The locally connected coclosure of a Grothendieck topos

JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied

The locally connected

Algebra

137 (1999)

coclosure

17-27

of a Grothendieck

topos

J. Funk* Department

of Mathematics

Communicated

and Statistics. McGill University, 805 Sherbrooke Quebec, Canada H3A ZK6

by M. Barr; received

10 February

1996; received

Street

West, Montreal,

in revised form 26 August

1997

Abstract Locally connected Grothendieck toposes are shown to be coreflective in Grothendieck toposes. We refer to this coreflection as locally connected coclosure. The locally connected coclosure of a localic topos is localic. A topos and its locally connected coclosure are seen to have equivalent categories coclosures

of Lawvere distributions. A class of locales is produced whose locally connected are trivial. We show how the locally connected coclosure can be described in terms

of the locally connected coclosure of a localic cover. @ 1999 Elsevier Science B.V. All rights reserved. 1991

Math.

Subj. Class.: 18B25

0. Introduction The full subcategory connected

of topological

is a coreflective

the locally

connected

as the original

space,

coclosure

This coreflection,

of a topological

but topologized

finer than the given one (this topology example,

spaces determined

subcategory.

by those which are locally which we shall refer to as

space, has the same underlying

with the smallest

locally

connected

set

topology

always exists - see [6] for further details).

For

the space

‘3

* E-mail: ~nk~scylla.math.mcgiIl.ca. 0022-4049/99/$ ~ see front matter @ 1999 Elsevier PII: SOO22-4049(97)00168-O

Science B.V. All rights reserved

(1)

18

J. FunklJournal

topologized

as a subspace

nected coclosure

of Pure and Applied

Algebra

137 (1999)

17-27

of the real plane, is not locally connected.

Its locally con-

is the coproduct

in the category of topological spaces. (ProoJ Let X denote the given space (1) with locally connected coclosure 2 (which always exists), and let Y denote the coproduct (2). Since Y is locally connected, P

y\P

there is a commutative

diagram

,x

J

&X

X

where i is a continuous bijection and where (the underlying map of) .a~ is the identity. To see that i, is an open map observe that the collection of sets {p-lb

1b is a quasi-component

is a base for Y. (In general,

(= component)

quasi-components

of an open of X}

determine

a coarser partition

than do

components, but in this case they agree.) This collection is a base because it is equal to the collection {c 1c is a component of an open of Y}. Since $ is a bijection, and sets of the form c;‘b, where b is a quasi-component of an open j(p-‘b)=c,‘b, set, are always open in the locally connected coclosure ([6, Proposition 5.111). Thus, Y is homeomorphic to 2.) Let & denote an arbitrary Grothendieck topos over Set (henceforth, ‘topos’ will always mean ‘Grothendieck topos’). The notion of a locally connected topos is well understood [2]. By definition, a geometric morphism 8 + 9 is locally connected if its inverse image hmctor has an p-indexed left adjoint. Then a topos is locally connected if its unique structure morphism to Set is locally connected. In this case, the left adjoint (if it exists) is automatically Set-indexed, so that a topos 8 is locally connected iff the inverse image function

A : Set --) d has a left adjoint. If 8 is locally connected, we denote this left adjoint by ~0. The locally connected coclosure of a topos 8 is, by definition, a locally connected topos C? equipped with a geometric morphism ~8 : lf-+ 6 such that for a locally connected topos 9, composition with E& gives an equivalence of categories of geometric morphisms GTop(P,& and GTop(F,c?). In this paper, the locally connected coclosure of an arbitrary topos is shown to exist. This is done by recognizing a connection with distributions in a topos, and with the theory of locally presentable categories. Lawvere [l l] has proposed the notion of a distribution in a topos 6, meaning a colimit preserving functor 6+ Set. If d is locally connected, then the left adjoint rco is a distribution. Moreover, 7~0 is stably terminal amongst all distributions, i.e., rco is the terminal Set-valued distribution, and furthermore, for a topos 9, the composite functor Ay 7~0 is the terminal g-valued distribution (this property characterizes

J. Funk1 Journal of Pure and Applied Algebra 137 (1999)

local connectedness distributions

[5, Theorem

3.101). By the work of Ulmer

in a topos is locally

there exists the terminal cally connected

presentable,

distribution

consequently

19

17-27

[ 141, the category

complete.

of

In particular,

from which we deduce the existence

of the lo-

coclosure.

The category of distributions in a topos is equivalent to the category of complete spreads over the topos [5]. Through this interaction with complete spreads we show in Section

2 that the symmetric

monad

[4, 51, the topos classifier

of distributions,

commutes with the locally connected coclosure. Consequently, a topos and its locally connected coclosure have equivalent categories of distributions. We also use complete spreads to show that if X is a complete metric space, then ?%3) ash, where 2 is the coclosure of X as a space, as described in the first paragraph. Unlike the spatial case, the locally connected coclosure of a topos can be trivial (Proposition 2.4). Our methods reveal that the locally connected coclosure of a locale is a locale. In Section 3, we take the opposite point of view and obtain a description of 6 in terms of a localic cover of 8. That is done as a typical application of descent theory [ 131. The procedure, which is well known [lo], consists of choosing a suitable localic cover X 4 8 which furthermore is of effective descent for toposes and geometric morphisms. Then d is obtained by ‘descending’ 2 +X.

1. Existence The calculations given here are based on the following result due to Ulmer [14]. We also refer the reader to Bird’s thesis [3]. For the definitions and the facts presented in Proposition 1.2 concerning the theory of locally presentable categories, the reader may wish to consult [l]. In the discussion to follow, the symbol LXwill always denote a regular cardinal. Theorem 1.1 (F. Ulmer).

Let cp5

$1 d -+ 93 denote a natural transformation be-

tween cocontinuous functors between locally u-presentable categories. If cp and $ preserve u-presentable objects, then the inverter of t, as constructed in categories, is a locally a-presentable category, and its inclusion into JZZis cocontinuous and preserves cc-presentable objects. Proposition 1.2. 1. Let D = (DO,01) denote a small category, and let P be an arbitrary object of the functor category SetD. Assume IDo1
SetD denote an arbitrary functor.

Then its left extension SetC +

SetD preserves a-presentable objects ifs for every object c E CO, Fc is a-presentable. We denote by DB the category of distributions in the topos 8. The fact that Db is locally presentable can be found in the aforementioned works of Ulmer and of Bird. We present here a calculation of the rank of presentability.

20

J. Funk I Journal of Pure and Applied Algebra

Proposition

137 (1999)

17-27

1.3. Let d denote a topos, with site (C,J).

We denote the set of covers of an object c by Jc, and the set of morphisms in C with codomain c by (C/c),. If

~~J~W+~

(1)

then D6 is locally u-presentable. Proof. Consider the topology J regarded as a presheaf on C, hence as a discrete fibration J -+ C. The set of objects Jo of J is the collection of all pairs (c,R) such that

R E Jc. A Set-valued mnctor on C which carries covers to colimiting cones is called a cosheaf: The category of cosheaves on C is thus the inverter in categories of the inclusion of cocontinuous functors cp L II/ : SetC --f SetJ such that Il/(d)(c, R) = C(d, c), 3 c 1 m E R}, and td(c, R) is the inclusion of the latter into the former. cp(d)(c,R) = {d Then for d E CO, we have

JO

CO

which when summed

over CO gives

flu $(d)(c,W2 fl Co JO

Jc

x (Clc)o.

co

If (1) holds, then

so that by Proposition

1.2, 11/preserves

a-presentable objects. The categories so that by Theorem 1.1, the category

cc-presentable

objects.

Similarly,

q preserves

SetC and SetJ are locally finitely presentable, of cosheaves

on C is locally a-presentable,

its inclusion into SetC is cocontinuous and preserves a-presentable objects. known that D6 is equivalent to the category of cosheaves on C. 0 The 2-category

of locales

[8, 121 is regarded

associating with a locale its topos of sheaves. 24mctor is referred to as localic.

and

It is well

as a full sub-2-category

of GTop by

A topos in the essential

image of this

Theorem 1.4. The locally connected coclosure of an arbitrary Grothendieck topos

exists. The counit of this corejlection is the terminal complete spread (with locally connected domain). The counit is a localic geometric morphism, and the locally connected coclosure of a localic topos is localic. Proof. Let d denote an arbitrary topos. Since DG is locally presentable, it is complete. In particular, there exists the terminal distribution T in G. Let C denote a

J. Funk/ Journal of Pure and Applied Algebra

small generating

full subcategory

which can be regarded

of E. The restriction

as discrete opfibration

137 (1999)

17-27

of T to C is a functor C t

21

Set

U : T --t C. We define ~8 : 8 + B as the

pullback

0

I I ”

Ik----+Set

C"P

where u is the geometric morphism induced by U (u* is pullback along U). L! is a locally connected topos, and ~8 is terminal amongst geometric morphisms F 4 8, with 9 locally connected ([5, Propositions 2.10 and 2.111). This proves the first statement of the theorem. Discrete opfibrations are faithful, so that the geometric morphism ~8 is localic as a faithful functor A + B induces a localic geometric morphism Set*‘” --f SetBop, and since localic geometric morphisms are pullback stable ([7, Propositions 3.1 and 2.11). Alternatively, one can argue that sg is a complete spread and that spreads are localic ([5, Propositions 2.10 and 1.31). Then the coclosure of a localic topos is localic since localic geometric

morphisms

are closed under composition

[7].

0

Remark 1.5. The topos 8 can be constructed

also as follows. The terminal distribution T : 8 + Set corresponds to a point T : Set +Md of the symmetric topos, and there is a bicomma object of toposes (see [5])

where 6 is the canonical essential inclusion. A result due to A.M. Pitts says that since 6 is essential, 6 is locally connected (see [5]). That 2 regarded this way has the desired universal property is immediate. Indeed, if g is a locally connected topos, and 9 -% B is a geometric morphism, then there is a unique natural transformation cp* .6* - Acf. T* obtained by adjointness from the unique where rra -I A/q and 61 -1 6*.

natural

transformation

rco . cp* -+ T? T* . S!,

2. Complete spreads We have two applications of the theory of complete spreads to the locally connected coclosure. The first application will be to show that the symmetric monad commutes

J. Funk1 Journal of Pure and Applied Algebra 137 (1999) 17-27

22

with the locally connected

coclosure.

We will use the following

facts concerning

com-

plete spreads.

Proposition 2.1. Let Yf LB L d denote geometric morphisms topos Y. Assume that % and 9 are locally connected (over 9). 1. If 4 and $ are complete spreads, then so is $ . 4. 2. If $ . 4 and $ are complete spreads, then so is 4. Proof. A proof of these facts for locales is given in [6, Proposition can be interpreted

for toposes.

over a base

4.81. That proof

Cl

The symmetric monad originally arose as the topos classifier of distributions [4]. It can also be regarded as the topos classifier of complete spreads with locally connected domain [5]. This means that for a topos b, the category of complete spreads over X x d with locally connected X-domain is naturally equivalent to GTop(Z,Md), where M denotes the symmetric Z-valued distributions

monad and X is an arbitrary in a topos &’ by ox(&).

topos. We denote the category of

Corollary 2.2. Let 3 denote a locally connected topos. Then composition with the inverse image functor of the geometric morphism &g: 8 -+ 6 induces a natural equivalence D,(d) rv Dy(&). We have M-8 ?Mk. Proof. Change of base topos preserves

complete spreads with locally connected domain [5]. Therefore, changing base along 9 -+ Set produces a complete spread 9 x d -+ 3 x d with locally connected g-domain. Furthermore, since we are assuming that 9 is locally connected, Y x k---f 93 x & is the terminal such complete spread. Thus, by Proposition 2.1, the category of complete spreads over 9 x & with locally connected g-domain

is canonically

with locally connected equivalences

GTop(9,Mk)

equivalent g-domain.

to the category

of complete

For locally connected

spreads

$9, we therefore

over 9 x k have natural

N GTop(Y,M6) z GTop($%%).

The first equivalence above shows that Dg (2) rv Dg( 6’). As 2 is locally connected, is Mk [5], and of course so is E. Therefore, z EM&. 0

so

For our second application of the theory of complete spreads to the locally connected coclosure, let lot: TSP+LOC denote the functor which carries a topological space X to the locale whose corresponding frame is the lattice of open subsets of X. Let 1. / denote the right adjoint of lot, i.e., 1. 1 is the ‘points’ ftmctor. Recall [8, 121 that a locale Y is said to be spatial if the counit ZoclY] --f Y is an isomorphism.

J. Funk1 Journal of Pure and Applied Algebra 137 (1999)

23

17-27

Proposition 2.3. Let X denote an arbitrary complete metric space. Then IlFXl-_f and l&?X G 10~2. So S@?)FSh(k). (Here, 2 is the spatial locally connected coclosure.) Proof. The local morphism recall Theorem

EX: l&k

+ 1ocX is a complete

6.1 of [6] which asserts that the domain

spread

of a complete

(1.4).

We now

spread over a

complete metric space is a spatial locale. Therefore, lz is spatial, so that ]&%I is locally connected, whence homeomorphic to 2. Then it follows that la ” loci, and that Sh^(x)-Sh(k).

0

The locally connected coclosure can be trivial. We conclude this section by producing a class of locales whose locally connected coclosures are trivial. (The frame corresponding to the trivial locale is the single element lattice.) First, observe that a locale with trivial locally connected coclosure could have no points. Let Y denote an arbitrary TI topological space. There is a sublocale Y, of Y such that Yk is the largest sublocale of Y with no points. The frame O(Yk) is the lattice of opens U of Y such that Y - U has no isolated points [12, p. 5241. When Y is Tt, the sublocale Yk also has the property that if it is trivial, then Y is scattered (i.e., the isolated points of every closed subspace are dense in that subspace). Proposition 2.4. 1. Let Y denote a totally disconnected topological space. Then Y, has no non-trivial connected sublocales. 2. Let Y denote a T1, totally disconnected space which is not scattered (examples of such spaces are easily produced). Then Fk is trivial (and Yk is non-trivial). Proof.

1. Let S denote a connected

sublocale

of Yk. To be shown is that S is trivial.

The closure S in Y is spatial since closed sublocales of a space are spatial. But S is also connected in Y, and therefore must be a singleton since Y is assumed to be totally disconnected. The only possibility left is that S is trivial because Yk has no points. 2. If ?jj is non-trivial, then let C denote a connected component of Fk. The image of C in Yk must be a non-trivial connected sublocale of Yk. That contradicts 1. 0

3. Calculating

the locally connected coclosure

In this section we show how calculating the locally connected coclosure can be reduced to the localic case. We will use descent theory for toposes in order to obtain a description of the locally connected coclosure of a Grothendieck topos in terms of that of a suitable localic cover of the given topos. If a geometric morphism F-8 is of efictive descent for toposes and geometric morphisms (in the 2-categorical sense), then in particular, the comparison 2-iimctor is

24

J. FunklJournal

(pseudo)

of Pure and Applied

fully faithful. This is equivalent

Algebra

137 (1999)

17-27

to asserting that whenever

we have a pullback

(2)

in GTop, the diagram 02 x 5 xx&Fxg.P-

PI2

6

, +~x&F4.~~~

(3)

+

+ 6

PI3

is a ‘2-coequalizer’ in GTop, i.e., it is universal for .%?+X which satisfy a unit and cocycle condition, so that Y is a ‘descent’ topos, i.e., its objects (morphisms) are those X6X equipped (commuting) with descent data 0:X” 0,*X. Here, 01 and 92 arise in the diagram

in which every face is a pullback. is the diagonal.

The morphisms

pt2 and pt3 are projections,

and 6

It is well known [ 10, 131 that geometric morphisms which are open surjections are of effective descent in the sense outlined above, so that whenever we have a pullback (2) with f an open surjection, (3) is a 2-coequalizer. For our present purposes, we will require

only the weaker

and more easily

surjections are of effective descent (locally We will also use the following. Lemma 3.1. If 9 L&T

is a pullback in GTop.

established

connected

fact that locally

geometric

morphisms

connected are open).

is a locally connected geometric morphism of toposes, then

J. Funk1 Journal of Pure and Applied Algebra

Proof. The pullback

137 (1999)

17-27

25

B in

is a locally connected topos since locally connected geometric morphisms are stable under pullback and closed under composition. Therefore, PI factors essentially uniquely through E.F, say p I E E,F. q. On the other hand, we have a universal morphism .& 5 .Y arising from f and e,p. It follows that qp g l.+ and pq ” 1,~. 0

A localic cover of a topos d is a geometric morphism X .‘& for which X is localic and f is a surjection. (In what follows, we denote with the same symbol a locale and its topos of sheaves.) That for an arbitrary topos a localic cover can be constructed is well known [lo], but we are interested also in the fact that f can be taken to be locally connected [9]. An explicit description of 8 can be given in terms of a localic cover X .f

d for which f is locally connected.

Since such a cover is of

effective descent, by specializing (3) with %‘Ld as 8 Lc? and F as X, we see that 8 can be described [lo] as the category whose objects are all pairs (EAX,cr), where

e is an &tale map of locales,

and Xl xxE A

E is an action

of (Xl ,X)

on

E AX. Here XI denotes the locale X x 6 X. Morphisms (E AX, a) + (F AX, ti) in this category are morphisms E + F which commute with the structure and action morphisms. The following theorem describes in a similar fashion the topos & as the category of those L -+X (not necessarily &tale) which are locally homeomorphic to 8 over X (so that L is a locally connected locale), and which are equipped with an action of (Xl, X).

Theorem 3.2. Let 6’ denote an arbitrary topos. Let X f B be a localic cover of B ,fbr which f is locally connected. Then 8 can be described as follows. Objects of 2: Pairs (L&X,0), where L is a locally connected locale, such that the unique factorization 6: L +k is &tale, and where a is an action of (X1,X) on LLX. a) + (K AX, IC) is a locale morphism Morphisms of 8: A morphism m: (L AX, L AK commuting with the structure morphisms and with the actions. The counit ~8 : C#+ 6: The inverse image functor 8: carries an object (E AX, to the pair (I? 5 E *X,a’), where the action a’ is the composite

X, xx,?“X,

xxEAE.

(r)

^n

(4)

26

J. Funk1 Journal of Pure and Applied Algebra

Proof.

It follows from Lemma 3.1 (applied

to X L

137 (1999)

17-27

8) and the remarks preceding

it

that Q2xx jx&Yxgx

6

p12

2 X&X 82 *

i

R L

2

01

PI3

is a 2-coequalizer in GTop. The description of b in the statement of the theorem follows from this. Concerning the counit ~8 : b -+ 6, first note that by Lemma 3.1 the square

is a pullback since e is locally connected (which is so since e is Ctale). Therefore, e^is Ctale, i.e., (bf-+E AX, a’) is an object of 8. Regarding the action D’, the canonical isomorphism Xi XX 8 E’xi FX E in (4) arises by Lemma 3.1 and the following pullback squares. * Xl xxE-+Xl

I E-E-X

!“I EE

In this diagram, nected f.

xxE-Xl

I e

81 is locally

connected

as it is a pullback

of the locally

con-

q

Acknowledgements The author has benefitted from and is grateful for conversations subject with Marta Bunge, Bill Boshuck and with Mamuka Jibladze.

concerning

this

References [I] J. Adamek, J. Rosicky, Locally Presentable and Accessible Categories, London Mathematical Society Lecture Notes Series, vol. 189, Cambridge University Press, Cambridge, 1994. [2] M. Barr, R. Pare, Molecular toposes, J. Pure Appl. Algebra 17 (1980) 127-152. [3] G.J. Bird, Limits in 2-categories of locally presentable categories, Ph.D. Thesis, University of Sydney, 1984. [4] M. Bunge, Cosheaves and distributions on toposes, Algebra Universalis 34 (1995) 469-484. [5] M. Bunge, J. Funk, Spreads and the symmetric topos, J. Pure Appl. Algebra 113 (1996) l-38.

J. Funk I Journal of Pure and Applied Algebra 137 (1999)

17-27

21

(61 J. Funk, The display locale of a cosheaf, Cahiers de Top. et Geom. Differential Categoriques 36 (I ) (1995) 53-93. [7] P.T. Johnstone, Factorization theorems for geometric morphisms I, Cahiers de Top et Geom. Differential Catigoriques 22 (1) (1981) 3-17. [8] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [9] P.T. Johnstone, How general is a generalized space?, in: I.M. James, E.H. Kronheimer (Eds.), Aspects of Topology, London Mathematical Society, Lecture Notes Series, vol. 93, Cambridge University Press, Cambridge, 1985, pp. 77-l 1 [lo] A. Joyal,