C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
A NDERS K OCK
The stack quotient of a groupoid
Cahiers de topologie et géométrie différentielle catégoriques, tome 44, no2 (2003), p. 85-104
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© Andrée C. Ehresmann et les auteurs, 2003, tous droits réservés.
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THE STACK QUOTIENT OF A GROUPOID
by Anders KOCK
CAHIERS DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XLIV-2 (2003)
Resum6. On decrit un sens 2-dimensionnel
pr6cis
danslequel le champ
des G-fibrésprincipaux
est unquotient
dugroupoide
G.L’outil cle a cette fin est une reformulation de la descente
(ou
donn6es de
coégalisation),
en termes de rel6vementssimpliciaux
de
diagrammes simpliciaux.
It is a well known
conception,
see[1],
Ex.4.8,
that the stackB(G.)
of
principal G,
bundles is in some sense aquotient
ofG..
I intendhere to make this into a more
precise statement,
and to prove it in aquite general
context -essentially
that of acategory
withpull-backs
and
equipped
with a class D of descentepis (as
in[6]
or[5]).
For an
equivalence
relation in acategory B,
it isunambiguous
whata
quotient
of it should be: Ifis an
equivalence
relation inB,
aquotient
for it is a map q :G0--> G-1 mediating,
for everyobject X,
abijection
where
Coeq(R, X)
is the set of X-valued"coequalizing
data" forR, meaning
maps p :G0 -> X
with p odo
= p od 1.
For
G,
agroupoid
inB,
and X a stack
(or just
a fiberedcategory)
overB,
we describe a cate-gory
(groupoid,
infact), Coeq(G.,X)
of X-valuedcoequalizing data koi
86
descent
data).
We describe in which sense the stackBG.
of"principal G.-bundles"
is a 2-dimensionalquotient (expanding
the basecategory
B into the
2-category
of stacks overB).
So we look for anequivalence _
there will be such an
equivalence, provided
X has a stackproperty,
in the sense we shall recall in Section 5 below. Theequivalence
is not itselfexplicit,
but isexpressed
in terms of twoexplicit equivalence
functors(cf. (22) below),
whose
quasi-inverses
are notcompletely explicit,
sincethey depend
onchosing cleavages
or solutions of descentproblems.
The first
equivalence, Coeq(G., X)-> Grpd(X)jG.,
is dealt with in Sections 2 and 3.Thus, also,
a reformulation of the notion of descent data isprovided. -
The universalcoequalizing data,
i.e. thecoequalizer itself,
can in terms ofgroupoids
inBG,
begiven completely explicitly:
it is Illusie’s
Dec. (G.).
Our formulations of fibration
theory,
and ofdescent,
are free of cleav-ages.
The
question
ofquotients
ofgroupoids
may be relevant to the for- mulations of intentionaltype theory
of e.g.[3], [7],
whoapproximate
the notion of
types-with-an-intentional-equality
in terms ofgroupoids.
I want to thank the
Mittag-Leffler Institute,
where the mainpart
of thepresent
work was carried out inMay
2001.1 Basics
onfibrations
This section is
mainly
to fix notation andterminology.
Consider afibration 7r : X -> B
(see
e.g.[5]).
For G anobject
inB, XG
denotesthe fibre. We consider
only
fibrations where all fibres aregroupoids;
this is well known to be
equivalent
to theassumptions
that all arrows inX are cartesian.
(Nevertheless,
we sometimes use the word"cartesian",
as a reminder of the universal
property.) By 2-category,
we understandhere a
2-category
where all 2-cells areinvertible; equivalently,
acategory
enriched in thecategory
ofgroupoids.
We consider the2-category
offibrations over
B,
denotedFibB. Morphisms
are functors over B(which
preserve the
property
ofbeing
a cartesian arrow; this is hereautomatic, by
ourassumption) .
And 2-cells are natural transformations all of whosecomponents
are vertical(mapping
to anidentity
arrowby r).
For twoobjects
X and Y inFibB, hom(X, Y)
denotes thehom-category (a groupoid,
infact, by
ourassumptions).
For any G E
B,
the domain functorB/G - B is a fibration,
de-noted
y(G);
such fibrations one callsrepresentable; they
have discretecategories
as fibres:( y(G) ) H
is the set of arrows from H to G. Anarrow in
B/G
over the arrowf : X1 --> Xo
is a commutativetriangle
h = g o
f , where g
and h have codomainG;
suchtriangle,
when viewedas an arrow in
B / G,
is denoted(g; f )
.We shall be interested in
morphisms in FibB
whose domain are rep-resentable fibrations
y(G), y(H),
etc. We collect some basic formulas.Note that since no
"cleavage"
or otherarbitrary things
arementioned,
the
principle
"whatever ismeaningful,
is true" islikely
to beapplicable.
(We
refer to these assertions as "Basic Item1.-4.".)
1. Let D :
y (G)-> X,
and let d : H -t G. Thecomposite
D oy(d),
is
given
onobjects
e E(y(H))K by
,and on
morphisms (e; f )
iny (H) by
it is an arrow in X over
f .
2.
Next,
we consider a 2-cellSo for d E
y(G)H,
thecomponent Cd : D(d) -> D’(d)
is an arrow inX,
vertical over H. For an arrow
(d; e) : f-> d
iny(G) (where f
=d o e),
the
naturality
square is3. We next consider the
composition ( "whiskering" )
of the formwhere d : H -> G in B. For an
object
e EKy(H),
thecomponent
of thewhiskering C
oy(d)
at e isgiven
as follows:it is a an arrow in
X,
vertical over K.4. Let D and D’ be as in item 2. above. From the
naturality
square exhibited in(4),
it is easy to conclude that if the values of D’ are(cartesian)
arrows, and if two2-cells C and 77 :
D -> D’ agree on theobject 1G (identity
map ofG),
thenthey
agreeeverywhere. For,
fromthe
naturality
squares(4) for C and n
withrespect
to(l; d) :
d -31,
itfollows that
D’(1 ; d) 0 Çd
=D’(1; d)
o nd, but twoparallel
vertical arrowswhich
postcompose
with some cartesian arrow togive
the same, areequal.
Remark.
( "Yoneda Lemma" )
There is anexplicit
functor evl(=eval-
uation at the
object 1G
iny(G)
=BIG),
and it is an
equivalence,
but aquasi-inverse
is notexplicit;
aquasi-
inverse amounts to
choosing
for each X EXG
and each h : H - Ga
(cartesian)
arrowh*(H) -> X
over h. So it amounts to a"partial
cleavage"
of X - B.We have the full
embedding
y of thecategory
B into the2-category
FibB. It
actually
factorsthrough
the fullsubcategory
of D-stacksSB,
to be described in Section 5 below. It is full and faithful on 1-cells
as well as on 2-cells
(viewing
B as alocally
discrete2-category,
in thesense that all 2-cells are
identities).
We sometimes omit the name of theembedding y
from the notation. We shall not discusscoequalizers
in
S
ingeneral,
butonly coequalizers
ofgroupoids G,
in BC S .
Inthe first
approximation,
this means of course adiagram
which
commutes,
and is universal(in
a suitablesense)
inSB
with thisproperty.
But since there are 2-cells available betweenparallel
arrowsto
X,
two-dimensional wisdom says that the notion "the twocomposites
are
equal"
should bereplaced by
"there is aspecified 2-cell 0 comparing
the two
composites" .
But wisdom also says thatspecifications
shouldcome
together
withequations
to besatisfied,
and here it is acocycle
condition
on ?P,
which involves the three mapsG2 -> G1.
To makebetter room for the
pasting geometry involved,
we exhibit the fork(6)
in terms of a square
Then the
equations
to be satisfied are acocycle condition,
and a unitcondition. The
cocycle
condition isexpressed
in terms ofcommutativity
of the 2-cells in a
cube,
The three faces
adjacent
to the vertex labelled X areequal,
andare all filled with the
(invertible)
2-cell1/J,
and the three otherfaces, adjacent
to the vertex labelledG2,
arestrictly commutative,
and express the threesimplicial
identities that obtains between thecomposite
faceoperators G2 - Go.
As apasting diagram,
it makes sense,(Y being
an oriented
2-cell;
there are in fact orientations on the threesimplicial
identities
making
this cube into a validpasting scheme, namely doso
-d0 d1, d1 d0
->doS2,
andd1 d1
-->d1 d2).
The
cocycle
conditionon Y
says that thepasting diagram
commutes.There is also a unit condition: it says that
pasting
the2-cell 0
withs :
Go - G¡ yields
anidentity 2-cell,
If X is
equipped
with acleavage,
so that one hasfunctors d0* : XG0 -->
XG, etc.,
the cubiccocycle
condition can be rendered in the usual formd2*(Y)
odo* (Y)
=d1*(Y)
for descent data.The collection of such data form a
groupoid Coeq(G., X),
whosearrows are 2-cells P - P’
compatible
with theY’s.
We may think of it as an alternative way ofdescribing
thecategory
of descent data for descentalong
e, ifdo, d1 happen
to be the kernelpair
of some mape.)
2 From coequalizing data to groupoids
We consider a fibration 7r : X->
B;
we assume that all arrows in Xare
cartesian,
so that the fibresXG (for
G EB)
aregroupoids.
We alsoassume that B has
pull-backs.
Then it follows that X haspull-backs,
and that 7r preserves them. Even more, 7r
reflects pull-backs,
in thesense that if a commutative square in X is
mapped
to apull-back by
7r,then it is itself a
pull-back.
A
groupoid object
in B may begiven
in terms of its nerveG.;
amore economic way of
giving
the data of agroupoid object
G is thefollowing
standard one: it consists of truncatedsimplicial data,
of face maps
satisfying
thesimplicial identities,
cf.Appendix,
fromwhere the notation is
taken, plus
a map s :G0 -> Gi, splitting
thetwo face maps
G1 --> Go (s "picks
outidentity arrows" ).
For such truncated data to be a
groupoid,
the three commutative squares thatrepresent
the threesimplicial
identities among face maps(see Appendix)
should bepull-backs; also,
with the middle of the three face mapsG2 -> GI
ascomposition,
thiscomposition
should be associa- tive and have s as unit. If these conditions aresatisfied,
its "nerve"G.
may be formed. It is a
full-fledged simplicial object,
of which thegiven
data then is a "truncation" . The
category
of smallgroupoids
becomes afull
subcategory
of thecategory
ofsimplicial objects. -
With the statedassumptions
on 7r : X ->B,
we then haveProposition
1 Let X be truncatedsimplicial
data in Xmapping
to agroupoid
G in B . Then X is agroupoid.
Proof. The
associativity
condition for thecomposition
mapd1 : X2 -> Xl
isexpressed
as anequality
between twoparallel
maps a1, a2 :X3 - Xl (where X3 = Xl
x X0Xl
x X0X1 ).
Now sinceX.
maps toa
groupoid G.,
where theassociativity
conditionholds,
and since 7rpreserves
pull-backs,
it follows that7r(al) = 7r(a2).
Since therefore al and a2 areparallel
maps over the same map, it suffices to see thatthey
become
equal
whenpost-composed
with some(cartesian)
map. Butclearly
for instancedo : X1-> Xo
will do thisjob.
So to construct a
groupoid
in X out ofcoequalizing
data P :Go - X, Y,
as in(7),
it suffices to construct truncated dataX2, Xl, Xo,
withthe relevant six maps in between. This is
completely explicitly done,
and exhibited in the
diagram (10)
below(as
far as the five face maps areconcerned). Namely,
we takeXo
:=P(1Go), Xl
:=P(do), X2
:=P(eo);
they
are theobjects
of the upper row in(10).
The five face maps arealso
present
in thediagram.
We use notation for face maps as in theAppendix,
N N and decorate the face maps in the NX.
under constructionby
6i, dj,
etc. Weput do := P(lGo; do),
andd1
:=P(1G0 ; d1) oY1G1.
Weput So
:=P(do; So), (note
thatby do
o80
= eo,(do; So) :
eo -do
iny(Go),
and similar for the other "semicolon"
expressions). Similarly,
weput
Jl
: =P(do; d1) ;
forJ2,
we needagain
to involveY : S2
:=P(do; b2)
o08,,;
Finally,
the construction iscompleted by putting
9 :Xo - X1 equal
toP(do; s) (note
that sincedo
o s =1, (do; s)
is amorphism
iny(Go)
from1 to
do).
The reader will find some of this data exhibited in thediagram
(the
1on 0
refers to1 G¡ ).
To prove the
simplicial
identities among the8i, dj
and 9 is easier the fewer’lj;’s
areinvolved,
i.e. the smaller the indices iand 3
are. Themethod is in any case the same, so we are
only going
topresent
one ofthem,
the "worst" one, - theonly
oneinvolving
thecocycle condition,
as well as the
identity involving
the unitcondition,
93-
So we calculate
(using naturality of 0
withrespect
to(1 ; d1) : 81
->1G1)
using functorality
of P on thecomposite (1; d1)
o(d1 ; b1) (1; e2).
Onthe other
hand,
by naturality of 0
w.r.to(1, d2) : S2
-lcl . By functorality
ofP,
this is
P(l; e2)
0Yd2
oYd0,
andby
thecocycle condition,
thisequals P(l; e2)
oYd1
as desired. - To prove the unit condition:do
o g = 1 istrivial
by functorality
ofP; dl
o 9 uses the unit condition forY, namely Ys=1.
3 From groupoids to coequalizing data
We consider a
groupoid X,
inX, mapping by
7r to the fixedgroupoid G,
inB,
andproceed
to constructcoequalizing
data(P : G0 -> X, 1/; )
out of this data. This is not a
completely explicit construction;
onepiece
of information is notcompletely explicit, namely
a functor(partial cleavage)
P :y(Go) --3
X withP(1)
=Xo (1 denoting
theidentity
map ofGo).
We assume such a P chosen.(For instance,
if X isequipped
with a
cleavage,
then we may for c : H -->Go
in B takeP(E)
to be thecleavage-chosen
cartesian arrowe*Xo - Xo
overe.) Being
a functorover
B,
we have for each e : H ->Go
in B agiven (cartesian)
arrowp(E) : P(e) - Xo
over it.We have to
provide
the naturaltransformation Y :
P oy(do)
-P o
y(dl)
between the indicated functorsy(G1) ->
X.(The simplicial operators
onG,
consist of mapsd;, Sj,
and ek, asbefore;
thesimplicial operators
onX.
are denotedsimilarly,
but with a tilde:d;, etc.)
So consider an
object d : H --> Gl
iny(Gi), then 1/;ó
should be avertical arrow in X over
H,
denoting do
o Jby
ta andd,
o Sby
Eb, we then constructOs by
thefollowing recipe:
Considerp(ea) : P(ea)
-->Xo;
then use thatdo
iscartesian,
so that we may consider thecomparison
arrow a :P(Ea)
->X,
over
d, arising
from the factorization Ea -do
o6; similarly,
considerP(Cb) : P(Eb) --> Xo:
then use thatd1
iscartesian,
so that we may consider thecomparison arrow B : P(C-b)
->Xl
over Sarising
from thefactorization Cb =
di
o J. Since both aand j$
live overJ,
and havecommon codomain
X1,
we may use that0
iscartesian,
toget
aunique
vertical
comparison
from a toB,
and this is to be our08,
soFor the convenience of the
reader,
we record therecipe
in adiagram:
The unit
condition 0
oy(s)
= 1 followsby contemplating
this dia-gram, with S = s, then the
long sloping
arrows will be1xo ;
so a = 9and
0
= 9by uniqueness
of cartesianfactorization,
andso 0
oy(s)
isthe
identity
2-cell of1X0.
To prove the
cocycle
condition(in
the "cube"form, (8)),
we need tocalculate the
whiskerings Y
oy(6;)
for i =0, 1, 2.
We claim
that,
for theircomponents
at theobject 1G2 (for
briefdenoted
1),
wehave,
for certain canonical vertical arrows co, el and c2 to begiven below,
Since natural transformations in this case are determined
by
their com-ponent
at theidentity
of thedomain,
theseequations
will establish thecocycle
condition forY,
(where
we used * rather than o to denote horizontalcomposition (whisker- ing).
The three calculationsproceed
in the same way, so we shallgive only
the one for(14).
We use the cartesianproperty
ofdo
to lift thefactorization
do
oS,
= eo to a factorization ofp(eo) through do,
saywith
r(a) = Si,
andsimilarly,
the factorizationd1
oS,
= e2 lifts to afactorization
of p(e2)
overdl
with
7r(Q) = di. Also, by
the definition of081,
with
1/;81 vertical,
for the a andB
of(16)
and(17).
Let ci denote theunique
verticalcomparison X2 - P(et)
withThen we claim
These are
parallel
arrows over the same arrow81
inB,
so it suffices toprove that
they
becomeequal by post-composition
with some(cartesian)
arrow;
here, do
will do thejob, since, by (16) do
o cx o co =p(eo)
o co =eo
=d0
od1.
We can now proveSince both sides of this
equation
arevertical,
it suffices to prove thatpost-composing
them with some(cartesian)
arrowgive
sameresult;
weshall utilize
p(e2),
so we intend to proveWe calculate
This finishes line prooi of (14). For the convenience 01 tne reader, we
compile
the data of theproof
of(14)
into adiagram.
Note that thecorresponding diagrams
for(13)
and(15)
will looksimilar,
but that thea and
B
will denotesomething
different(whereas
thecz’s
remain thesame).
We now prove that the two processes
(of
Section 2 and the currentpart
of thepresent Section)
are inverse of eachother,
up to canonicalisomorphism.
If we start withcoequalizing
data(P, Y),
P :y(Go) -+
Xin
particular
is apartial cleavage
of X with codomainXo (so P(1)
=Xo,
1
denoting 1Go);
thegroupoid
constructedgives
riseto, possibly
new,coequalizing
data(P’, Y’),
whose construction starts out withchoosing
a
partial cleavage
P withP(1)
=Xo
=P(1).
Hence there is aunique isomorphism
betweenthem,
and thecompatibility with 0
means anassertion of
equality
of two natural transformations with domainy(Gi).
From Basic Item
4,
it suffices to seeagreement
on1G1,
which is easy.Conversely,
if we have agroupoid X.
in X overG,
inB,
andproduce
coequalizing data, by
somepartial cleavage P, (with P(1)
=Xo)
thenwe have the vertical
comparisons
co :X1 -> P(do)
and c2 :X2 - P(e2);
and
by
theconstruction,
thesecomparisons
areimmediately compatible
with the face maps,
except possibly
with the last onesdl
and32,
whosedefinition involved
1/;,
cf. thedisplay
in(10).
Butcontemplate
the con-struction
of 0,
in terms of thegroupoid,
cf.(12):
in thatdiagram,
thecomparison
a isjust
the inverse of thecomparison
co,and 0 similarly
for cl, the
unique comparison
fordl,
soY1
o co = cl, and then the de- siredcompatibility
is clear. For thecompatibility
of theS’s,
one canutilize that we are
dealing
withgroupoids
over the samegroupoid G.,
and prove the desired
equality by post-composition
with some suitable(cartesian)
arrowX1 -> Xo (take dl).
Summing
up, we thus have our reformulation ofcoequalizing
data(and
hence of descentdata):
Theorem 1 For any
groupoid G.
in B and anyfibration
withgroupoids
as
fibres
X -B,
theexplicit functor
described in Section 2is an
equivalence.
If
G.
is a smallgroupoid (identified
with its nerve, which is asimpli-
cial
set),
aprincipal G,
bundle is asimplicial
set over p :G., E. -> G.
such that
1)
all the squares,expressing that p
commutes with the face-operators,
arepull-backs,
and2) E.
is the(nerve of)
anequivalence relation,
withcoequalizer E0-> E_1,
say, called theaugmentation.
Wesay that
E.
is aprincipal G.-bundle
onE-1.
Thecategory
ofprincipal G.-bundles,
withaugmentation E. --> E-1
aspart
of thedata,
form afibered
category
overB,
7r :B(G.)-> B,
where7r(E.
-E_1)
=E-1.
All arrows in
B(G.)
are cartesian(equivalently,
the fibres are(large) groupoids.) (It
isactually
even a stack in the sense of Section5,
pro- vided that the structural maps(face operators)
ofG.
are D-epis.)
A
particular object
inB(G.)
is Illusie’sDec1 (G.),
orDecl,
forshort,
since
G,
will befixed;
it is aprincipal
bundle overGo,
and isgiven by
Decn
=Gn+1.
It isdepicted
in row number two from below in thediagram
98
The row above that is called
Dee2,
and above that(not depicted) Dec3,
etc.Although
there are three maps fromDec3
toDec2,
and twomaps from
Dec2
toDec1, they
all compose togive,
for each n,exactly
one map from Decn to
G..
In fact this map makes Decn into aprincipal G.-bundle
overGn-1
for n > 1.Altogether,
the various Deen’s fittogether
into asimplicial object
ofprincipal bundles, augmented
overthe
simplicial object G,
in theright
hand column. Since all squares insight
arepull-backs,
this means that the Decn’s form agroupoid Dec.(G.)
inB(G.),
over thegroupoid G.
in B.4 Stacks
The notion of stack that we shall use is relative to a class D of "descent
epis"
in the basecategory
B. If B is atopos,
D could be taken to be the class of allepimorphisms.
In thecategory
of smoothmanifolds,
itcould be taken to be the class of
surjective
submersions. An axiomatic treatment of theproperties
of such D has beengiven
mostsuccinctly
in
[6];
see also[5].
We shall not need to bespecific here.
Suffice it to say that anypull-back
of aD-epi
isagain
aD-epi,
and allrepresentable
fibrations
y(B)
are D-stacks.Let q :
G0 --> G-i
be a map inB,
withsimplicial
kernelG. =
Let X - B be a fibration. We then have an
explicit
functorFor,
let P Ehom(G-1, X),
i.e. P is a functory(G-1)
=B/G_1 -+
Xabove B. Now
G,
is asimplicial object
inB/G_1
above thegroupoid G.,
so it goesby
the functor P :B/G-1->
X to asimplicial object
inX,
aboveG,; by Proposition 1,
thissimplicial object
is agroupoid
inX.
Definition 1 The
fibered category
X -+ B is a stackif for
all q ED,
the
functor (21)
is anequivalence.
Proposition
2 Let rr : X -> B be astack,
and letbe arrows in X with common
domain,
and with7r( 8) = 7r(S)
aD-epz*
q :
G0 --> G-1.
Then there is aunique
verticalisomorphisms ç : X-1-->
X’ 1. satisfying C
0 8 = S’.Proof. Choose a
cleavage (-)*
of X - B. Then weget
a functorq* : XG-1 -> GrPd(X)/G" namely
the one which to anobject X
EXG-1
associates the(cartesian)
lift with codomain X of thesimplicial
kernel
G,
of q :Go - G_1.
This functor makescommutative up to
isomorphism.
Since ev, isalways
anequivalence,
andC is an
equivalence by
the assumed stackproperty
ofX,
we concludethat
q
is anequivalence.
Inparticular,
it is full and faithful. Now there is aunique
verticalcomparison, : q*(X-1)
->Xo with d o y equal
to thecleavage
chosen liftq*(X-1)
->X-1
of q.Similarly,
there is acomparison
y : q*(X’-1 )
->Xo.
These twocomparisons
compose(inverting Y)
to acomparison C0 : q* (X-1)
-q*(X’-1) .
We may continuesimilarly
for thelifts of
Gi-> G-1,
andtogether,
thisprovides
anisomorphism
in
Grpd(X)IG.,
andby
fullness ofq*,
it comes from a vertical isomor-phism C : X-1 -> X’-1.
The various arrows mentioned here sit in adiagram
in which the arrows x and x’ are the
cleavage-chosen
lifts. It follows fromcommutativity
of theremaining triangles
in thisdiagram
that alsothe
triangle C
0d= d’ commutes. - Theuniqueness
ofsuchC
followssimilarly
from the faithfulness ofq*.
For a
groupoid X.
overG.,
anyaugmentation X0 --> X-1
over q deserves the name of solutionof
the descentproblem posed by X.. By Proposition 2,
such solution isessentially unique.
5 The coequalizer
We are now
going
to makeprecise
in which sense andwhy B(G.)
is acoequalizer
of thegroupoid G..
This first of all means that one shouldspecify
the2-category
in whichthings
takeplace;
this is the2-category
SB,
the fullsubcategory
of stacks inside the2-category FibB. Secondly,
one should
specify
the map q :G0-> B(G.),
which is to be the "co-equalizing map" , together
with a2-cell §
betweenq o do and q
odl .
Themap q is
going
to beDecl(G,),
moreprecisely,
somepartial cleavage
ofB(G.)
with codomainDec1(G.).
Andq, 0
isgoing
to be "the"object
in
Coeq(G., B(G.))
whichcorresponds
to thegroupoid Dec.(G.)
overG. in B(G,),
under thecorrespondence
of Sections 2 and 3.(One
reason forreformulating coequalizing data/descent
data interms of
groupoids
is thatDec’(G,)
is acompletely explicit piece
ofdata, involving
nochoices,
orquotation
marks around definitearticles.)
So
consider,
a fixedfibration-in-groupoids
7r : X ->B,
and also afixed
( "small" ) groupoid G,
in B. We have thefollowing categories
andfunctors
The
categories
are,respectively,
thecategory
ofcoequalizing
data(p : G0-> X, 0),
asexplained
in Section2,
thecategory of groupoid
ob-jects X.
inX,
overG,,
andhom(B(G.), X)
is thecategory
of(cartesian)
functors between
fibrations-in-groupoids,
over B. All threecategories
are in fact
(large) groupoids.
The functors
displayed
are allequivalences;
the full arrows are ex-plicit,
the dotted ones arequasi-inverses,
anddepend
on choice(say,
ofpartial cleavages
and solutions of descentproblems);
the functorrequires
for its construction that X is a stack. The two functors onthe left in
(22)
are those that have beenexpounded
in theprevious
sections. The
functorality
of theexplicit
functors in(22)
is:pasting
with F : X -3 Y on the left
corresponds
toapplying
F ongroupoid objects
in X. Theexplicit
functor on theright is just
"evaluate atDec*";
for,
a functor overB,
say Y -X, clearly
takesgroupoid objects
overG,
in Y togroupoid objects
overG,
in X. This inparticular applies
tothe
groupoid object Dec.(G.)
inB(G.).
So the
remaining
task is toprovide
the functor(23), provided
thatX is a
stack,
and prove it to bequasi
inverse to the evaluation at Dec*.When this has been carried
out,
we have theright
to assertTheorem 2 Let q :
Go - B(G.), 0 be
thecoequalizing data,
corre-sponding
under theleft
sideequivalence of (22)
to thegroupoid object
Dec.(G.)
inB(G.).
Thenfor
any stack X overB, pasting
with q pro- vides anequivalence
This is
exactly
to say that suchq, O
is acoequalizer,
in the 2-dimensional sense, of
G., recalling
that universalproperties
2-dimensi-onally
should beexpected
toclassify "up
toequivalence",
not"up
toisomorphism" .
So let us construct a functor
(23).
LetX,
be agroupoid
overG.,
in
X,
assumed to be a stack. To construct itsimage
under the functor(23)
means to construct a functor overB,
The construction is
going
to involve somechoosing
of(cartesian) lifts;
a
partial cleavage B/Go -
X will suffice.Also,
it involveschosing
solutions for descent
problems
in X. So we assumegiven
anobject E. -> E-1
on the left handside, i.e.,
aprincipal G.-bundle
withquotient E-1 ;
so there is inparticular
asimplicial
map a. :E. - G..
For eachn, we take a cartesian lift over an with codomain
Xn, say an : X’n --> Xn . (Such
lifts can be obtainedcanonically by comparison
with the chosen lift of d o an :En-> Gn - Go,
where d :Gn - Go
is thecomposite
of astring
ofdo’s say.)
Nowby using
the cartesianproperty
of thean’s,
andcomparing
with thesimplicial
mapE. - G,,
one obtains a series offace
operators
between theX’n’s, making X’
into asimplicial object
inX above the
groupoid E..
But such data is nowprecisely
descent data for descentalong
theaugmentation E0 -> E’ 1,
so since X is astack, X;
descends to anobject X’-1
inXE-1.
The processE. -> X’ 1
thusdescribed is the
requisite
functorB (G.) ->
X.We now prove that the two processes are inverse to each
other,
up toisomorphism.
Let us start with agroupoid X,
overG,
in the stackX;
we want to evaluate the
resulting
functorB(G.)
- X onDec.( G.).
ButDec.(X.)
sits in X aboveDec.(G.)
inB,
so it follows fromProposition
2 that
Dec1(X.) --> Xo
is a solution of the descentproblem posed by Decl(X.),
andsimilarly
forDeC2 (X.)
->Xl , etc.,
so up toisomorphism,
we recover the
groupoid X..
,Conversely,
let us start with a functor P :B(G.) --> X,
and evaluate it atDec.(G.),
so as toget
agroupoid P(Dec.(G.))
inX; by
therecipe provided,
thisgroupoid gives
rise to a functor P :B(G.) -> X,
whosevalue at a
principal G.-bundle E. - El
may be desribed as follows: it amounts to use the stackproperty
of X to descend a certainequivalence
relation in X
along E0 --> E-i
inB,
and thisequivalence
relation is described in terms of its nerve, which issimply
but since P preserves
pull-backs
and solutions of descentproblems (this
follows from
Proposition 2),
this solution is(isomorphic to) P(E.).
Appendix. The faces of a triangle
For a
simplicial object X.
in anycategory,
we shall be interested in its lowest dimensionalparts,
The three face
operators X2 -> Xl
we denoteSo, S,
andS2,
and the twoface
operators X1 -> Xo,
we denotedo
andd 1.
For thecalculations,
itis also convenient to have names for the three
composites X2 -> Xo,
wecall them eo, el and e2,
they
are definedby
thefollowing
basicequations
For the case where
X,
is the nerve of a smallcategory
and we considera
2-simplex
x,i.e.,
acomposable pair
JO(X)
=f, d1(x)
= g of, 12(z)
= g, and for instance the middle equa- tion can be renderedverbally:
the domainof
the second arrow g is the codomainof the first
arrowf -
andeo(x)
=A, el(x)
=B, e2(x)
= C.The commutative square
expressed by
the middleequation
is apull- back, by
definition of"composable pair";
the commutative squares ex-pressed by
the two otherequations
arepull-backs precisely
whenX,
isa
groupoid.
104
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Address of author: Mathematics
Institute, University
ofAarhus,
DK 8000 Aarhus
C,
Denmarke-mail: kock@imf.au.dk