C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
M. G RAN
E. M. V ITALE
On the exact completion of the homotopy category
Cahiers de topologie et géométrie différentielle catégoriques, tome 39, no4 (1998), p. 287-297
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© Andrée C. Ehresmann et les auteurs, 1998, tous droits réservés.
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ON THE EXACT COMPLETION OF THE
HOMOTOPY CATEGORY
by
M. GRAN and E.M VITALECAHIERS DE TOPOLOGIE ET
GEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XXXIX-4 (1998)
R6sum6. On montre que la
completion
exacte de lacat6go-
rie de
1’homotopie
des espacestopologiques
est unpr6topos.
Pour
cela,
on determine une condition n6cessaire et suf- fisante pour que lacompletion
exacte d’unecat6gorie
a li-mites finies faibles soit extensive.
Introduction
The exact
completion (Top) ex
ofTop,
thecategory
oftopological
spaces and continuous maps, has been
recently
studied withencreasing
interest. The reason for this lies in the
deep
connection between(TOP) ex
and the
category
ofequilogical
spaces(cf. [10]
and[1]).
It has beenproved
that(Top)ex
is alocally
cartesian closedpretopos (cf. [2], [4]
and
[9]).
Since the
category HTop
oftopological
spaces andhomotopy
classesof continuous maps is
equivalent
to a fullsubcategory
of(Top)ez and,
moreover, the full inclusion behaves well with
respect
to weak limits(cf. proposition 3.3),
it is natural to ask if also(HTop)e,,
is alocally
cartesian closed
pretopos.
In this note we show that
(HTop)ex
is apretopos (i.e.
an extensiveexact
category).
Thisproperty
isquite simple
to prove in the case of(Top)ex,
becauseTop
itself is extensive and has finitelimits;
on thecontrary,
theproof
is more delicate if one works withHTop,
whereonly
weak limits are available: To overcome thisdifficulty
we proposea weakened notion of
lextensivity;
this notion can beexpressed
in anyweakly
left exactcategory C,
and it turns out that it is necessary and sufficient forCex
to be lextensive(i.e.
left exact andextensive).
For the notion of
extensivity
the reference is[3].
Thetheory
of theexact
completion
of aweakly
left exactcategory
can be found in[5] (see
also
[7])
or, in a shorterform,
in[11].
We work with finite sums; thegeneralization
to small sums isstraightforward.
1 Weakly lextensive categories
From
[3]
we recall that acategory
C with sums is extensive if it haspull-backs along injections
in a sum and thefollowing
condition holds:in the commutative
diagram (where
the bottom row is asum)
the
top
row is a sum if andonly
if the two squares arepull-backs.
Inthe
following
we refer to the(if) part
as condition I and to the(only if)
part
as condition II. Condition I is known as theuniversality
of sums.If C has
pull-backs along injections,
theextensivity
isequivalent
to have
disjoint
and universal sums, wheredisjointness
of sums is thefollowing
condition III:III.a: the
injections
in a sum X -> X + Y - Y aremonic;
III.b: if 0 is an initial
object,
then thefollowing diagram
is apull-back
Moreover,
if C is leftexact,
condition I isequivalent
to condition IV and to condition V:IV: if the first two
diagrams
arepull-backs,
then also the third one isa
pull-back
V.a: the canonical
morphism (X
xY)
+(X
xZ) -
X x(Y
+Z)
is anisomorphism (i.e.
C isdistributive);
V.b: if the first two
diagrams
areequalizers,
then also the third one isan
equalizer
Recall now from
[5]
that one can not build up all weak finite limitsstarting
from weakpull-backs
and weakterminals;
on thecontrary,
it ispossible
to do itstarting
from weak finiteproducts
and weakequalizers.
Keeping
this situation inmind,
wegive
thefollowing
definition.Definition 1.1 Let C be a
weakly
left exactcategory
with sums; we say that C isWeakly
lextensive if thefollowing
conditions hold:III: sums are
disjoint;
V.aw: for each choice of weak
products
X x Y and X xZ,
the sum(X
xY)
+(X
xZ),
with the obviousmorphisms,
is a weakproduct of X and Y + Z;
v.bw:
(with
the same notations asv.b)
for each choice of weakequalizers Ex
andEY,
the sumEx
+Ey
is a weakequalizer;
VI: initials are strict.
A first
example
ofweakly
lextensivecategory
isgiven by
the sum-completion Fam(C)
of a smallcategory
C. This can bedirectly
checked ordeduced from
proposition 2.1,
because the exactcompletion
ofFam(C)
is
equivalent
to thetopos
ofpresheaves
on C(cf. [5]).
We list now some
properties
ofweakly
left exact andweakly
lexten-sive
categories.
Proposition
1.21)
let C beweakly
leftexact;
condition V.aw can beequivalently
restated
replacing
"for each choice"by
"there exists achoice";
the same holds for condition
V.bw;
2)
let C beweakly
leftexact;
conditions V.aw and V.bwimply
theweakened version IV.w of condition IV:
(yvith
the same notationsas in
IV)
for each choice of weakpull-backs Px
andPY,
the sumPx
+Py
is a yveakpull-back;
3)
let C beweakly lextensive;
then condition II ofextensivity holds;
4)
let C beweakly lextensi ve;
if the first two sq uares are weakpull- backs,
then also the third one is a weakpull-back
5)
let C beweakly
left exact withproducts;
if sums are universal inC,
then conditions V.aw and V.bw hold. In
particular,
aweakly
leftexact and extensive
category
withproducts
isweakly
lextensive.Proof: 1)
and2)
are a routine calculation with weak limits.3):
we have to prove thatare
pull-backs;
sinceiA
andiB
aremonic,
it isenough
to show thatthey
are weak
pull-backs.
To find a weakpull-back
ofix
andf
+ g we canuse condition
IV.w;
thepull-back
ofix
andiA . ( f
+g) = f. ix
is Abecause
ix
is a mono; thepull-back
ofix
andiB ( f
+g)
= g.iy
factorsthrough
thepull-back
ofix
andiy,
and then it is 0by disjointness
ofsums and strictness of initials.
4):
similar to3).
5): universality
of sumsimplies
theparticular
case of condition IV where the arrow X + Y - B is theidentity
of X + Y.But, by
associa-tivity
of weakpull-backs,
thisparticular
caseimplies
condition IV.w.Moreover,
ifproducts exist,
one can check that IV.wimplies
V.aw andV.bw.
2 The exact completion
Proposition
2.1 Let C be aweakly
left exactcategory
with sums and r: C -Cex
its exactcompletion; Cex
is extensive if andonly
if C isweakly
lextensive.Proof: (only if):
recallthat,
up to the fullembedding
r: C -Cex, C
is a
projective
coverof Cex,
that is eachob j ect
of C isregular pro j ective
in
Cex
and eachobject
ofCex
is aregular quotient
of anobject
of C.Then,
a weak limit in C can be recoveredperforming
thecorresponding
limit in
Cex
and thencovering
it with anobject
of C.Moreover,
thefunctor r preserves all the sums which turn out to exist in C. From
this,
the firstimplication easily
follows.(if): Cex
has sums:given
twoobjects
inCex,
that is twopseudo equiv-
alence relations in
C,
ro, rl: R => X and so, sl: S=>Y,
its sum inCex
isro + so, rl +
s1 : R
+ S x X+ Y;
the fact that it is apseudo equivalence
relation follows from
part 4)
ofproposition
1.2.To prove that in
Cez
sums aredisjoint
anduniversal,
we need an ex-plicit description
ofpull-backs
inCex.
The idea isquite simple:
writedown what a commutative square in
Cex
is and thentake,
at eachstep,
a weak limit in C. Given two arrows in
Cex
we can choose a weak limit
(P, f ,
p,g)
as in thefollowing diagram
and then a weak limit
(E,
p, eo, el,T)
as in thefollowing diagram
The
required pull-back
inCex
is thenObserve that a weak limit P can be obtained
taking
three weakpull-
backs. Also the construction of E can be
split
into twosteps:
first takea weak limit L over the
zig-zag
(which
can becomputed
with iterated weakpull-backs)
and then takea weak
equalizer
E ->L x
T.In
Cex
sums are universal: we have to calculate thepull-back
inCex
ofFollowing
theprevious description
andusing points 2)
and4)
of propo-sition 1.2 for any weak
pull-back,
and condition V.bw for any weakequalizer,
weexactly
obtain the sum of thepull-backs
ofIn
Cex
sums aredisjoint:
we have to calculate thepull-back
inCex
ofBy part 3)
ofproposition
1.2 andby disjointness
of sums inC,
thebottom
part
of thepull-back
isand, by
strictness ofinitials,
thetop part
is also 0.It remains to prove that in
Cex injections
are mono. This follows from the fact thatthey
are mono inC, using
onceagain part 3)
ofproposition
1.2.
Corollary
2.2 Let C be aweakly
left exactcategory; Cex
is extensive if andonly
if theCauchy completion
of C isweakly
lextensive.Proof:
theCauchy-completion cc(C)
of C isequivalent
to the fullsubcategory
ofregular projectives objects
ofCea:. Conversely,
one hasthat
(cc(C))ex
isequivalent
toCez..
Corollary
2.3 Let C beweakly lextensive;
then T: C->Cex is
thepretopos completion
of C.Proof:
recall that the functor T preserves all the sums which turn out to exist in C. Now anexchange argument
between sums andcoequalizers
shows
that,
for any exactcategory
with sumsA,
the exactextension P
of a left
covering
functor F preserves sums if andonly
if F preservessums
Corollary
2.4 A left exactcategory C
is extensive if andonly
if it isweakly
lextensive.Proof:
Oneimplication
is contained inpoint 5)
ofproposition
1.2.’Conversely,
assume that C isweakly
lextensive.By proposition 2.1,
it isa full
subcategory,
closed under sums and finitelimits,
of the extensivecategory Cex.
3 The exact completion of HTop
Proposition
3.1 Thecategory HTop
is extensive.Proof:
SinceTop
is extensive and sums inHTop
aretopological
sums, it is
enough
to show thattopological pull-backs along injections
are also
pull-backs
inHTop.
Sinceinjections
inTop
arefibrations,
thentopological pull-backs along injections
arehomotopical pull-backs (cf.
[8]
and[6])
and then weakpull-backs
inHTop. They
are"strong" pull-
backs in
HTop
becauseinjections
are fibrations and monic inTop,
sothat
they
are monic inHTop.
Corollary
3.2HTop
isweakly lextensive,
and(HToP)ex is a pretopos.
Proof: HTop
hasproducts,
which aretopological products,
and isextensive. It remains
only
to prove that it isweakly
left exactand,
in view of
proposition 3.3,
we recall here anexplicit description
of ho-motopy equalizers (which
are weakequalizers
inHTop).
Consider twocontinuous maps
f, g: X :::4
Y and the evaluations evo, evi:Y [0, 1]
:::4 Y.An
homotopy equalizer
e: L -> X off and g
isgiven by
thefollowing topological
limitIt remains to clear up the relation between
HTop
and(Top) ex .
Forthis,
consider the functor £:Top - (Top)ex
which sends each space X into thepseudo equivalence
relation evo, evi:X[0,1]
=4 XProposition
3.3 The functor £respects homotopy
and its factoriza-tion £’:
HTop ->(Top)ex
isfull,
faithful and leftcovering.
Proof:
The leftcovering
character of S’ followscomparing
the de-scription
of weakequalizers
inHTop just given
with thedescription
ofequalizers
in the exactcompletion.
This latter can be found follow-ing
the same idea used for thedescription
ofpull-backs
in theproof
ofproposition
2.1. The rest of the statement is obvious.Remark: it remains for us an open
problem
to determine if(HTop)ex
is cartesian closed. The
preliminary
work done in[9]
for a lextensivecategory
can begeneralized
to aweakly
lextensive one, but we are not able to conclude because of the lack of agood
factorizationsystem
inHTop.
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[10]
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spaces andequiva-
lence
relations, preprint
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E.M. VITALE: Localizationsof algebraic categories,
J. PureAppl. Algebra
108(1996)
315-320.Marino
Gran,
Enrico M. Vitale :D6partement
deMath6matique,
Universitécatholique
deLouvain,
Chemin du
Cyclotron 2,
1348Louvain-la-Neuve, Belgium.
e.mail :