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Representing top oi by top ological group oids
Carsten Butz and Ieke Mo erdijk, Utrecht
Abstract
It is shown that every top os with enough p oints is equivalent to the clas-
sifying top os of a top ological group oid.
1 De nitions and statement of the result
We recall some standard de nitions [1, 5 , 9] . A top os is a category E whichis
equivalent to the category of sheaves of sets on a small site. Equivalently, E is
a top os i it satis es the Giraud axioms [1], p. 303 . The category of sets S is a
top os, and plays a role analogous to that of the one{p oint space in top ology. In
particular, a point of a top os E is a top os morphism p: S!E. It is given bya
functor p : E!Swhich commutes with colimits and nite limits. For an ob ject
sheaf E of E , the set p E is also denoted E , and called the stalk of E at p.The
p
top os E is said to have enough points if these functors p , for all p oints p, are jointly
conservative see [9], p. 521, [1] . Almost all top oi arising in practice have enough
^
p oints. This applies in particular to the presheaf top os C on an arbitrary small
category C , and to the etale top os asso ciated to a scheme. In fact, any \coherent"
top os has enough p oints see Deligne, App endix to Exp os eVIin[1] .
We describ e a particular kind of top os with enough p oints. Recall that a groupoid
is a category in whicheach arrow is an isomorphism. Such a group oid is thus given
by a set X of ob jects, and a set G of arrows, together with structure maps
s
/
u m
o /
1
G G
G X:
X
?
/
071234
t
i
Here s and t denote the source and the target, u x 2 G is the identityat x 2 X ,
1
i g =g is the inverse, and m g; h =g h is the comp osition. A topological
groupoid is such a group oid in which X and G are each equipp ed with a top ology,
for which all the structure maps in 1 are continuous.
Given such a top ological group oid, a G{sheaf is a sheaf on X equipp ed with a
continuous G{action. Thus a G{sheaf consists of a lo cal homeomorphism p: E ! X
together with a continuous action map E G ! E , de ned for all e 2 E and
X x
g : y ! x in G, and denoted e; g 7! e g ; this map should satisfy the usual identities
for an action. 1
The category Sh X ofallsuch G{sheaves, and action preserving maps b etween
G
them, is a top os. It is called the classifying topos of the group oid G X .Such
a classifying top os always has enough p oints. In fact, any ordinary p oint x 2 X
de nes a p ointx: S!Sh X , by
G
1
x E =E = p x :
x
The collection of all these p ointsx is jointly conservative.
Our main aim is to provethatevery top os with enough p oints is, up to equiva-
lence, the classifying top os of some top ological group oid:
Theorem 1.1 Let E be any topos with enough points. There exists a topological
groupoid G X for which thereisanequivalence of topoi
Sh X : E
=
G
We end this intro ductory section with some comments on related work. Repre-
sentations of categories of sheaves by group oids go back to Grothendieck's Galois
theory [4] . In [8], a general theorem was proved, which is similar to our result,
and which states that for every top os E not necessarily with enough p oints there
is a group oid G X in the category of lo cales \p ointless spaces" for whichthere
Sh X . This theorem was sharp ened, again in the context is an equivalence E
=
G
of lo cales, in [7]. The basic idea for our construction comes from the latter pap er.
We wish to p ointout,however, that our result for top oi with enough p oints is not
a formal consequence of any of these theorems. Moreover, our pro of is di erent. The
pro ofs in [8] and [7] dep end essentially on change{of{base techniques, the internal
logic of a top os, and the b ehaviour of lo cales in this context. These techniques
cannot b e applied to the present situation. In fact, we b elieve that the pro of of our
theorem is much more accessible and direct.
2 Description of the group oid
Let E b e a top os with enough p oints. We recall the de nition of the space X = X
E
from [2], x2, and showthatitispartofagroupoid G X . First, although the
collection of all p oints of E is in general a prop er class, there will always b e a set of
p oints p for which the functors p are already jointly conservative [5], Corollary 7.17.
Fix such a set, and call its memb ers smal l points of E . Next, let S be an ob ject
n
of E with the prop erty that the sub ob jects of p owers of S , i.e., all sheaves B S
for n 0, together generate E .For example, S can b e the disjointsumofallthe
ob jects in some small site for E . Let I b e an in nite set, with cardinalitysolarge
that
card S card I
p
for all small p oints p of E . 2
In general, if A is any set with card A card I , wecallan enumeration of A
1
a function : D = dom ! A, where D I and a is in nite for each a 2 A.
These enumerations carry a natural top ology, whose basic op en sets are the sets
V = f j u g; 2
u
here u is any function fi ;:::;i g!A de ned on a nite subset of I , and u
1 n
means that i 2 dom and i =u i , for k =1;:::;n.Leaving the index set
k k k
I implicit, we denote this top ological space by
En A ;
and call it the enumeration space of A.
The space X ,involved in the group oid, is de ned by gluing several of these
enumeration spaces together. A p ointofX is an equivalence class of pairs p; ,
where p is a small p ointof E and 2 En S isanenumeration of the stalk S .
p p
0 0
Twosuch pairs p; and p ; are equivalent, i.e., de ne the same p ointof X ,
0 0
if there exists a natural isomorphism : p ! p for which = . Note that
S
0
for sucha , its comp onent is uniquely determined by and , b ecause is
S
surjective. In what follows, we will generally simply denote a p ointofX by p; ,
and we will not distinguish such pairs from their equivalence classes whenever we
can do so without causing p ossible confusion. The top ology on the space X is given
n
by the basic op en sets U , de ned for any i ;:::;i 2 I and any B S ,as
i ;:::;i ;B 1 n
n
1
U = f p; j i ;:::; i 2 B g: 3
i ;:::;i ;B 1 n p
n
1
0 0
Observe that this is well{de ned on equivalence classes; i.e., if p; p ;
0
0
, where we write by an isomorphism as ab ove, then i 2 B i i 2 B
p p
0
i = i ;:::; i and similarly for .
1 n
Next, we de ne the space G of arrows. The p oints of G are equivalence classes
of quintuples
p; ! q; ;
where p; and q; are p oints of X as ab ove, and : p ! q is a natural
isomorphism. Wedonot require that = . Twosuch p; ! q;
S
0
0 0 0 0
and p ; ! q ; represent the same p ointofG whenever there are isomor-
0 0 0 0
phisms : p ! p and : q ! q such that = and = ,
S S
0
while in addition = . The top ology on G is given by the basic op en sets
V = V de ned by
i ;:::;i ;B ;j ;:::;j ;C i;B ;j;C
n n
1 1
V = f p; ! q; j i 2 B ; j 2 C ; and i = j g:
i;B ;j;C p q
Here wehave again used the shorter notation i for i ;:::; i , etc. Note, as
1 n
ab ove, that these basic op en sets are well{de ned on equivalence classes. It remains
to de ne the structure maps x1 1 of the group oid. For an arrow g =[ p; !
q; ], its source and target are de ned by
s g = p; and t g = q; : 3
The maps s and t are well{de ned on equivalence classes, and are easily seen to
b e continuous for the top ologies on X and G as just de ned. For two arrows g =
0 0 0 0
[ p; ! q; ] and h =[ q ; ! r; ] for which[q; ]= [q ; ] as p oints of X ,
0 0
the comp osition h g in G is de ned as follows: since q; q ; , there is an
0 0
isomorphism : q ! q so that = . De ne h g to b e the equivalence class
S
of
S
p; ! r; :
It is easy to check that his de nition do es not dep end on the choice of ,isagain
well{de ned on equivalence classes, and is continuous for the given top ology on
G and the bred pro duct top ology on G G. Finally, the identity u: X ! G
X
id
and the inverse i: G ! G are the obvious maps u p; =[ p; ! p; ] and
1
i[ p; ! q; ] = [ q; ! p; ].
This completes the de nition of the top ological group oid G X .
3 Review of lo cally connected maps
Before we turn to some basic prop erties of the group oid G X ,we need to recall
some elementary prop erties of lo cally connected or \lo cally 0-acyclic" [10 ] maps
between top ological spaces. These prop erties are all analogous to well-known prop-
erties of lo cally connected maps of top oi. For spaces, however, the de nitions and
pro ofs are much simpler, and it seems worthwhile to give an indep endent presenta-
tion.
A continuous map f : Z ! Y of top ological spaces is called local ly connected l.c.
if f is an op en map, and Z has a basis of op en sets B with the prop erty that for
1
any y 2 Y , and any basic op en set B 2B, the bre B = f y \ B is connected
y
or empty.
Lemma 3.1 i The composition of two local ly connected maps is l.c.
ii In a pul lback square
0 /
Z
Z
0
f
f
/
0
Y;
Y
0
if f is l.c. then so is f .
iii Any local homeomorphism sheaf projection is l.c.
iv If the composition f = p g : Z ! E ! Y is l.c. and p: E ! Y is a local
homeomorphism then g is l.c.
Proof. These are all elementary.We just remark that for i , one rst proves
that for a l.c. f : Z ! Y with basis B as ab ove, f S \ B is connected for every op en
B 2B and any connected subset S f B . Then, if g : Y ! W is another l.c. map 4
1
with basis A for Y , the sets B \ f A , for B 2B and A 2Awith A f B , form
a basis for Z witnessing that g f is l.c. 2
Prop osition 3.2 For any l.c. map f : Z ! Y there exists a unique up to homeo-
morphism factorization
c
p
Z f ! Y; f = p c;
0
where p is a local homeomorphism and c is a l.c. map with connected bres.
Proof. We de ne the space f : the p oints are pairs y; C where y 2 Y and
0
1
C is a connected comp onentoff y . To de ne the top ology on f , let B be
0
the collection of all those op en sets B Z for which B is empty or connected
y
8y 2 Y . Then B is a basis for Z . The basic op en sets of f are now de ned to
0
b e the sets
B = f y; [B ] j y 2 f B g;
y
1 1
where [B ] is the connected comp onentof f y whichcontains B = f y \ B ,
y y
and B ranges over all elements of B .
To see that this is a basis, supp ose y; C 2 B \ A . Thus, A; B 2Band
C A \ B . Since C is connected, there is a chain of basic op en sets
y y
A = B ;B ;:::;B = B
0 1 n
in Z with the prop ertythat B \ B \ C 6= ; i =0;:::;n 1 . Nowlet
i i+1
n 1
T
1
D = B [[B \ f f B \ B :
0 n i i+1
i=0
Then D 2B,and y; C 2 D B \ A .
Now f : Z ! Y factors into a map c: Z ! f , c z = f z ; [z ] where [z ]is
0
1
the comp onentof f f z containing z , and a map p: f ! Y , p y; C =y .This
0
map p restricts to a homeomorphism B ! f B for each B 2B, hence is a lo cal
homeomorphism. Thus c is a lo cally connected map by 3.1 iv and the b ers of c
1 1
are evidently connected, since c y; C = C f y .
The uniqueness of this factorization is easy,andwe omit the pro of. 2
Corollary 3.3 i Let f : Z ! Y be a l.c. map. Then the pul lback functor of
sheaves
f :Sh Y ! Sh Z
has a left adjoint f :Sh Z ! Sh Y .
!
ii For any pul lback square of topological spaces
b
/ 0
Z
Z
0
f
f
0 /
Y
Y
a
0
and f l.c. the projection formula a f = f b holds.
!
! 5
One also says, that the \Beck{Chevalley condition" holds for this square.
Proof. In the pro of, weidentify the category of sheaves E over Z with that of
lo cal homeomorphisms p: E ! Z .
Ad i . For a lo cal homeomorphism p: E ! Z , the comp osite f p is l.c. by
Lemma 3.1 i and iii , so factors uniquely as f p = E f p ! Y asin
0
Prop osition 3.2. De ne f E to b e the sheaf f p ! Y .Thus, by construction,
! 0
1
the stalk of f E aty is the set of connected comp onents of f p y ,
!
1
f E = fp y : 4
! y 0
For adjointness, let q : F ! Y b e a sheaf on Y ,andlet : E ! f F = F Z be
Y
a map. Then = : E ! F isamapover Y ,i.e.,q = f p. Since the bres
1
1
of q are discrete, is constant on the connected comp onents of each bre fp y ,
hence factors uniquely as a map f E ! F .
!
Ad ii . For a sheaf E on Z , the adjointness of part i provides a canonical map
0
f b E ! a f E . It follows from 4 that this map is an isomorphism on each
!
!
stalk, hence an isomorphism of sheaves. 2
4 Prop erties of the group oid
In this section, we present some prop erties of the group oid G X , de ned in x2,
which will enter into the pro of of the theorem, to b e given in the next section.
Before going into this, we recall from [2] that the enumeration spaces En A
describ ed in x2 are all connected and lo cally connected; in fact, each basic op en set
of the form V is connected. For a small p oint p: S!E,theenumeration space
u
En S iscontained in our space X , via the obvious map
p
i :En S ! X; i = p; :
p p p
This map is a continuous injection but not necessarily an emb edding .
Prop osition 4.1 The bres of the source and target maps s; t: G ! X areenu-
1
meration spaces: for a point p; 2 X , thereare homeomorphisms s p;
=
1
t p; . En S
=
p
Proof. It suces to prove this for the source map. Fix p; 2 X . Note rst
id
1
that any p oint p; ! q; in G is equivalent to the p oint p; ! p; ;
S
in other words, eachequivalence class has a representation of the form
id
p; ! p; : 5
id
Thus, the evident map j :En S ! G de ned by 7! [ p; ! p; ] is a
p
p;
1
bijection into s p; . Now consider a basic op en set V in G as describ ed in x2, of
the form
V = f p; ! q; j i 2 B ; j 2 C ; i = j g:
p q 6
Representing equivalence classes in the form 5 , we can also write
id
V = f p; ! p; j i 2 B ; j 2 C ; i = j g:
p q
Thus, for p; xed, and for s = i ,:::, s = i , we see that
1 1 n n
1
V =f 2 En S j s = i ;:::;s = i g; 6 j
p 1 1 n n
p;
if s ;:::;s 2 B \ C , and empty otherwise. But the right hand side of 6 exactly
1 n p p
describ es a standard basic op en set x2 2 in the enumeration space En S . 2
p
Prop osition 4.2 The source and target maps arelocal ly connected.
Proof. Again, it suces to do one of the two, say the source map. Consider a
basic op en set V G as in the previous pro of,
id
V = f p; ! p; j i 2 B ; j 2 C ; i = j g:
p q
Then s V = f p; j i 2 B and 9 2 En S i = j 2 C g = f p; j
p p p
i 2 B \ C g, and this is the union of basic op en sets V in X , where u ranges over
p p u
all partial functions u: fi ;:::;i g!S with u i = u i ;:::;u i 2 B \ C .
1 n p 1 n p p
Thus s is an op en map. Furthermore, the pro of of Prop osition 4.1 shows that under
1 1
s p; , the set V \ s p; corresp onds to a standard the identi cation En S
=
p
1
basic op en set in En S . In particular, V \ s p; is connected. This shows that
p
s is a lo cally connected map. 2
Next, we recall from [2] that there is a top os morphism
':Sh X !E;
describ ed at the level of the stalks by
' E = E ;
p; p
for each p oint p; 2 X . The following lemmawas proved in [2]:
Lemma 4.3 The functor ' : E!Sh X has a left adjoint ' . Furthermore, for
!
each smal l point p: S!E, thereisacommutative up to isomorphism square
i
p
/
Sh X Sh En S
p
'
p
/
S E
for which the projection formula
i = p '
! !
p
holds.
Here S = Sets =Sh pt , while and i areinduced by the continuous maps of
p
i
p
! X . spaces pt En S
p 7
Prop osition 4.4 The source and target maps s; t: G X t into a square of topos
morphisms
t
/
Sh X Sh G
'
s
'
/
Sh X
E
t ' . Moreover, the which commutes up to a canonical isomorphism : s '
=
projection formula holds for this square, i.e., the induced natural transformation
: s t ! ' '
! !
is an isomorphism.
Proof. For a p oint g =[ p; ! q; ] of G, and for any ob ject E of E ,we
have
s ' E = E ;
g p
t ' E = E ;
g q
and the stalk of the isomorphism : s ' E ! t ' E atthepoint g is de ned to
E
b e the isomorphism : E ! E . It is easy to checkthat is continuous, using
E p q E
the explicit description of the top ology on ' E given in [2].
Next, weprove for eachsheaf F on X that s t F =' ' F or more precisely,
! !
that the canonical map s t F ! ' ' F is an isomorphism. It suces to check
! !
that s t F = ' ' F for the stalks at an arbitrary p oint x = p; inX .
! x ! x
Consider for this the diagram
j
p;
t
/ /
Sh En S
Sh G Sh X
p
7
'
s
'
x
/ /
Sh X
S E
Here the left-hand square comes from a pullback of top ological spaces Prop osi-
tion 4.1 , and j is as de ned in the pro of of 4.1. Since s is lo cally connected by
p;
Prop osition 4.2, Corollary 3.3 ii gives the pro jection formula
8 x s = j
! !
p;
for the left hand square in 7 . Moreover, since t j = i and ' x = p, Lemma 4.3
p
p;
gives that
' x ' = t j 9
! ! p;
for the comp osed rectangle. Thus
s t F = x s t F
! x !
= j t F by 8
!
p;
= x ' ' F by 9
!
' F ; = '
! x
2 8
5 Pro of of the theorem
We will now prove the theorem stated in x1, and rep eated here in the following form.
Theorem 5.1 The functor ' : E!Sh X induces an equivalenceofcategories
E Sh X .
=
G
Proof. By de nition of the set of small p oints p of E , the functor ' is faithful.
It follows that ' induces an equivalence b etween E and the category of coalgebras
for the comonad ' ' on Sh X see e.g. [9] . By standard category theory [3, 9]
the latter category is in turn equivalent to that of algebras for the monad ' '
!
on Sh X . Thus,toprove the theorem, it suces to show that for any sheaf F
on X , algebra structures
: ' ' F ! F 10
!
are in bijective corresp ondence to group oid actions
: F G ! F: 11
X
By the pro jection formula ' ' F = s t F , maps as in 10 corresp ond to maps
! !
s t F ! F over X , and hence to maps ~: t F ! s F over G, since s is left
! !
adjointto s . By comp osing with the pro jection s F = G F ! F , these maps
X
~ corresp ond to maps
: F G = s F ! F:
X
For an arrow g in G andapoint 2 F ,we write
s q
g = ; g : 12
def
To prove of the theorem, it now suces to verify that in 10 satis es the unit
and asso ciativity axioms for an algebra structure if and only if the corresp onding
multiplication 12 satis es the unit and asso ciativitylaws for an action.
To this end we rst make the corresp ondence b etween 10 and 12 more explicit:
For a p oint p; 2 X ,wehaveby 4.3 and 4.4,
' ' F = i F
! !
p;
p
F . = the set of connected comp onents of i
p
So for a p oint p; 2 X ,any 2 F de nes a connected comp onent[ ] 2 i F ,
p;
p
and [ ] then de nes a p ointinF . Nowletg be anyarrowinG. Since
p; p;
id
p; ! q; is equivalent to i.e., de nes the same p ointof G as p; !
p; wemay represent g in the form
S
id
g =[ p; ! p; ]:
Then, for 2 F , the action 12 is de ned from by
p;
g = [ ] : 13
p; 9
For , the laws for an algebra structure assert that for any 2 F ,
p;
[ ] = 14
p;
[ [ ] ] = [ ] : 15
p; p; p;
id
But clearly, 14 states that 1= , where 1 is the identity arrow[ p; ! p; ]
in G, while 15 states that h g = h g , where g and h are the arrows in
G represented by
id id
[ p; ! p; ] and [ p; ! p; ];
resp ectively. Since any comp osable pair of arrows in G can b e represented in this
form, 15 is equivalent to the asso ciativity condition for the action by G on F , and
the theorem is proved. 2
6 The action by the group Aut I
We conclude this pap er with some remarks on the action by the group H =Aut I
of bijections : I ! I , from the set I to itself. There is a natural continuous action
of H on the space X , de ned explicitly by
p; = p; ;
and whichiswell{de ned on equivalence classes.
Let Sh X denote the top os of H {equivariantsheaves on X .Weobserve rst
H
that each sheaf ' E onX carries a natural action by H , so that from ' one obtains
a top os morphism :
:Sh X !E:
H
Explicitly, E is the same sheaf on X as ' E ,
E = f p; ; e j p; 2 X; e 2 E g
p
and H acts on E by acting trivially in the e{co ordinate.
We will prove the following prop osition:
Prop osition 6.1 The morphism :Sh X !E has the fol lowing properties:
H
i is ful l and faithful.
ii commutes with exponentials, i.e., the canonical map
E E
F ! F
is an isomorphism, for any two sheaves E and F in E . 10
iii is bijective on subsheaves, i.e., for any sheaf E in E , induces an iso-
morphism
Sub E ! Sub E :
E H
Here Sub E is the set of subsheaves of E in E , while Sub E is the set
E H
of H {invariant subsheaves of E .
Prop erty i is actually a consequence of prop erty iii . Using standard terminol-
ogy [6] , i expresses that is connected, iii that it is hyp erconnected. Since any
hyp erconnected in fact, any op en morphism with prop erty ii is lo cally connected,
we can rephrase Prop osition 6.1 as
Corollary 6.2 The morphism :Sh X !E is local ly connected and hypercon-
H
nected.
Note that Sh X isan etendue [1] . Thus any top os with enough p oints
H
admits a lo cally connected hyp erconnected cover from an etendue. This is related
to a result of [12], stating that any top os not necessarily with enough p oints admits
ahyp erconnected morphism from a \lo calic" etendue.
Proof of Proposition 6.1. As said, i is a consequence of iii . For ii , note rst
that ' naturally factors as
/
Sh X Sh X
H
L
L
L
L
L
L
L
L
'
L
L
L
E :
The inverse image simply `forgets' the H {action. In particular, is conservative
i.e., re ects isomorphisms . Moreover, since H is a discrete group, preserves
exp onentials. In fact, is an `atomic' morphism. To showthat preserves
exp onentials, it therefore suces to showthat' do es. We prove this in a separate
lemma.
Lemma 6.3 The functor ' : E!Sh X preserves exponentials.
Proof. This follows from Theorem 5.1 and [11], Theorem 3.6 b , b ecause
s; t: G X are lo cally connected. Alternatively, from the explicit description
of ' in [2] together with Lemma 4.3, one easily checks that the canonical map
!
' S ' E ! ' S E is an isomorphism for eachsheaf S on X and each E 2E.
! !
It then follows in the standard wayby adjointness that ' preserves exp onentials.
2
We return to Prop osition 6.1, and prove part iii . Let E 2E, and let S ' E
b e an arbitrary subsheaf. WehavetoshowthatifS is H {invariant, then S = ' U
for a necessarily unique subsheaf U E . By Theorem 5.1, it suces to prove that
if S is H {invariant then it is also G{invariant. To this end, consider any arrow g in G.
id
As in the the pro of of 5.1, it can b e represented in the form g =[ p; ! p; ]. 11
Let e 2 E and assume that s = p; ; e 2 S E ' E .Wehave to show
p p
p; p;
that s g = p; ; e 2 S . Cho ose a section : U ! S through s,de nedon
i;C
p;
a basic op en neighb ourho o d U of p; . By the description of the top ology on
i;C
' E in [2], wemayassumethat is of the form q; =f i where f : C ! E
q
in E . In particular, s = p; = f i .
p
Nowcho ose 2 H so that i = i . Then p; i 2 U ,so
i;C
p; ; e = p; 2 S .Byinvariance of S under the action by , also
p;
p; ; e 2 S ,aswas to b e shown. 2
p;
Acknowledgements. The research is part of a pro ject funded by the Netherlands
Organization for Scienti c ResearchNWO.
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