Higher order principal bundles
J.F. Jardine*and Z. Luo
February 7, 2004
Introduction
Suppose that G is a sheaf of groups on a topological space X, and that i : Y !
X is a principal Gbundle for X. The bundle is locally trivial, so there is an
open covering {Uff} of X such that the restriction of the bundle to each member
of the covering admits a section
_Y>>
__
___ i
__ fflffl
Uff_____//X
On intersections Uff\Ufiof the covering family, the restrictions may be differe*
*nt
but they are related by multiplication by a unique element gfffiof the sections
G(Uff\ Ufi) of the group G. The collection of all elements gfffidefine a cocycle
for the covering with coefficients in the sheaf of groups G, and cohomologous
cocycles correspond to isomorphic bundles. In this way, the set of isomorphism
classes of G bundles i which trivialize over the covering {Uff} is isomorphic t*
*o a
set of naive homotopy classes of maps ß(Uo, BG) of simplicial sheaves from the
C~ech resolution Uo corresponding to the covering to the classifying simplicial
sheaf BG. This line of argument is classical and well known and, subject to
placing oneself in the context of simplicial sheaves, is most of the proof of t*
*he
theorem that says that there is a natural bijection
H1(X, G) ~=[*, BG]
relating the nonabelian H1 invariant associated to G (aka. isomorphism classes
of principal Gbundles) with morphisms in the homotopy category of simplicial
sheaves on X from the terminal object * to the classifying simplicial sheaf BG.
The link between H1(X, G) and the homotopy theory object [*, BG], or rather
the desciption of it arising from the generalized Verdier hypercovering theorem,
amounts to the observation that the fundamental groupoid of a hypercover is a
C~ech resolution.
____________________________*
This research was supported by NSERC.
1
The argument is also universal, in that it gives a bijection
H1(E, G) ~=[*, BG] (1)
relating isomorphism classes of Gtorsors with morphisms in the simplicial sheaf
homotopy category, which holds in any Grothendieck topos E and for all groups
G in the topos. This result has been known for some time now [6].
The purpose of this note is to give a description of the corresponding homo
topy theoretic invariant ___
[*, W G] ~=[*, dBG]
when G is either a presheaf of simplicial groups or G is a presheaf of groupoids
enriched in simplicial sets. Of course, the last case is the most general _
a simplicial group is a simplicial groupoid with one object _ but a separate
development is given for presheaves of simplicial groups in the first two secti*
*ons
of this paper. ___
The_objects W G and dBG are models for a classifying space construction for
G: W G is the universal cocycles construction, and as such is a generalization *
*of
a classicial construction of EilenbergMac Lane [5], while dBG is the diagonal *
*of
the bisimplicial object arising from standard nerve functor applied to a simpli*
*cial
groupoid G. These constructions are shown to be weakly equivalent in the final
section of this paper. They have been used to create model structures for
various flavours of sheaves and presheaves of simplicial groupoids [3], [9], [1*
*1],
all of which give models for the homotopy category of simplicial sheaves and
presheaves.
The following observation is a central idea of this paper. If G is an ordina*
*ry
sheaf of groups on a Grothendieck site C, then a Gtorsor is a sheaf X admitting
a free (or principal) Gaction such that the coinvariant sheaf X=G is a copy of
the terminal sheaf * up to isomorphism. Backing up a bit, one knows that,
in the presence of a principal Gaction on a sheaf Y , the corresponding Borel
construction EG xG Y is a simplicial sheaf which is weakly equivalent to the
discrete object Y=G. In fact, the converse is true: if EG xG Y ! Y=G is a
local weak equivalence then the Gaction on Y is principal. Thus a Gtorsor
is a sheaf X with a Gaction such that the Borel construction EG xG X is
locally weakly equivalent to the terminal sheaf * in the sense of simplicial sh*
*eaf
homotopy theory.
The analogue of a group action for a presheaf of groupoids G enriched in sim
plicial sets is a simplicial functor X defined on G and taking values in simpli*
*cial
presheaves. Each such simplicial functor X has a homotopy colimit holim!GX,
and one says, by direct analogy with the Borel construction (aka. homotopy
colimit) description of ordinary torsors, that X is a Gtorsor if this homotopy
colimit is locally weakly equivalent to the point *. There is a category of G
torsors G  Tors which has a class ß0(G  Tors) of path components, and the
main result of this paper (Theorem 17) asserts that there is a natural bijection
[*, dBG] ~=ß0(G  Tors).
2
The special case of Theorem 17 corresponding to the case where G is a
presheaf of simplicial groups is given its own proof in the second section of t*
*his
paper, and appears as Corollary 10.
This is done to display a quick application of our main new technical device,
which is an expanded notion of cocyle which appears in Lemma 8. Cocycles
have previously been interpreted (in the most general formulation) as maps
defined on hypercovers. Hypercovers are most precisely defined as locally fibra*
*nt
presheaves which are weakly equivalent to the terminal object *. As such, they
are examples of simplicial presheaves U which are weakly equivalent to a point,
and the magic thing here is that when one looks at the path components of the
category Triv=Y of all morphisms U ! Y (no homotopy classes), one gets a
class ß0(Triv=Y ) which is isomorphic to [*, Y ]. Furthermore this result holds
in great generality: Y can be any member of an arbitrary right proper model
category having a cofibrant terminal object.
We also give a new demonstration of the bijection (1) relating isomorphism
classes of Gtorsors with homotopy classes of maps [*, BG] for sheaves of groups
G, in Remark 11.
With the new approach to cocycle theory in hand, Theorem 17 is a rather
easy consequence of a result of Joyal and Tierney [9] which asserts that their
homotopy category of sheaves of simplicial groupoids is equivalent to the ho
motopy category of simplicial sheaves via the classifying space functor dB. The
point is that when G is a simplicial groupoid with discrete objects and X is a
Gtorsor, the homotopy colimit holim!GX can be taken apart and put back to
gether again with well known results of Quillen. There is no appeal to amenable
objects [9] in the proof of Theorem 17. Also, our torsors do not coincide with
the pseudotorsors of [8] (see Remark 19).
One can define Gtorsors, and one has an analog of Theorem 17 for all
presheaves of simplicial groupoids G. In the case where G has discrete objects,
a result of Moerdijk [13] can be used to show that a Gtorsor X is locally a
copy of the loop space object dBG of the classifying space. In particular, in
this case, any morphism of Gtorsors is a local weak equivalence. This is the
analogue of the well known observation that any morphism of ordinary torsors
for a sheaf of groups is an isomorphism.
There is another antecedent for our theory in the description of torsors for*
* a
sheaves of groupoids which appears in [7], and Theorem 17 is a generalization of
[7, Th.14]. The reader should be aware that the proof of the older result conta*
*ins
an error which is fixed in the proof of Theorem 17 _ see the explanation in
Remark 18 at the end of the third section.
The final section of this paper contains a first application: Theorem 23
identifies the class of path components of the category of Ggerbes for a sheaf*
* of
groups G with set of isomorphism classes of right torsors over the automorphism
2groupoid object Aut (G). In the world of ordinary groups, the automorphism
2groupoid of a group H has one 0cell, 1cells given by the automorphisms
of H and 2cells given by their homotopies. A gerbe is a locally connected
stack, and a Ggerbe is a gerbe which is locally equivalent to either G or its
associated stack of Gtorsors. Our classification theorem can be inferred from a
3
result of Breen [2, Prop.7.3] and its proof employs similar constructions, but *
*we
avoid a discussion of bitorsors and do not encounter homotopy coherence issues.
Theorem 23 is a direct homotopy theoretic classification of Ggerbes up to local
equivalence.
The results of Sections 14 of this paper appeared in preliminary form in
the thesis of Zhiming Luo (the second author), [12].
Contents
1 Torsor categories for presheaves of simplicial groups 4
2 Cocycles 7
3 Torsors for presheaves of simplicial groupoids 11
4 Universal cocycles for simplicial categories 16
5 Gerbes 18
1 Torsor categories for presheaves of simplicial
groups
Suppose that G is a presheaf of simplicial groups on a (small) site C, and write
Gs Pre(C) for the category of all simplicial presheaves X admitting a Gaction
G x X ! X. We shall also call these objects simplicial Gpresheaves.
Lemma 1. There is a cofibrantly generated closed model structure on the cat
egory G  s Pre(C) of simplicial Gpresheaves, where a map f : X ! Y is
a fibration (respectively weak equivalence) if the underlying map of simplicial
presheaves is a global fibration (respectively local weak equivalence).
Proof. The product G x X is the free simplicial Gpresheaf on a simplicial
presheaf X. It follows that the free simplicial Gpresheaf functor preserves
cofibrations and weak equivalences. Colimits in simplicial Gpresheaves are
formed as in simplicial presheaves. Thus, if i : A ! B is a trivial cofibration*
* of
simplicial presheaves and the diagram
G x A _____//X
1xi i*
fflffl fflffl
G x B ____//_Y
is a pushout in the category of simplicial Gpresheaves, then the map i* is
a weak equivalence. It follows that the generating set A ! B for the class
of trivial cofibrations of simplicial presheaves determines a generating family
4
G x A ! G x B for the trivial cofibrations of simplicial Gpresheaves. The
set of morphisms G x K ! G x LU ( n) of simplicial Gpresheaf morphisms
induced by the simplicial presheaf inclusions K LU ( n) is a generating set *
*__
for the class of all cofibrations of simplicial Gpresheaves. *
*__
As usual, LU ( n) denotes the simplicial presheaf which is freely generated
by an nsimplex in Usections.
Say that G acts freely on X or that X is Gfree if the simplicial group G(U)
acts freely on the simplicial set X(U) for all U 2 C.
Suppose that G acts freely on X. Let X=G be the quotient simplicial presheaf
and let ß : X ! X=G be the canonical map. Then the lifting OE exists in the
diagram
t:X:t
tt
tOEttt ß
tt fflffl
LU ( n)__x__//X=G
for each simplex x 2 X=G(U) for all U 2 C. Multiplying the map OE by the
action of G determines a simplicial presheaf map G x LU ( n) ! X, and it is
easy to see that the freeness of the action implies this map factors through an
isomorphism ~
OE* : G x LU ( n) =!LU ( n) xX=G X.
The cofibrant simplicial Gpresheaves X in the model structure of Lemma 1
are (sectionwise) principal Gbundles X ! X=G, on account of the following:
Lemma 2. Suppose that X is a cofibrant simplicial Gpresheaf. Then G acts
freely on X in all sections.
Proof. The maps GxK ! GxLU ( n) generate the cofibrations of the category
of simplicial Gpresheaves, and any pushout
G x K _______//_Z
 
 
fflffl fflffl
G x LU ( n)_____//W
has the effect of adding some freely generated G(U)space to Z(U) for each
U 2 C. The cofibration ; ! X has a factorization
;_____//?V
?? 
??? ß
?ØØ?fflffl
X
where ß is a trivial fibration and the map ; ! V is a transfinite colimit of
pushouts of the above form. It follows that G acts freely on V . But then, by a
standard argument, X is a retract of V (since ß is a trivial fibration)_so that*
* G
acts freely on X. __
5
Remark 3. The model structure of Lemma 1 specializes to the standard model
structure for Gspaces in the case where the site C is a point, and where G is a
simplicial group. In the case of Gspaces, Lemma 2 has a converse [5, V.2.10].
It follows that a simplicial Gpresheaf X having free Gaction is a diagram of
G(U)spaces X(U), each of which is cofibrant. It is not clear that X itself is
cofibrant.
Write G  Torsfor the category of cofibrant simplicial Gpresheaves X such
that the canonical map X=G ! * is a hypercover (ie. a local trivial fibration).
A morphism f : X ! Y of G  Tors is just a Gequivariant map of simplicial
presheaves. Write G  Tors for the corresponding category.
Choose a factorization
; ___i_//BEG
BB 
BBB ß
BB!!fflffl
*
where i is a cofibration and ß is a trivial fibration in the category of simpli*
*cial
Gpresheaves. Write BG = EG=G. Observe that BG is a presheaf of Kan
complexes on account of Lemma 2, [5, V.2.7], and [5, V.3.7].
The homotopy type of BG is independent of the choice of the object EG.
If E0G is a second such choice with quotient B0G = E0G=G then there is a G
equivariant homotopy equivalence EG ! E0G since both objects are fibrant and
cofibrant. This homotopy equivalence induces a homotopy equivalence BG !
B0G of the quotients.
Remark 4. The EilenbergMac Lane object W G has a free Gaction, and the
map W G ! * is a sectionwise trivial fibration. There are maps of simplicial
Gpresheaves
W G pW~ G j!EG
such that p is a trivial fibration and W~G is cofibrant, and such that j is a
trivial cofibration and EG is fibrant. The map p is in particular a sectionwise
weak equivalence of sectionwise cofibrant_Gspaces, and therefore induces a sec
tionwise weak equivalence W~G=G ! W G. The map j is a trivial cofibration
of cofibrant simplicial Gpresheaves, and therefore induces a weak equivalence
W~ G=G ! EG=G = BG.
A similar argument works for the diagonal map d(EG) ! d(BG) induced
by the standard bisimplicial sheaf map EG ! BG, because the induced map of
diagonal simplicial objects is a sectionwise principal Gfibration and the obje*
*ct
d(EG) is weakly equivalent to a point. It follows that d(BG) ' BG for the two
different senses of BG.
Remark 5. Every trivial cofibration i : A ! B of simplicial Gpresheaves
induces a trivial cofibration i* : A=G ! B=G. In effect, i has the left lifting
property with respect to all global fibrations p : X ! Y of simplicial presheav*
*es
with trivial Gaction.
6
Suppose that X is a cofibrant simplicial Gpresheaf such that the induced
map X=G ! * is a local weak equivalence. Find a trivial cofibration j : X ! ~X
in the category of simplicial Gpresheaves such that X~ is fibrant. Then the
induced map j* : X=G ! ~X=G is a trivial cofibration of simplicial presheaves,
and ~X=G is a presheaf of Kan complexes so the map ~X=G ! * is a hypercover.
Write G  Tors0 for the category of cofibrant simplicial Gpresheaves X such
that X=G ! * is a local weak equivalence. Then the inclusion
G  Tors G  Tors0
induces an isomorphism
ß0(G  Tors) ~=ß0(G  Tors0).
Remark 6. Write GTors 1for the category of simplicial Gpresheaves Y such
that the canonical map d(EG xG Y ) ! * is a local weak equivalence, where
d(X) denotes the diagonal of a bisimplicial object X. Then there is an inclusion
G  Tors0 G  Tors1
since the canonical map d(EG xG Z) ! Z=G is a sectionwise weak equivalence
if Z is cofibrant. On the other hand, if X is a simplicial Gsheaf such that
d(EG xG X) ! * is a weak equivalence, there is a trivial fibration Z ! X of
simplicial Gpresheaves such that Z is cofibrant. The induced map d(EG xG
Z) ! d(EG xG X) is a local weak equivalence, so that Z is an object of
G  Tors0. It follows that there is an isomorphism
ß0(G  Tors0) ~=ß0(G  Tors1).
2 Cocycles
Suppose that M is a model category with a terminal object *, and let X be an
object of M. Write Triv=X for the category whose objects are all morphisms
W ! X of M such that the map W ! * is a weak equivalence. Observe that
there is a function
_X : ß0(Triv=X) ! [*, X]
which is defined by associating to an object W ! X the composite
* ' W ! X
in the homotopy category.
Lemma 7. Suppose that M is a right proper model category with terminal object
*. Suppose that the map g : X ! Y is a weak equivalence. Then the induced
function
g* : ß0(Triv=X) ! ß0(Triv=Y )
is a bijection.
7
Proof. The function g* is induced by a functor which is defined by associating
to the object W ! X the composite
W ! X g!Y.
Suppose that v : U ! Y is an object of Triv=Y . Choose a factorization
j
U ____//_@@V
@@ p
v@@ØØ@fflffl
Y
of v, where j is a trivial cofibration and p is a fibration. Form the pullback
g*
X xY V _____//V
 p
 
fflffl fflffl
X ___g___//_Y
Then the map g* is a weak equivalence by the right properness assumption, so
that the projection X xY V ! X is an object of Triv=X. The path component
of this object is independent of the choices made, and is independent of the
choice of representative for the path component of U ! Y .
In effect, if
U ______ff____//_@@U0
@@ """"
v @@ØØ@~~v0"""
Y
is a morphism of Triv=Y and v0= p0. j0 is a factorization of v0with j0 a trivial
cofibration and p0a fibration, then there is a commutative diagram
j0ff
U ____//_V>0>"
j!"""" p0
fflffl"fflffl"
V __p__//Y
and so there is a commutative diagram
X xY VH _____!*_____//_X xY V 0
HHH uuuu
HHH uuu
HH## zzuu
X
It follows that there is a welldefined function
g0: ß0(Triv=Y ) ! ß0(Triv=X).
The composite functions g0. g* and g* . g0 are both identities. _*
*__
8
Lemma 8. Suppose that Y is an object of a right proper model category M in
which the terminal object * is cofibrant. Then the function
_Y : ß0(Triv=Y ) ! [*, Y ]
is a bijection.
Proof. By Lemma 7, it is enough to suppose that Y is fibrant. Then the function
ß(*, Y ) ! [*, Y ]
is a bijection since * is cofibrant. Here, ß(*, Y ) denotes homotopy classes of
maps with respect to a fixed cylinder object I of *. If two maps f, g : * ! Y
are homotopic, then there is a diagram
* @
 @@@f
d0 @@
fflfflØØ@
IO____//_YO??~
d1 ~~g~~
 ~~
*
Then the morphisms d0 and d1 are weak equivalences, so that f and g are in
the same path component of Triv=Y . It follows that there is a well defined
function
OE : ß(*, Y ) ! ß0(Triv=Y )
and that the diagram
~=
ß(*, Y )_______//[*, Y ]
MMM OO
MMM _Y
OEMMMM&&
ß0(Triv=Y )
commutes. Finally, if U ! Y is an object of Triv=Y , there is a factorization
j
U ____//_AV
AA 
AAA p
AA__fflffl
*
where j is a trivial cofibration and p is a trivial fibration. The fibration p *
*has
a section s : * ! V since * is cofibrant, and the map U ! Y extends to a
map V ! Y since j is a trivial cofibration and Y is fibrant. It follows that the
function OE is surjective, and is therefore a bijection. __
The map _Y is therefore a bijection if Y is fibrant. __
9
Lemma 9. Suppose that G is a presheaf of simplicial groups. Then there is a
bijection
[*, BG] ~=ß0(G  Tors0).
Recall that G  Tors0 is the category of cofibrant simplicial Gpresheaves X
such that the map X=G ! * is a local weak equivalence.
Proof. We establish the existence of a bijection
ß0(Triv=BG) ~=ß0(G  Tors0).
Then the desired result follows from Lemma 8.
First of all, there is a function
ß0(Triv=BG) ! ß0(G  Tors0)
which is defined by associating a cofibrant model Z(X) of the simplicial G
presheaf X xBG EG to the object X ! BG of Triv=BG.
Here, one means that a choice of trivial fibration Z(X) ! X xBG EG is
made in the category of simplicial Gpresheaves such that Z(X) is cofibrant.
This can be done functorially since the model structure on the category of
simplicial Gpresheaves is cofibrantly generated. Observe that the induced map
Z(X)=G ! X is a sectionwise weak equivalence since X xBG EG is Gfree.
Suppose that Z is a cofibrant simplicial Gpresheaf such that Z=G ! * is
a local weak equivalence. Then there is a Gequivariant map Z ! EG and an
induced map Z=G ! BG. The class of the object Z=G ! BG in ß0(Triv=BG)
is independent of the choices that have been made: any two Gequivariant maps
Z ! EG are naively homotopic and so the induced maps Z=G ! BG are naively
homotopic and hence represent the same element of ß0(Triv=BG). It follows
that there is a well defined function
ß0(G  Tors0) ! ß0(Triv=BG)
and this function is the inverse of the function in ß0 which is induced by_the
functor of the previous paragraph. __
Corollary 10. There is a bijection
[*, BG] ~=ß0(G  Tors1).
Recall that the objects of the category G  Tors1are simplicial Gpresheaves Z
such that d(EG xG Z) ! * is a local weak equivalence. Lemma 9 is equivalent
to Corollary 10, by Remark 6.
Remark 11. If G is a sheaf of groups, then a Gtorsor X is naturally a member
of G  Tors1 after identification of X with a constant simplicial Gsheaf, and
in this way the category G  torsof ordinary Gtorsors imbeds in G  Tors1.
We claim that the induced function
ß0(G  tors) ! ß0(G  Tors1) (2)
10
is a bijection.
Suppose that X is a simplicial Gpresheaf such that the map d(EGxG X) !
* is a local weak equivalence, or that X is a member of G  Tors1. Then the
canonical map d(EG xG X) ! BG is a local fibration with fibre X according
to Lemma 12 below. The total space object d(EG xG X) is locally weakly
equivalent to a point by assumption, so that X is nonequivariantly locally
equivalent to BG ' G, where the sheaf of groups G is identified with a constant
simplicial sheaf. It follows in particular that the Gequivariant map X ! ß0X
is a local weak equivalence. The sheaf of groups G acts on the associated sheaf
ß~0X, and the composite
X ! ß0X ! ~ß0X
is a Gequivariant local weak equivalence. The induced map
d(EG xG X) ! EG xG ~ß0X
is also a local weak equivalence, so that EG xG ~ß0X is locally equivalent to a
point. This last statement means precisely that the sheaf ~ß0X is a Gtorsor:
the freeness of the Gaction is the vanishing of the sheaf ~ß1(EG xG ~ß0X), and
ß~0(EG xG ~ß0X) ~=(~ß0X)=G ~=* as a sheaf. All constructions are natural, so
the function
ß0(G  tors) ! ß0(G  Tors1)
is a bijection with inverse specified by X 7! ~ß0X.
3 Torsors for presheaves of simplicial groupoids
Write sGpd 0to denote the category of presheaves of groupoids enriched in
simplicial sets, and write sGpd for the full category of presheaves of simplic*
*ial
groupoids. A groupoid enriched in simplicial sets is a simplicial groupoid with
discrete objects, and the two ways of describing such an object will be used
interchangeably. All sheaves or presheaves in this section are defined on a fix*
*ed
small Grothendieck site C. ___
The purpose of this section is to analyze the set of morphisms_[*, W G] for a
presheaf of simplicial groupoids with discrete objects. Here, W G is the univer*
*sal
cocycle construction of [5] and [11] _ see also Section 4. It_is_also shown in
Section 4 that there is a natural weak equivalence j : dBG ! W G, where dBG
denotes the diagonal of the usual bisimplicial nerve BG. The homotopy type of
dBG is also insensitive to whether or not G is a sheaf, and we shall therefore
focus attention on computing [*, dBG] when G is a sheaf of groupoids with
discrete objects.
Joyal and Tierney have a model structure for sheaves of simplicial groupoids
[9] for which a map G ! H is a weak equivalence if and only if the induced
map dBG ! dBH is a local weak equivalence of simplicial sheaves. The Joyal
Tierney model structure is proper [9, Th.9]. They also show [9, Th.12] that the
functor dB determines a functor
dB : sGpd =G ! sShv =dBG
11
which induces an equivalence of homotopy categories. It follows from Lemma 8
that the functor dB induces an isomorphism
dB : ß0(Triv=G) ~=ß0(Triv=dBG)
for all sheaves of simplicial groupoids G. If one says that a map f : H ! H0
of presheaves of simplicial groupoids is a weak equivalence if the induced map
dBH ! dBH0 is a local weak equivalence of simplicial presheaves, then it's
clear that the functor dB and the associated sheaf functor H 7! H~ together
induce a commutative diagram
ß0(Triv=H) __dB//_ß0(Triv=dBH)
~= ~=
fflffl ~= fflffl
ß0(Triv=H~)__dB//_ß0(Triv=dBH~)
The function dB in the diagram is therefore a bijection for all presheaves of
simplicial groupoids H.
The following result is a restatement of a theorem of Moerdijk, specifically
Theorem 2.1 of [13]. It can also be proved with the techniques used to prove
the group completion theorem in [5]. As Moerdijk observes in [13], the group
completion theorem is a consequence of this result.
Lemma 12. Suppose that C is a category enriched in simplicial sets and that
X : C ! S is a simplicial functor taking values in simplicial sets. Suppose that
all arrows a ! b of C0 induce weak equivalences X(a) ! X(b). Then the map
X(a) ! Fa taking values in the homotopy fibre over a of the simplicial set map
d(holim!CX) ! d(BC) is a weak equivalence.
The object holim!CX is the bisimplicial set with simplicial set
G
X(a0) x G(a0, a1) x . .x.G(an1, an)
(a0,a1,...,an)
in horizontal degree n. In vertical degree m, it is the simplicial set holim*
*!GmXm .
Corollary 13. Suppose that G is a groupoid enriched in simplicial sets, and
that X : G ! S is a simplicial functor taking values in simplicial sets. Then
the map X(a) ! Fa taking values in the homotopy fibre over a of the simplicial
set map d(holim!GX) ! d(BG) is a weak equivalence.
A simplicialFfunctor X : G ! S can alternatively be described as simplicial
set X = a2Ob(G) Xa fibred over the object set Ob (G) in the sense that there
is a simplicial set map f : X ! Ob (G) which collapses summands to points.
Suppose that the simplicial set X xsMor (G) is defined by the pullback diagram
X xs Mor(G) _____//Mor(G)
 s
 
fflffl fflffl
X _____f____//_Ob(G)
12
where s is the source map. Then the other piece of data required for the sim
plicial functor X is a simplicial set map m : X xs Mor(G) ! X which fits into
a commutative diagram
X xs Mor(G) ___m___//X
 
 f
fflffl fflffl
Mor (G)____t___//Ob(G)
where t is the target map. The map m must respect identities of G is an obvious
way.
There is a canonical diagram
X _______//d(holim!GX) (3)
 
 
fflffl fflffl
Ob (G)________//d(BG)
and then Corollary 13 has the following equivalent formulation
Corollary 14. Suppose that G is a groupoid enriched in simplicial sets, and
that X : G ! S is a simplicial functor taking values in simplicial sets. Then
the diagram (3) is homotopy cartesian.
Lemma 12 can be expressed in terms of a similar homotopy cartesian dia
gram.
Example 15. Suppose that H is a simplicial groupoid with discrete objects and
let f : U ! H be a morphism of simplicial groupoids (U does not necessarily
have discrete objects). Take a 2 Ob (H) and write f # a for the simplicial
category given in degree n by the comma category fn # a arising from the
functor fn : Un ! Hn. Then the functors Hn ! catgiven by a 7! fn # a define
a simplicial functor dB(f # ) : H ! S. The forgetful functors fn # a ! Un also
assemble to define a weak equivalence
holim!HB(f # ) ff!BU
The simplicial sets dB(f # a) therefore become identified with the homotopy
fibres of the diagonal simplicial set map associated to the canonical bisimplic*
*ial
set map
holim!HdB(f # ) fi!BH
In öh rizontal degree" n, this map can be identified with the projection
dB(f # a0) x H(a0, a1) x . .x.H(an1, an) ! H(a0, a1) x . .x.H(an1, an)
13
Remark 16. Suppose that C is a small category, and consider the simplicial
set maps
G fi G
BC ffd( B(C # x0) ! d( *) = BC
x0!...!xn x0!...!xn
arising from the simplicial set construction underlying Example 15. In other
words the map ff is induced by the forgetful functors C # a ! C, while fi is the
canonical map induced by the simplicial set maps B(C # x0) ! *. Both maps
are weak equivalences.
The object G
X = d( B(C # x0))
x0!...!xn
is the simplicial set consisting of strings (y, x) of arrows
y0 ! . .!.yn ! x0 ! . .!.xn
of length 2n+1 in C, and the map ff takes this string to the string y0 ! . .!.yn
while fi maps this element to the string x0 ! . .!.xn. The nsimplices of
the simplicial object X can therefore be identified with functors n * n ! C
defined on the poset join n * n, and with simplicial structure maps induced by
precomposition with maps ` * ` : m * m ! n * n. The maps ff and fi are
induced by the inclusions n ! n * n of the left and right substrings of length n
respectively.
There is a poset map hn : n x 1 ! n * n which is defined by
(
(i, ffl) 7! i if ffl = 0, and
n + i if ffl = 1.
As a picture, hn is the diagram
y0 _____//y1___//_._._._//yn
  
  
fflffl fflffl fflffl
x0 _____//x1___//_._._._//xn
The maps hn are natural in ordinal numbers n. It follows that the composites
n x 1 hn!B(n * n) (y,x)!BC
together define a simplicial set map X x 1 ! BC from ff to fi. This construc
tion is natural in all small categories.
Suppose that G is a presheaf of simplicial groupoids with discrete objects.
A torsor for G is a simplicial functor X : G ! s Pre(C) taking values in sim
plicial presheaves such that the associated simplicial presheaf d(holim!GX) *
*is
weakly equivalent to a point. A morphism f : X ! Y of Gtorsors is a natural
14
transformation of simplicial functors; it may also be described as a simplicial
presheaf morphism
f
X _________________//FY
FF xxx
FFF xxx
F""F __xx
Ob(G)
which respects the Gstructure. It is an immediate consequence of Corollary 14
that any such map f must be a local weak equivalence. This map also induces
weak equivalences of all local choices of fibres.
Write G  Tors for the category of Gtorsors, and let ß0(G  Tors ) denote
its set of path components. There is a welldefined function
OE : ß0(G  Tors) ! ß0(Triv=dBG)
which is induced by associating to a Gtorsor X the element represented by the
map d(holim!GX) ! dBG.
Note that the map OE is morally induced by the elts map of Joyal and Tierney
[9, p.288], although it is defined on enriched diagrams of presheaves rather th*
*an
sheaves.
There is a function
_ : ß0(Triv=G) ~=ß0(Triv=dBG) ! ß0(G  Tors)
which is defined as follows. Let f : U ! G be an object of Triv=G and perform
the construction of Example 15 sectionwise to construct the diagram
dBU oo'__ d(holim!GdB(f # ))

 f*
 
fflfflff fflffl
dBG oo'__ d(holim!GdB(G # ))
'fi
fflffl
dBG
Then the simplicial Gfunctor a 7! dB(f # ) is a Gtorsor. This construction is
functorial and defines the function _.
The composites fi .f* and ff.f* are homotopic by the construction of Remark
16. It follows that the canonical map fi . f* and the original map dBU ! dBG
represent the same element of ß0(Triv=dBG), and so the composite OE . _ is the
identity function.
If X is a Gtorsor, then the canonical map d(holim!GX) ! dBG is induced
by a morphism f : EG X ! G of presheaves of simplicial groupoids (where
EG X is the translation category for the functor Xn : Gn ! Set in each degree.
There is a Gnatural functor f # a ! Xn(a) which induces a map (also a weak
equivalence)
dB(f # a) ! Xn(a)
15
for all n and a, and hence determines a map of Gtorsors
dB(f # ) ! X
It follows that the composite _ . OE is the identity function. We have therefore
proved the following
Theorem 17. Suppose that G is a presheaf of simplicial groupoids with discrete
objects. Then the natural function
OE : ß0(G  Tors) ! ß0(Triv=dBG) ~=[*, dBG]
is a bijection.
Proof. The displayed isomorphism is a consequence of Lemma 8. The proof __
that OE is a bijection is given above. __
Remark 18. Theorem 17 generalizes Theorem 14 of [7]. The proof of Theorem
17 also implicitly fixes an error in the proof of that result, which does not
properly take into account the phenomenon discussed in Remark 16.
Remark 19. A simplicial sheaf of groupoids G for which the coequalizer c(G)
of the source and target maps s, t : Mor (G) ! Ob (G) Ob (G) ! ~ß0(Ob (G))
is simplicially discrete is said to be locally transitive in [8]. All sheaves *
*of
simplicial groupoids with discrete objects are locally transitive in this sense*
*.Joyal
and Tierney define a Gpseudo torsor for a locally transitive object G to be a
simplicial sheaf X on which G acts freely (in each simplicial degree), and such
that the colimit X=G is locally weakly equivalent to a point. All Gpseudo
torsors are Gtorsors in the sense of this paper for G with discrete objects, b*
*ut
the class of Gtorsors is larger. Theorem 24 of [8] gives a homotopy classifica*
*tion
of pseudotorsors, and implies that the categories of Gpseudo torsors and G
torsors have isomorphic presheaves of path components in the case where G is
a simplicial sheaf of groupoids with discrete objects.
4 Universal cocycles for simplicial categories
___
The simplicial set W G for a groupoid enriched in simplicial sets (aka. simplic*
*ial
groupoid with discrete objects), is defined as a space of universal cocycles in
[5, V.7]. We show here how to extend the definition of this construction_to all
simplicial categories C, and we construct a_comparison_map j : dBC ! W C.
We show in Lemma 20 that the map j : dBG ! W G is a weak equivalence for
groupoids G enriched in simplicial sets.
Suppose that C is a simplicial object in the category of small categories.
Write EC for the following variant of the Grothendieck construction: the set
of objects of EC consists of all pairs (x, n) with x 2 Cn, and a morphism
(f, `) : (x, m) ! (y, n) is a pair consisting of an ordinal number map ` : m ! n
and a morphism f : x ! `*y of Cm . There is an obvious forgetful functor
ß : EC ! which takes values in the ordinal number category .
16
The segment category Seg(n) of subintervals [j, n] of n = [0, n] can be iden
tified with the opposite nop via the functor [j, n] 7! j. There is a functor
cn : nop! which is defined by j 7! n  j.
An ncocyle taking values in the simplicial category C is a functor X : nop!
EC which is a lifting of cn in the sense that the diagram of functors
EC==
zz
Xzzzz ß
zz fflffl
nop _cn__//_
commutes. This is a generalization of the definition of an ncocycle taking
values in a groupoid enriched in simplicial sets, in view of the identification*
* of
the categories Seg(n) and nop.
The ncocycle X : nop! EC is otherwise described as a string of arrows
(x0, n) (x1, n  1) . . .(xn, 0)
each of which has the form (ffi, d0), with ffi : xni ! d0(xni1). This means
that the string consists of objects xi2 Cni and morphisms xi! d0(xi1).
Every ordinal number map ` : m ! n induces a commutative diagram
~=
m  i_______//[i,_m]___//m

`i `i `
fflffl fflffl fflffl
n  `(i)_~=_//[`(i),_n]_//_n
and there is a corresponding diagram
(`*0x`(0), m)oo___(`*1x`(1), m o1)o_. .o.o____ (`*mx`(m), 0)
(1,`0) (1,`1) (1,`m)
fflffl fflffl fflffl
(x`(0), n  `(0))oo_(x`(1), n  `(1))oo._.o.o__(x`(m), n  `(m))
The string on top is denoted by `*X._ ___
In this way, a simplicial set W C is defined, with W Cn given by the set of
ncocycles in C. The functoriality follows from the relations
`ø(i)øi= (`ø)i
associated to composable ordinal number maps
k ø!m `!n.
There is a function
___
j : dBCn = (BCn)n ! W Cn
17
which sends a string
x0 x1 . . .xn
in Cn (note the BousfieldKan indexing [1, p.328]) to the cocycle consisting of
the objects dj0xnj 2 Cnj and the induced morphisms
dj0ffnj : dj0xnj ! dj0xnj1 = d0dnj10xnj1,
or rather to the string
(x0, n) (d0x1, n  1) . . .(dn0xn, 0)
in the Grothendieck construction EC .
Suppose that ` : m ! n is an ordinal number map. One checks that the
composite
___ `* ___
dBCn j!W Cn ! W Cm
sends the string of arrows x0 x1 . . .xn in Cn to the string
(`*0d`(0)0x`(0), m) (`*1d`(1)0x`(1), m  1) . . .(`*md`(m)0x`(m), 0)
while the composite *
___
dBCn `!dBCm j!W Cm
sends that same string in Cn to the string
(`*x`(0), m) (d0`*x`(1), m  1) . . .(dm0`*x`(m), 0).
Then `*id`(i)0x`(i)= di0`*x`(i), and it follows that the maps j respect the sim*
*pli
cial structure.
Lemma 20. Suppose_that G is a groupoid enriched in simplicial sets. Then the
map j : dBG ! W G is a weak equivalence.
___
Proof. The functors dB and W preserve homotopy equivalences_and disjoint
unions. If H is a simplicial group, the map j : dBH ! W H classifies the H
bundle dEH ! dBH, and so j is a weak equivalence for simplicial groups. Every
simplicial groupoid G is homotopy equivalent to a disjoint union of simplicial_
groups. __
5 Gerbes
In homotopy theoretic terms, but according to the standard definition [2], [4],
[10], a gerbe is a locally connected stack on a (small) Grothendieck site C.
If G is a sheaf of groups on C, then a Ggerbe (following [2]) is a stack D
such that there is a covering family U ! * of the terminal sheaf such that there
are equivalences
DU ! St(GU )
18
for each U in the covering family, where St(GU ) is the stack completion (stack
of GU torsors) of the restricted sheaf of groups GU . In this case, the stac*
*k D
is automatically locally connected.
We can alternatively say that a stack D is a Ggerbe if there is a covering
family V ! * such that there are local equivalences
GV ! DV
for each V in the cover. In effect, locally, there is an object x of (GU )  t*
*ors
which lifts to DU up to isomorphism, any equivalence of groupoids induces
isomorphisms of automorphism groups, and the sheaf of automorphisms of any
Gtorsor is isomorphic to G.
A presheaf of groupoids E is said to be a Ggerbe if there is a covering
W ! * of the terminal object by objects of C such that there are local weak
equivalences
GW ! EW
for each W in the covering. Write G  gerbe for the corresponding category
of Ggerbes and morphisms E ! E0 of presheaves of groupoids which are local
weak equivalences. We shall only be interested in local weak equivalence classes
of Ggerbes, so it will be irrelevant whether our gerbes are sheaves or preshea*
*ves
of groupoids. Note that the definition of Ggerbe works equally well when G is
a presheaf of groups, and that the following is easily proved:
Lemma 21. Suppose that G is a presheaf of groups, and let ~Gdenote its as
sociated sheaf. Then the natural functor ~G gerbe ! G  gerbe defined by
restriction of structure induces a bijection
ß0(G~ gerbe) ~=ß0(G  gerbe).
If H is a group, write Aut(H) for the 2groupoid with one object, a 1cell
for each automorphism of H and a 2cell for each homotopy (conjugation by an
element of H) between automorphisms. One can check that the object Aut(H)
is a group object in groupoids, so the simplicial groupoid (automorphisms and
all their strings of homotopies) corresponding to Aut(H) is a simplicial group,
and there is a natural inclusion
Aut(H) hom (BH, BH).
We shall identify the 2groupoid Aut(H) with this simplicial group.
In fact, Aut(H) can be characterized as the subcomplex of hom (BH, BH)
which consists of those nsimplices (functors) H x n ! H whose vertices are
automorphisms. Observe that the evaluation
hom (BH, BH) x BH ! BH
restricts to an action
Aut(H) x BH ! BH
19
of the simplicial group Aut(H) on the nerve BH.
Suppose that G and G0 are presheaves of groupoids. The simplicial set
equi (G, G0) is the subobject of hom (BG, BG0) consisting of all functors G x
n ! G0such that all restrictions to vertices
G ~=G x 0 1xi!G x n ! G0
are local equivalences of presheaves of groupoids. Any f : G ! G0 which
is homotopic to a local weak equivalence must be a local weak equivalence,
so it suffices that there is some restriction to a vertex which is a local weak
equivalence. It follows also that the simplicial set equi(G, G0) is the nerve of
a groupoid whose objects are the local weak equivalences G ! G0 and whose
morphisms are the homotopies between them.
The simplicial presheaf Equi(G, G0) is defined by
Equi (G, G0)(U) = equi(GU , G0U )
for each object U of the underlying site C. Then Equi(G, G0) is the nerve of a
presheaf of simplicial groupoids, in an obvious way. Write also
Aut (G) = Equi(G, G),
and
aut(G) = equi(G, G).
Then
Aut (G)(U) = aut(GU )
for each object U of the site C.
Lemma 22. Suppose that G is a sheaf of groups with associated stack morphism
j : G ! St(G). Then the map j induces local weak equivalences
*
Equi (G, G) j*!Equi(G, St(G)) j Equi (St(G), St(G)).
Proof. Note first of all that any local weak equivalence G ! G0 of fibrant
presheaves of groupoids is a homotopy equivalence, since the associated map
BG ! BG0is a homotopy equivalence. It follows that if G ! G0is a different
choice of fibrant model for G, then there is a map G0! St(G) which induces a
homotopy equivalence
equi(G, G0) ! equi(G, St(G))
It follows that the induced map
Equi(G, G0) ! Equi(G, St(G))
is a sectionwise weak equivalence.
20
We can therefore assume that St(G) is a sheaf of groupoids. Write * for the
image of the unique object of G under j (the trivial torsor). Then since j is a
local weak equivalence and St(G) is a sheaf of groupoids the induced map
j* : G ! hom (*, *)
is an isomorphism of sheaves of groups.
Suppose that f : G ! St(G) is a local equivalence of sheaves of groupoids,
and let x = f(*) be the image of * in global sections. Then St(G) is locally
connected, so there is a covering family of objects U ! * of C, and a morphism
xU ! * for each member of the covering family. Then fU is homotopic to a
composite of the form 0
GU f!GU j!St(G)U
for each U in the covering family. This is true in all sections, so it follows *
*that
the induced sheaf map
~ß0Equi(G, G) ! ~ß0Equi(G, St(G))
is an epimorphism. This map is also a monomorphism, on account of the sheaf
isomorphism G ~=hom (*, *) which is induced by j. That same sheaf isomor
phism induces a sheaf of fundamental groups isomorphism
~ß1(Equi (G, G), ff) ~=~ß1(Equi (G, St(G), jff)
for all (local) choices of base points ff. The map
Equi(G, G) j*!Equi(G, St(G))
is induced by a morphism of presheaves of groupoids, and is therefore a map of
presheaves of Kan complexes. It follows that j* is a local weak equivalence.
The diagram
Equi (St(G), St(G))___//_Hom(B(St(G)), B(St(G)))
j* j*
fflffl fflffl
Equi (G, St(G))________//Hom(BG, B(St(G)))
is a pullback since j is a local equivalence. The map j* of function complex
presheaves is a trivial fibration since j is a trivial cofibration and the simp*
*licial
presheaf B(St(G)) is fibrant. It follows that the map
j* : Equi(St(G), St(G)) ! Equi(G, St(G))
is a trivial fibration, and is therefore a local weak equivalence. *
*___
21
Suppose that H is a simplicial group and that simplicial sets X and Y are
chosen such that X has a right Haction and Y has a left Haction. Then H
acts on the left on X x Y via
(g, (x, y)) 7! (xg1, gy),
and the resulting bisimplicial set EHxH (XxY ) has horizontal path components
isomorphic to X xH Y (balanced product), where X xH Y = (X x Y )= and
the indicated equivalence relation is generated by the relation (xg, y) ' (x, g*
*y).
Note that HxH Y ~=Y in the special case where the group H is interpreted as
having a right Haction by the group multiplication. In this case, the canonical
map
d(EH xH (H x Y )) ! H xH Y ~=Y
is a weak equivalence. In effect, the path component in EH xH (H x Y ) corre
sponding to a fixed vertex (e, x) has objects consisting of all pairs (g, g1x),
and the map g : (g, g1x) ! (e, x) is uniquely determined. The function
H ! H x Y which is defined by g 7! (g, g1x) induces an isomorphism of
categories of EH xH H (right action) with the path component of (e, x). It
follows that if H acts freely on X, then the map
d(EH xH (X x Y )) ! X xH Y
is a weak equivalence.
The corresponding (opposite) simplicial group Ho is obtained by reversing
all arrows in H all simplicial degrees. A right (aka. contravariant) action
of the simplicial group H on a simplicial set X corresponds to a left action
Ho x X ! X.
Suppose that G is a sheaf of groups and that F is a right Aut (G)torsor,
meaning (see Remark 5) that F is a cofibrant Aut (G)oobject, and the map
F=Aut (G) ! * is a local weak equivalence.In particular, F has a free right
Aut (G)action. Remark 4 and Lemma 9 together imply that there are bijections
___ o o
[*, dB(Aut (G)o)] ~=[*, W (Aut (G) )] ~=ß0(Aut (G)  Tors).
The remainder of this section consists of the proof of Theorem 23, which asserts
that these objects are in bijective correspondence with the set ß0(G  gerbe)
of path components (ie. local equivalence classes) of Ggerbes.
The simplicial sheaf of groups Aut (G) acts on the simplicial sheaf BG via
the composition
Aut(G) x BG ! Hom (BG, BG) x BG ev!BG
where ev is the evaluation map. The canonical map
d(EAut (G) xAut(G)(F x BG)) ! F xAut(G)BG (4)
is a local weak equivalence by the previous paragraphs.
22
Since the map F=Aut (G) ! * is a local weak equivalence, there is a covering
family of maps U ! * with U 2 C such that there are sections
F??
oe~~~~
~~ 
~ fflffl
U _____//*
These sections induce Aut (G)equivariant equivalences (maps of right torsors)
oe* : Aut (GU ) ! F U for all maps U ! * in the covering family. The induced
maps of balanced products
BGU ~=Aut (GU ) xAut(GU)BGU ! F U xAut(GU)BGU
are local weak equivalences for all U ! * in the covering family by the previous
paragraphs, so that F xAut(G)BG is locally equivalent to BG. It follows that
the stack completion St(ß(F xAut(G)BG)) of the corresponding fundamental
groupoid is a Ggerbe. The fact that the maps (4) are weak equivalences for
all right Aut (G)torsors also implies that any map F ! F 0of Aut (G)torsors
induces a local weak equivalence
St(ß(F xAut(G)BG)) ! St(ß(F 0xAut(G)BG))
of Ggerbes.
Suppose that E is a Ggerbe, interpreted as a stack which is locally equival*
*ent
to G. Then there is a covering family U ! * by objects U 2 C such that there
are equivalences ffU : GU ! EU for all U ! * in the covering family. Since
EU is a stack there are equivalences GU Tors ! EU such that the diagrams
GU ___ffU__//_EU99
sss
j  ssss0
fflfflffUsss
GU  Tors
commute. In the composite
0
Equi(GU , GU ) j*!Equi(GU , GU  Tors) ffU*!Equi(GU , EU )
the map ff0U*is a homotopy equivalence since ff0Uis a weak equivalence of stack*
*s,
and the map j* is a local weak equivalence by Lemma 22. These maps are
equivariant for the action by Equi (GU , GU ) on the right. It follows that t*
*he
map
EAut (G) xAut(G)Equi (G, E) ! *
is a local weak equivalence, so that Equi (G, E) represents a right Aut (G)
torsor. The corresponding torsor is an Aut (G)cofibrant model
ß : Equi(G, E)c ! Equi(G, E).
23
Note that the cofibrant object Equi (G, E)c has a free Aut (G)action, so that
any restriction Equi(G, E)cU has a free Aut (GU )action.
Suppose that there is a covering family U ! * of objects U 2 C such that
there are local weak equivalences ffU : GU ! EU for all U ! * in the covering
family. Then there is a diagram
Equi(G, G)cU xAut(GU)BGU___'__//Equi(G, E)cU xAut(GU)BGU
'  
fflffl fflffl
Equi (GU , GU ) xAut(GU)BGU___//Equi(GU , EU ) xAut(GU)BGU
~= 
fflffl fflffl
BGU ______________'_____________//_BEU
The top horizontal map is a local weak equivalence since the map
Equi (G, G)c ! Equi(G, E)c
is a local weak equivalence of simplicial presheaves having free Aut (G)action*
*s.
Similarly, the map Equi(G, G)c ! Equi(G, G) is a weak equivalence of simpli
cial presheaves having free Aut (G) actions, so the corresponding vertical map
is a weak equivalence. It follows that the induced composite
Equi (G, E)cxAut(G)BG ! Equi(G, E) xAut(G)BG ! BE
determined by the evaluation map is a local weak equivalence. In particular,
there is an induced natural local equivalence
St(ß(Equi (G, E)cxAut(G)BG)) ! E
of stacks.
Suppose that F is a right Aut (G)torsor. Then there is an Aut (G)equivar
iant map
F ! Equi(G, St(ß(F xAut(G)BG))) (5)
which is adjoint to the canonical map
F xAut(G)BG ! B St(ß(F xAut(G)BG)).
Locally, the map (5) has the form
Aut (G) ! Equi(G, St(ß(Aut (G) xAut(G)BG))) (6)
Thus, if we show that all instances of (6) are local weak equivalences, then all
instances of (5) are local weak equivalences.
The evaluation isomorphism Aut (G) xG BG ! BG is adjoint to the iso
morphism (identification) Aut (G) ! Equi(G, G), and there is a commutative
24
diagram of local weak equivalences
Aut (G) xAut(G)BG _____//B(St(ß(Aut (G) xAut(G)BG)))
 
 
fflffl fflffl
BG ____________j_______//B(St(ß(BG)))
It follows that there is a commutative diagram
Aut(G) ______//_Equi(G, St(ß(Aut (G) xAut(G)BG)))
~= '
fflffl fflffl
Equi (G, G)_______j*___//_Equi(G, St(ß(BG)))
The map j* is a local weak equivalence by Lemma 22, so the desired (top
horizontal) map is a local weak equivalence.
We have therefore proved the following:
Theorem 23. Suppose that G is a sheaf of groups on a small Grothendieck site
C. Then there are bijections
___ o
[*, dB(Aut (G)o)] ~=[*, W (Aut (G) )] ~=ß0(G  gerbe).
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26