C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
P ANAGIS K ARAZERIS
Compact topologies on locally presentable categories
Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no3 (1997), p. 227-255
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© Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés.
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COMPACT TOPOLOGIES ON LOCALLY
PRESENTABLE CATEGORIES
by Panagis
KARAZERISCAHIER DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
J Illume XXXVIII-3 (1997)
RESUME. Les
topologies
sur lescategories
localementpr6sentables g6n6rallsent
les notions famill6ressuivantes,
d’unepart
lestopologies
de Grothendieck sur despetites categories,
d’autrepart
lestopologies
de Gabriel sur descategories
abéliennes àgenerateurs.
Dans cet article on introduit unecondition, qui peut
être v6rifi6e pour les
topologies pr6c6dentes, appel6e "compacit6".
Dans le cas des
topologies
deGrothendieck,
cette conditionsignifie qu’un
recouvrementquelconque
a un sous-recouvrementfini. Les
topologies compactes correspondantes
ont des localisations ferm6es dans lacat6gorie
donnee pour des colimites filtrantesmonomorphiques.
On examine aussi la fermeture des
objets s6par6s
et desfaisceaux pour les colimites filtrantes. Les
topologies compactes
surune
cat6gorie
localement depresentation
finie forment un locale. Sicette
cat6gorie
est untopos coherent,
alors le locale est compact etlocalement compact. Si le topos est celui des faisceaux sur un
espace
coherent,
alors le locale enquestion
est celui des ouverts pour latopologie
recoll6e("path topology")
sur cet espace.1 Introduction
By
atopology
on alocally presentable category
we mean a universalclosure
operator
on such acategory:
Basic Definition A
topology
on acategory
withpullbacks
is aprocess, which associates to each
subobject
S >-> Aof
anobject A,
another
subobject jA(S)
>-> Af-1(jA(S)
=jB(f- 1(S)), for
all arrowsf:
B -+ A in thecategory
and
f-1 denoting pulling
backalong f.
This notion has become a standard
category
theoretic tool since the1970’s,
but someexamplification
of it may still beenlightening:
Tostart with a somewhat trivial case, when the
category
inquestion
isa lattice
A,
atopology
on it isnothing
else but a nucleus(as
in[10],
p.
48). Indeed,
there is a one to onecorrespondence
between thetwo, sending
anucleus j
on A to the closureoperator which,
to a bassigns jb(a)
=j(a)
A band,
on the otherhand, sending
a universal closureoperator
to the nucleus that has the closure of a in thetop
element ofA as value on a E A.
On the
category
of abelian groups we may consider thetopology,
which
assigns
to asubgroup
H of G the closurej(H) = {g
E GI
3n EN s.t n.g E
H},
i.e the elements of G that are torsion withrespect
to H. Thisexample
may be heldresponsible
for the term "torsiontheory"
used in
ring theory
to describetopologies
onmodule,
or moregenerally,
abelian
categories.
It alsoprovides
for onepossible justification
for theuse of the word
topology
in thissetting:
Such apiece
of data renders thering
over which we consider the moduleslinearly topological.
The otherand
probably
more burdensomejustification
is the use of the wordby
Grothendieck in
generalizing
the notion ofcovering family
fromtopolog-
ical spaces to small
categories,
in view of thecorrespondence
betweenuniversal closure
operators
onpresheaf categories
and suchcovering
families on the
indexing category, leading
to thedevelopement
oftopos
theory. 1
In a manner similar to that of
topos theory,
universal closure op- erators on alocally a-presentable (I.a - p) category correspond
to en-domorphisms, satisfying
the Lawvere-Tierney axioms,
of anobject
Qthat serves as a
subobject
classifier for thecategory ([6]).
Weknow, by
thegeneral theory
of thesecategories given
in[9],
that such an £ isequivalent
to acategory
of those ,S’et-valued functors from a smallcategory
C with colimits of size less than a(a
aregular cardinal)
thatpreserve the colimits in
question.
Theobject Q, then,
lives in the fullpresheaf category SetCoP.
When thecategory happens
to be atopos,
theobject Q
is thesubobject
classifier for thetopos.
A universal closure
operator
on alocally presentable category
cor-responds
also to afamily
ofsubobjects
of thegenerators, satisfying properties analogous
to those of a Gabrieltopology
on an abelian cat-egory with
generators ([17]),
or of a Grothendiecktopology
on a smallcategory (or
better on thecorresponding presheaf category) ([14]).
Anextra
property
isrequired
to ensurethat,
whenviewing
theobjects
ina
locally presentable category
as left exactfunctors,
the closure of asubobject
is left exact([5]). (We
refer to any of the above variants of auniversal closure
operator
as atopology.)
Contrary
to the classical cases of abeliancategories
ortoposes,
atopology,
in the above sense, does notalways give
rise to alocalization, although
a localizationuniquely
determines atopology.
Atopology
ona
locally presentable category
determines a fullsubcategory
ofit,
theone
consisting
of theobjects having
theunique
extensionproperty
withrespect
to dense monos. Such asubcategory
is reflective([8])
but thereflection now need not be left exact.
In the first section we introduce a condition on a
topology
on alocally finitely presentable (l. f . p) category,
that we callcompactness,
since for1 We should also mention here the work by Castellini, Dikranj an, Giuli and others
(cf.
for example Dikranjan and Giuli, Closure Operators I, Topology and Applica-tions 27
(1987), 129-143)
which places the study of closure operators in categoriesin a more general framework and succeeds, among other things, in accounting for
the ordinary closure operators of General Topology as well. We do not pursue these
ideas here though.
Grothendieck
topologies
itsimply
means that everycovering
has a finitesubcovering.
Such kind of finiteness conditions on atopology
have beenconsidered in the
past,
either for Grothendiecktopologies ([21]),
or inthe abelian context
([17], [18]).
Here we elaborate on that notion ofcompactness.
We start from whatcompactness
shouldintuitively
mean:when the supremum of a
family
ofsubobjects
is dense(with respect
to the
topology),
then so is the supremum of some finitesubfamily.
We do not
necessarily require
thatproperty
forsubobjects
ofarbitrary objects,
but forsubobjects of,
say, therepresentables
or of the freeobject
on one
generator,
in other words forsubobjects
ofobjects
in a set ofregular presentable generators.
We arrive at an invariantequivalent
formulation: A
topology
iscompact if,
viewed as a closureoperator,
itcommutes with directed suprema of
subobjects
of anyobject.
What wesaid here holds for I.a - p
categories
and notions of0-compactness (in
the obvious
generalized sense),
when¡1
islarger
than a.The condition was considered in connection with the
problem
of find-ing
when thecategory
of sheaves is closed under filtered colimits(similar
considerations led to it in
[21], [17], [18]).
Thisproblem
is taken up inthe second section.
First,
we show that thecompactness
of atopology implies
that thesubcategory
ofseparated objects
is closed under filtered colimits. We believe that this result has not been noticed even for thecase of Grothendieck
topologies.
On that we base the rest of the results in that section.Concerning
the fullsubcategory
ofsheaves,
we have thefollowing:
It is closed undermonomorphic
filtered colimits when thetopology
iscompact and, conversely,
thetopology
inducedby
a local-ization is
compact
when the localization is closed undermonomorphic
filtered colimits. When the ambient
category
islocally coherent,
mean-ing
thatfinitely generated subobjects
offinitely presentable
ones arefinitely presentable ([7]),
the sheaves are closed under filtered colimits.A
condition, stronger
thancompactness,
sufficient for thepreservation
of filtered colimits
by
the inclusion of sheaves is also discussed. That conditiongeneralizes
one, which is known to beequivalent
to the preser- vation of filtered colimits in the abelian case([18]). Still,
the results of the second section are valid forl.a - p categories
and/3-compact topolo- gies, ¡1 larger
than a.The set of localizations of a
topos
or of acategory
of modules over aring; dtdd+qd
uiidorinclusion
isthe
dual of a frame.Or, equivalently, )he
set
of topologies, with the order
inducedby
the order ofsubobjects, is
aframe.
Similar results hold for the
localizations of alocally presentable
category
in which finite limits commute with filtered colimits.Despite
the ramification of
concepts
related tolocalization,
the above results hold for each of the two branches: In alocally presentable category satisfying
the exactnessproperty
mentionedabove,
localizations form the dual of a frame([3]),
whiletopologies
form a frame([6]). The
com-pact topologies
on anl. f.p category
constitute a subframe of that of alltopologies.
When theL. f.P category
is a coherenttopos
then the frame inquestion,
which in this connection can bethought
off as the frame of coherent extensions of the coherenttheory
classifiedby
thetopos ([13]),
is a
locally compact, compact
one. Morespecifically
when thetopos
is that of sheaves on a coherent space, the frame can be identified as the frame of opens for thepatch topol9gy
on thegiven
coherent space.We should
point
out here that the aforementioned frame can be identified with other well known constructions inspecial
cases,notably,
when the
l. f .p category
is R-modules over a commutativering R,
withthe dual Zariski
spectrum
of thering ([11]).
This factthough
can beexpressed
in a morecomprehensive
wayusing
the formalism of Gabrieltopologies
onalgebraic quantales.
For that we intend topresent
this connection elsewhere.The work
presented
here ispart
of the author’s Ph.D thesis([11]).
Iwish to express my sincere
gratidude
to my teacher Anders Kock for his skillfulteaching
and hissupport.
I would also like to thank Professors F. Borceux and F. W. Lawvere for comments that contributed to theshaping
of thiswork,
as well as M. Adelmann and J. Schmidt for useful discussions.2 Compact Topologies
When we are
dealing
with a Grothendiecktopology
or atopology
on acategory
of modules we want to be able toverify
whether atopology
has a cerain
property by looking
at apresentation
of it in terms of densesieves or dense
(topologizing,
in thering-theoreric terminology)
ideals.In
particular
the intuitive idea ofcompactness -
afamily
ofsubobjects
has dense supremum if a finite
subfamily
of it does - should bereferring
to
subobjects
ofobjects
of a certaintype (free, representable,
theground ring etc.),
without on the other handdepending
on the choice of it. This is the aim of the nextProposition.
2.1
Proposition.
Let E be alocally
a-presentable (l.a - p) category, j
atopology
on it and M a setof regular generators,
which are a-presentable.
Thefollowing
areequivalent:
(i)
Whenever a0-directed family of subobjects of
anobject
in M hasj-dense
supremum, there is a memberof
thefamily
which isj-
dense.
(ii)
Whenever a/3-directed family of subobjects of
anobject
in £ hasj-dense
supremum, there is a memberof
thefamily
which isj-
dense.
(iii) Any j-dense subobject of
ana-presentable
P contains aR-
gener- atedsubobject
thatis j-dense
in P.(iv)
The closureoperator
commutes with/3-directed
supremaof
subob-jects of a-presentable objects.
(v)
The closureoperator
commutes with/3-directed
supremaof
subob-jects of
allobjects.
Proof: The
implications (v)
=>(iv)
and(ii)
=>(i)
aretrivial,
while(iv)
=>(iii)
and(ii)
=>(iii)
followimmediately
from the fact that in al.a-p category
everyobject
can be written as the0-directed
supremum of the0-generated subobjects
of it. Also theimplication (iii)
=>(ii)
follows
immediately
if we note that a/3-directed
supremum ofsubobjects
is the
monomorphic /3-filtered
colimit of thosesubobjects.
(i)
=>(ii):
Recall from[9] (Satz 7.6)
that in al.a - p category
everya-presentable object
can be written as a retract of a colimit of size less than a ofobjects
inM,
where .M is as in the statement above. Let then P be ana-presentable object,
m: P -colimjgk
asplit
monicwith section e, with
Gk
E.M,
cardJ a. LetfAi
>->Pli
EI}
a0-directed family of subobjects
ofP,
the supremum of which is dense in P. Consider thefollowing pullback:
By
theuniversality
of the closureoperator
and the commutation of0-directed
suprema withpullbacks
in E we haveThe iso in the lower row
gives
an iso in thecorresponding
upper row and then thecompactness hypothesis gives
an isoj(Az
X pGk) = Gk,
for some 1 E I. For each k E
J,
now, we have amorphism
fromGk
to aj (Ai) (the
inverse for the iso we found followedby
theprojection
of thepullback
to j(Ai)) and,
since there are less than a many k’s and I is0-directed,
we can fix 1 E I so that for all the
Gk’s
in the colimit there isGk -> j(Ai),
hence there is a factorization
g: colim jGk -> j(Ai).
In that way g.mbecomes a
right
inverse for the monoj(Ai)
>->P,
soAi
>-> P is dense.(iii)
=>(iv).
Here we have toappeal
to therepresentation
of Eas the
category
of a-lex Set-valuedfunctors, E
= a -Lex(COP, Set).
In the
representation just
mentioned there is a canonical choice forC, namely
C =£a-p.
Then thea-presentable objects
of E areexactly
the
representables. Subsequently
wedeliberately
avoid todistinguish
between the two. Let us then be
given
a0-directed family
ofsubobjects {Ai >-> Pli
EI}
of ana-presentable object
P. Notice then that theinequality B/Ijp(Ai) J?(V7 Ai) always
holds. For the otherinequality
recall that the closure of a
subobject
A >-> P of arepresentable
iswhere
by Dj(Q)
we mean the setof j-dense subobjects
ofQ (cf. [5]).
Then x E
jp(VIAi)(Q)
iffx-l(Ai)
=x-’(Ai)
EDj(Q),
in which case(ii)
tells us that there is i E I withx-1(Ai) E Dj(Q),
so that x Ejp(Ai)( Q).
That w4y we have shown thatjp(V I Ai) V I jp(Ai)
in thefull
presheaf category Set cop
. Since the inclusion E ->Set cop preserves 0-filtered
colimits the sameinequality
holds in S.(iv)
=>(v):
We consider anobject
A of E and afamily {Ai >->
A]
z E
I}
ofsubobjects
ofit,
which is0-directed.
Write A as the a-filteredcol-imit of
a-presentable objects
above it. Consider then thepullback
ofalong
aninjection
The
following
hold:using appropriately
theuniversality
of the closureoperator,
the com-mutation of the closure with
0-directed
suprema ofsubobjects
of a-presentable objects
and the commutation in E ofpullbacks
with/3-
directed suprema.
Moreover,
the commutation in E ofpullbacks
witha-filtered colimits
gives:
So
having
an arrowj(VI Ai)
xA P =VIj(Ai)
xA P -VI Ai
foreach P E
Ea-p/A
weget
a factorizationthrough
the colimitj(VIAi),
g :
j(VI
->Ai) VIj (Ai). Chasing
thediagram
we show for the inclu-sions
i 1 :
and theprojection
that
ii.g.sp
=Z2-SP
for allP, giving il.g
=Z2,
SO 9 is an iso asrequired.8
The
previous equivalences
allow us now to say what acompact topol-
ogy is.
Definition Let E be
l.a - p category, j
atopology
onit, /3
acardinal,
a
0.
We saythat j
is/3-compact if
itsatisfies
anyof
theequiv-
alent conditions
given
inProposition
1.1 above. We reserve the wordcompact for
the case a =/3
=Ro.
Examples:
a.Any
Grothendiecktopology
on a small Cgiven
in termsof
coverings
with theproperty
that everycovering family
contains a fi-nite
subfamily
which covers, is acompact topology
on thel. f.p category Setc°P.
See also the discussionfollowing Proposition
3.4.b.
Any
locale with theproperty
that any supremum can be attainedby using
less than K elements(K
aregular cardinal) gives
rise to a K-compact topology
when viewed as a site(with
the canonicaltopology).
If the locale
happens
to be a booleanalgebra,
then the conditionjust
stated is the K-chain condition.
c. The classical closure
operator (torsion theory)
on thel. f.p.
cate-gory of abelian groups,
assigning
to asubgroup H
>-> G the closurej(H) = fg
E GI
3n E N such that n.g EH},
is acompact
one.d. It was shown in
[4]
that when £ is a l.a - p.category
withuniversal
colimits,
then the a-lex closureoperator
is a universal one(the
a-lex closure
assigns
to a subfunctor G of an a-lex functor F:EoPa- P
->Set the smallest a-lex subfunctor of F that contains
G).
In the inductiveconstruction of the a-lex closure each
step
commutes with a-directed suprema.Hence,
the a-lex closure commutes with a-directed supremagiving
rise to ana-compact topology
in this case.e. Since a
locally presentable category E
iswell-powered
anytopol- ogy j
on it will be,8-compact
forsome ,8 sufficiently large (in partcular
it will be
,8-compact
fora ,8
> a, where a is thepresentability
rank ofE).
3 Preservation of filtered colimits
We consider the
following proposition
to be the basic one of thissection,
since it is the main technical means for
arriving
at a characterization ofcompactness
in terms ofpreservation
ofmonomorphic
filtered colimits.3.1 Theorem. Let .6 be a l.a - p
category, j a /3-compact topology
on it with a
/3.
Then the inclusionof j-separated objects
intof, sepjf
->£,
preserves,8-filtered
colimits.Proof: The
following things,
well known fromtopos theory (see
e.g[12]),
aregeneral "closure-theoretic"
and hold in our case.First,
ifthe
diagonal
of anobject
isj-closed
then theobject
isj-separated
andsecondly,
if a map P -> X x X factorsthrough
the closure of thediagonal
ofX,
then theequalizer
is
j-dense
in P. Let I-> sepjE
be a0-filtered diagram
ofj-separated objects.
Let X =
colimX2,
where the colimit is calculated in S. We want to show that X isj-separated.
For that it suffices to show that thediagonal
of X is
j-closed.
Since E islocally presentable
we check elements definedover the
(a-presentable) generators.
Let(x, y):
P -+ X x X be such anelement
belonging
to the closure of thediagonal (so
that we know thatthe
equalizer
E of xand y
isj-dense).
We want to show that x = y.Since I is
3-filtered
and Pa-presentable, x and y are
defined at some(common,
as it can bechosen) stage k
E I andrepresented by
elementsxk and yk,
respectively.
It suffices to exhibit a laterstage
of definitionwhere xk
and yk
are identified. Consider thecategory
of indices belowk, k/I.
It is still0-filtered. By
a theorem of Grothendieck and Verdier(proved
in detail in[20])
there is a0-directed poset li
cofinal ink/I.
Consider now the
equalizer diagrams
of the arrows xi =
tik.xk, yi
=tik.yk, where tik
the transition maps. The commutation of/3-filtered
colimits withequalizers
in Egives
the third
equality provided by cofinality
of K ink/I.
But nowE, being
the
equalizer
of an element in theclosure
of thediagonal
ofcolim¡Xi,
is
j-dense. By
the0-compactness assumption
one of theEi
-> P isj-dense
and since theXi’s
arej-separated
the two elements xi, y2 have to beequal.
This concludes theproof.
Remark: A
special
case of the theorem above is known. In[16]
anotion of B-valued set was
introduced,
for B acomplete
booleanalgebra,
which is
essentially
the same as aseparated presheaf
on B(seen
as asite with the canonical
topology).
It is shown there that thecategory
ModB of B-valued sets is l.k - p if B has the K-chain condition. This amounts to
showing
that the inclusion ofseparated presheaves
preserves K-filtered colimits when the canonicaltopology
on B isk-compact (after Example (b.)
in section1).
Theproof
there isgiven by
adirect, long
calculation
involving
thedescription
of colimits in ModB.Except
forthat
special
case above we believe that the result of Theorem 1 is new even for the case of a Grothendiecktopology.
For that reason we stateit
separately.
3.2
Corollary. If J is a
Grothendiecktopology
on a smallcategory C,
which is
a-compact,
then the inclusionsep(C, J)
->Set cop
preservesa-filtered
colimits and hencesep(C, J)
is l.a - p.We know of no direct
proof
of the fact that for atopology 3
on alocally presentable category sepj£
is reflective in £.Combining though
theresult of Theorem 2.1 with one of Adamek and
Rosicky ([1],
see also[15]
for aspecial case),
that asubcategory
of alocally presentable
one£’,
closed underarbitrary
limits and¡1-filtered colimits,
is reflective inE,
we obtain:3.3
Corollary.
Let E be alocally presentable category, j
atopology
onit. Then
sepj£
isreflective
in E and hence it isitself locally presentable.
Proof: As we mentioned in
Example (e.) above,
thetopology j
willbe
0-compact
forsome 0 sufficiently large
thussepj£
will be closed in E under/3-filtered
colimits. On the other handsepje
istrivially
closedin E under
arbitrary limits,
so we can concludeby
the aforementionedresult. 8
As far as sheaves are concerned the
following
holds:3.4
Proposition.
Let £’ be a l.a - pcategory, j
a/3-compact topology
on it. Then the inclusion
shj£
-> E preservesmonomorphic /3-filtered
colimits.
Proof: Let I ->
shjE
be amonomorphic /3-filtered diagram
ofj-
sheaves and X =
colimlX2 .
We wish to show that X is aj-sheaf.
We
already
know that it isj-separated.
We also know that in order to be a sheaf it suffices to beorthogonal
to dense monics with codomaina-presentable.
So consider such a monic A >->P,
and anarbitrary
A >-> X. We want to
produce
an extension of the latter(which
willthen be
unique).
But thesubobject
A >-> P can be refined to aj-dense
F - P one, with Fbeing 0-generated:
We can
produce
an extension of F >- > A -> Xalong
F >->P,
as the
following
calculation shows :horraE(F, X) = colimIhomE(F, Xi) = colimIhomE(P, Xi) = home (P, X) (where
the middlebijection
is due to the fact that theXi’s
aresheaves).
Then, by
theseparatedness
ofX,
it turns out that this is also an exten-sion of A -> X
along
A >->P, finishing
theproof. 8
If the ambient
category E
has suitableproperties, compactness
can havestronger implications concerning
thepreservation
of filtered col- imitsby
the inclusion of sheaves.Namely,
in some I.a - pcategories 0-generated
may coincide with/3-presentable (a 0). Categories
withthis
property
are calledlocally /3-noetherian. Examples
of such cate-gories
are these of booleanalgebras
andR-algebras
over a noetherianring R (for
a== f3 == Ro),
smallcategories (for a = No, /3= N1)
andcommutative
C*-algebras (for
a =/3
=N1) (see
e.g[7],
Ch.5).
Inother L.a - p
categories a-generated subobjects
ofa-presentable
onesare
a-presentable.
Suchcategories
are calledlocally
a-coherent and have been characterisedby
S. Fakir([7], chapter 9).
Atypical example
is that of R-modules over a coherent
ring
R.Finally,
for in acategory
ofpresheaves
on a smallcategory
withpullbacks, finitely generated
subob-jects
of therepresentables
arefinitely generated (See [21],
Remark15,
p.
301).
This is sobecause,
if a sieve R >->hc
isfinitely generated (meaning
that there arefinitely
manyCi
-> Cgenerating R),
then thereis an
epi
If the site has
pullbacks
thatepi
is thecoequalizer
of thediagram
(the diagram being
the kernelpair
ofLJni= 1 hc,
->>R),
hence R isfinitely presentable
as anobject
ofSetCop By
a similarproof
we can showthat in a coherent
topos finitely generated subobjects
ofrepresentables
are
finitely presentable.
In all the above cases thecompactness
of thetopology implies
thepreservation
of filtered colimitsby
the inclusion of sheaves.Remark. In the case of sheaves for a site
(with pullbacks)
the resultof
Proposition
2.4 is known(cf. [21],
Theorem11).
It follows from aneasy calculation
involving
the "doubleplus"
construction of the associ- ated sheaf functor. We do not know which fullsubcategories
of alocally presentable one
closed under limits and/3-filtered colimits,
for some(3,
arise ascategories
of sheaves for atopology.
Weknow, though,
thatlocalizations of such an E arise as
categories
of sheaves. In connection with that we have thefollowing:
3.5
Proposition.
Let E be al.a - p category
and £ -4 E a localizationof
it such that the inclusion preservesmonomorphic (3-filtered
colimits(we
assume that a(/3).
Then thetopology
on Einducing
,C is(3- compact.
Proof: Recall from
[6] (Proposition 1.5)
that thetopology inducing
the localization i : £ 9
£,
with left exact reflection r,assigns
to asubobject
S >-> A thepullback
of the monic
(by
the left exactness ofr)
irS >-> irAalong
theunit of the
adjunction
77A: A -> irA. Given now a3-directed family
ofsubobjects {Sk
>-> Ak
EIl
the closure of(since
i preservesmonomorphic
filtered colimits and the latter commute withpullbacks),
asrequired, t
Summarizing
the facts we have seen sofar,
we obtain:3.6 Theorem. When £ is a
l.a - p category
such thattopologies
on itagree with
localizations,
then/3-compact topologies (a {/3} correspond
to localizations
preserving monomorphic /3-filtered
colimits.We are
presented naturally
with theproblem
ofcharacterizing
thosetopologies, which,
in the abovecontext, correspond
to localizations closed under/3-filtered
colimits. Theproblem
has been considered for abelianlocally finitely presentable categories by
M. Prest in[18].
Thenon-abelian formulation of the condition
given
there turns out to be sufficient for thepreservation
of filtered colimitsby
the inclusion ofsheaves,
but not necessary.Yet,
as it is weakerthan,
forexample,
re-quiring
the local coherence of the ambientcategory,
so wegive
aproof
for its
sufficiency.
3.7
Proposition.
Let E be a l.a - pcategory and j
atopology
on itsatisfying
thefollowing
two conditions(i) j
is/3-corrapact (a /3)
(ii) If
F is a/3-generated subobject of
ana-presentable Q,
then itadmits a
presentation by a coequalizer
where P is
/3-presentable
and K contains a/3-generated
densesubobject.
Then the inclusion
shje
-4 E preserves/3-filtered
colimits.Proof: Let I ->
shj£
be a/3-filtered diagram
ofj-sheaves
and X =colz*mjxi .
We wish to show that X is aj-sheaf.
Wealready
know thatit is
j-separated.
It suffices to checkorthogonality
withrespect
to a densesubobject
of ana-presentable
one A >->Q.
As inProp. 2.4,
since X is
separated
it suffices to take A =F,
a0-generated subobject
of
Q.
There is apresentation
of itwith Ii’
having
a0-generated
densesubobject.
Consider a map A -> X.In the
following diagram
we
get
a factorization of P -> F -> Xthrough
anXi,
since P is(3- presentable.
The twocomposites
are
equal (upon replacing Xi
with a furtherXk
in thediagram)
sincethey
agree whenpreceeded by
aninjection
into the colimit. But G >-> I1 is dense andXi
is asheaf,
so the twocomposites
are
equal,
sincethey
agree whenthey
are restrictedalong
a dense mono.That way we obtain a factorization of P -
Xi through
F ->Xi.
Thereis a
unique
extension of F -Xi along
F >->Q
and thecomposite Q - Xi -> X
is therequired
extension to thegiven
map F -X. I 8
The
necessity
of the condition in the abelian case seems tohinge
onthe
good
combination of exactness and finitenessproperties
availablefor an abelian
locally finitely presentable category. Namely,
the factthat the kernel of an
epi
betweenfinitely presentable objects
isfinitely generated (cf. [19] Prop. 1.3.2),
as thefollowing proof
shows.3.8 Theorem.
(M. Prest, [iB],
Th.2.3)
A localizationof
alocally finitely presentable
abeliancategory
is closed underfiltered
colimitsiff
the
corresponding
Gabrieltopology
iscompact
and has thefurther
prop-erty that, if a
map betweenfinitely presentable objects
has denseimage,
then its kernel contains a
finitely generated
densesubobject.
Proof: The
sufficiency
isclearly implied by
thepreceeding proposi-
tion. For the
necessesity,
let P ->Q
be amorphism
betweenfinitely
presentable objects
with denseimage,
and let Ii be its kernel.Then, applying
the localization functor to the sequencewe
get
a sequencewhere rIi remains
the
kernel of rP ---rQ,
rP andrQ
remainfinitely presentable
and rP ---rQ
is anepi,
since theimage
of P -Q
isdense.
So,
rIi isfinitely generated,
which means that there is afinitely generated subobject
of Ii dense inK. 8
4 The frame of compact topologies
The fact
that,
under theassumption
that filtered colimits commute with finitelimits,
the set of alltopologies
on a l.a - p is a frame was shown in[6].
It is theanalogue
of theimportant
resultgiven
in[3] that,
underthe same
assumptions,
the localizations of alocally presentable category
form a co-frame.
4.1 Theorem. Let £ be a
l. f.p category.
Then the setof compact topologies
is asubframe of
theframe of
alltopologies.
Proof: As mentioned in the introduction to this
section, topologies
aregiven pointwise by acting
onsubobjects
ofa-presentable objects.
Thebinary
infimum oftopologies
is thengiven pointwise by
intersection ofsubobjects.
The supremum,however,
of a set J oftopologies
isgiven by
the
following
construction(cf. [6],
Theorem3,
p.313):
For asubobject
A >- > P of an
a-presentable object
definej’P(A) = VKjk1O...Ojkn(A),
where the supremum ranges over the set Ii of all the finite sequences of J
(hence
it is a directedone).
Then werepeat
the process overj’P(A)
and so on. The process terminates because
locally presentable categories
are well
powered.
We first show that the set ofcompact topologies
isclosed in the frame of all
topologies
underarbitrary
suprema. Let theJ be a set of
compact topologies
and{Ai
>-> Pz (E 11
a directedset of
subobjects
of afinitely presentable object
P. Since eachtopology j
E J commutes with directed suprema, and suprema commute with each other we have:Iterating
theapplication of j’
we willalways
begetting
the supremum outside theargument
ofit,
soeventually
as
required.
It is also closed underbinary
infimabecause, if j
and k arecompact topologies
and theAi’s
are asabove,
we have:since in E directed suprema commute with intersections. The latter double supremum,
though,
can be substitutedby
asingle
one,again by
directedness. So
finally
i-e j
A k iscompact,
which concludes theproof. 8
Remark. The above
proof
showsactually
that in alocally presentable category,
where filtered colimits commute with finite limits(so
thatthere is available a frame of
topologies),
thetopologies commuting
withdirected suprema of
subobjects
form a subframe of that of alltopologies.
We
might
also considertopologies
on aspecial
kind ofl.f.p categories, namely algebraic
lattices. As we noted in theintroduction, topologies
on such a
categories
arenothing
else but nucleui. Notice also that under thisidentification, orthogonal
elements for the closureoperator
corre-spond
to closed elements for the nucleus. Acompact topology
on analgebraic
latticeis, then,
a nucleuscommuting
with directed suprema.A frame on the other hand is a
locally presentable category,
where fil- tered colimits commute with finitelimits,
so we have:4.2
Corollary.
The setof directed-suprema-preserving
nucleiNw,(A)
on a
frame
or analgebraic
lattice A is asubframe of
theframe N(A) of
all nuclei.
We continue with a calculation
concerning
directed suprema of com-pact topologies,
which we aregoing
to use in thesequel
instudying
theframe of
compact topologies
on a coherenttopos.
4.3
Proposition. If
J is a directedfamily of compact topologies
on alocally finitely presentable category,
andif
A >-> P is anysubobject of
afinitely presentable object,
then.
Consequently,
the setof j-dense subobjects of
afinitely presentable object
is the unionof
the setsof j -dense subobjects, j
E J.Proof: In the