ex ^1Ey,
52
for points x, y of H0. The class of this cocycle is trivial under the pull-back*
* (8.3.3),
and moreover we know the pulled-back extension is actually a direct product,
rather than a semi-direct one only, since the composition _ O ~ is trivial in t*
*he
full butterAEy. A moment's thought reveals the (G1, G0)-torsor determined by
a direct product extension is in fact trivial, i.e. of the form (G1, 1), hence *
*we
must have coherent isomorphisms
ffih: E@h -~!G1,
where of course E@h is the ivaluej of the pulled back cocycle at h.
At a point (y, h) of H0 n H1, the pull-backs of E along the two face maps
di:H0 n H1 -! H0, i = 0, 1,
d0(y, h) = y@h, and d1(y, h) = y, are:
d*0E(y,h)= Ey@h, d*1E(y,h)= Ey.
Using the cocycle condition and the triviality argument above, we have an iso-
morphism
G1 1^ffih
Ey@h fly,@h---!Ey ^ E@h ---! Ey
at each point (y, h) of H0 n H1. Thus, we have obtained an isomorphism of
extensions, and hence of G -torsors, or again gerbes bound by Go, over the orst
stage U1o.
Similar arguments, this time using the coherence_of_fl and ffi, would show_t*
*he
axioms of a 2-descent datum with respect to BH0 ! W H_oare satisoed. |__|
___
Let us denote by E the descended gerbe over W H_o. Finally we have:
8.3.2 Assertion. The class of E determines the simplicial map (8.3.1).
Sketch of the proof.After sections 3 and 5, the class of a gerbe is eoeectively*
*_a
simplicial map of the sought-after type. |__|
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