Download Building closed categories

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ICHAEL BARR
Building closed categories
Cahiers de topologie et géométrie différentielle catégoriques,
tome 19, no 2 (1978), p. 115-129.
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Vol. XIX - 2 (1978 )
CAHIERS DE TOPOLOGIE
ET GEOMETRIE DIFFERENTIELLE
BUILDING CLOSED CATEGORIES
by Michael
BARR
*
examining and generalizing the construction
of the cartesian closed category of compactly generated spaces of GabrielZisman [1967 . Suppose we are given a category 21 and a full subcategory
5 . Suppose E is a symmetric monoidal category in the sense of EilenbergKelly [1966], Chapter II, section 1 and Chapter III, section 1. Suppose there
I
am
interested here in
is, in addition,
«Hom» functor lil°P X 5--+ 6 which satisfies the axioms of
a
closed monoidal category insofar
Eilenberg-Kelly
for
Then
that, granted certain
this
we
can
of these
show
a
be extended
to a
closed monoidal
objects of 21 which have
thesis suffices
to
One
original product in 5
is the cartesian
structure on
Z-presentation.
category is
cartesian closed. As
a
result,
only of compactly generated
simplicially generated, sequentially generated, etc...
spaces
derive the cartesian closedness
but also of
the full
subcategory
additional hypo-
a
show that when the
product, then the resultant
we
they make sense.
reasonable hypotheses on lbl and 21,
as
1. STRUCTURE ON
even
not
5.
(1.1) We suppose that T is equipped with
symmetric monoidal structure.
This consists of a tensor product functor - O - : 5 x 5 --+ 5, an object I and
coherent commutative , associative and unitary isomorphisms. See EilenbergKelly for details.
We suppose also a functor ( - , - ) : 5oP X 21 --+21 and natural equivala
ences
*
I would like
to
thank the National Research Council of Canada and the Canada
Council for their support.
115
C,
Here
A E 21 . The
C 1 , C 2 E 5,
first and third combine
give also
to
( 1.2 ) We make one further hypothesis about the relationship between T and
U . Namely that for every object A e 1 there be a set Cw --+ A} of maps with
domain in T such that every C - A with C E 5 factors through at least one
effect, demanding that each category of 5-objects over 21
weak initial family. Such a family is called a terminal 5-sieve for 21.
am, in
Cw --+ A . I
have
a
( 1.3 ) We suppose, finally, that I has regular epimorphism/ monomorphism
factorizations of its
5-PRESENTED
2.
morphisms.
OBJECTS.
( 2.1 ) I define A c a
be
to
5-generated
if there is
a
regular epimorphism :
Because of the cancellation
that if there is
I
GY --+ A}
some
is
Cw--+
a
properties of regular epimorphisms, it follows
any regular epimorphism of that kind we may suppose the
A factors through
terminal 5-sieve for A . For each
CY--+
A and
so
regular epimorphism
factors :
5-presented provided
there is
the
( 2.2 ) WOe
say that A is
Assuming
that A is
5-generated by Xi CY--+
of that map it is sufficient that there be
if there is such
case
a
terminal
( 2.3 )
that
ever,
as
be
happens that 6 -
there is
will be
(2.4) One
diagram
and that B is the kernel
A
epimorphism I Cç
it is clear that it suffices
4B.
seen,
word of
5-presented
no
in
I alone - is
or even
distinction between
being Z-presented
Z-generated
is the
a
pair
In any
take for
Cç}
generator for
21. In
to
Z-sieve for the kernel pair B .
It often
case,
a
an
coequalizer diagram
a
and
5-presented.How-
important thing.
warning is in order. It is entirely possible that an object
a full subcategory of 1 which contains Z but not in 21.
116
This is because the
coequalizer condition
is less
exigent
in
a
subcategory.
( 2.5 ) Suppose {Cw--+ A is a terminal 6-sieve for A , Ao = Z Cw, A1 =
I
and IC6-+A, XA Ao 1 is a terminal 5-sieve. Then we have a diagram:
Ctfr
This
is
diagram
call it
a
a
coequalizer
6-presentation
iff A
is
5-presentable
and in that
case we
of A .
( 2.6 ) Let f: A --+ B . Whether
or
not A or B is
5-presentable
we
can
find
diagrams
of the above type where
For all
hence
we
we
have
commutes.
are
equal
through
get that the map
For
an
CO) 4 A 4
fo : L Cw--+
any Vi
the
two
ZCç
Cç
through
some
Cç--+ B and
such that
maps
and hence determine
some
B factors
and results in
a
a
map
CY --+ Bo XBBo
commutative
Ao - Bo is another choice for fo,
map Ao 4 Bo X B Bo which similarly has
is that the map f induces a map, «unique
If go :
117
the
a
up
which has
diagram
maps
homotopy » from
lifting
=
together
lifting h : Ao - Bi .
two
to
a
determine
The
a
upshot
A1==++ Ao
to
B1 ==++Bo;
is induced
functor
now
the
on
let TT A
we
unique
a
( 2.7 ) It is
if
map,
and 7TB be the
naturally
denoted
standard argument
a
category U
respective coequalizers,
rf:
to see
TT A 4
that
7T
which lands in the full
It is clear also that there is
77
B.
is
a
there
well defined endo-
subcategory
5-present-
of
natural TT A 4 A which is
an isoobjects.
morphism iff A is 5-presented and, finally, that 7T determines a right adjoint to the inclusion of that subcategory. We let r21 denote the full subcategory and let 7T denote also the retraction 21 --+ r 21 .
ed
(2.8)
PROPOSITION.
Let
is
isomorphism for
all Cc 5. Then
an
PROOF.
imply
are
{ Cw --+ A} be
{Cw A 4 B} is
Let
that
and hence
rr f
is
(2.9)
an
for
pairs
they
are
have the
same
map such that
a
77f
is
5-sieve
for B . Let
maps,
for A . Then the
Ao = Z Cw
.
hypotheses
Now both
namely
isomorphic for all C e 6 . Hence, Ao XA Ao
terminal T-sieves and it follows immediately
an
Under the
same
hypotheses, if
isomorphism.
3. INDUCED STRUCTURE ON
r 21 .
( 3 .1 ) We define
There is
isomorphism.
an
and
that
isomorphism.
COROLLARY.
ted, f is
one
isomorphic
also
be
terminal
a
--+
the kernel
A0 XB A0
f: A- B
a
a
1-1
correspondence
between maps
With A E 21 ,
118
A and B
are
Z-presen-
Finally,
which
we
have
gives,
upon
We prove it is
application
of r ,
a
map
isomorphism by using ( 2.9 ). To
corresponds uniquely, using right adjointness of rr
( 3.2 ) Since
a
an
regular image
of
a
ular factorization is inherited
21-sieves
is evident. Hence
PROPOSITION.
by
r
4. INDUCED STRUCTURE ON
(4.1 ) We will
noidal
now
poses of this
In view of
The
(3.2)
( A, B )
same
21
21 .
able
C --+ [ C1 O C2 , A ]
is
(0..presented,
the reg-
The continued existence of terminal
to
conclude :
satisfies
all the
hypotheses of Section
1.
7T U.
given
1 . In order
to
tensor
and internal hom
simplify
notation
we
to a
closed
mo-
will, for the pur-
Section, suppose that
this is
( 4.2 ) Now choose
and let
extend the
structure on 7T
7r
map
several times :
5-presented object
we are
The category
a
a
permissible.
T-presentation ( see ( 2.5 ) )
be defined
argument
(A ’, B) --+ (A, B)
as
in
as
the
equalizer
( 2.6 ) shows that
of
any
map A 4
and this is well defined and functorial.
(4.3) Similarly define A OB by choosing
119
5-presentations
A’ induces
a
map
and then A OB is the
( 4.4 )
P ROP OSITION.
coequalizer
For
of
fixed CE5,
the
functor (C,-)
commutes
with
projective limits.
PROO F . V’e
use
( 2.9 ) . Suppose
We get
and
conversely.
(4.5)PROPOSITION. Let CE5. Then
PROOF. Let
be
is
a
5-presentation. Then
an
equalizer. But
Since
is
an
(4.6)
equalizer,
so are
PROPOSITION.
PROOF. We
use
For A
(2.9). Let
fixed, ( A, - )
commutes
B = projlim Bw .
120
If
with
projective limits.
conversely.
and
( 4.7 )
PROPOSITION.
For any
A1, A2, BE 21,
P ROO F. Let
be
a
T-presentation
from which the
(4.8)
with
of
Al .
Then
we
have
equalizers
isomorphism follows.
PROPOSITION.
For B
fixed,
the
functor (-, B :
21°p --+21
commutes
projective limits.
PROOF.
Let A
= ind lim AO) .
and the result f ollow s from
(4.9)
P ROPOSITION.
For
we have for any C
a
6-presentation.
is
a
coequalizer
so
E,
C 4
proj lim( AO) , B ) ,
( 2.9 ).
C E 5 , (C O A, B) = (C, (A, B)).
P ROOF. Let
be
c
Then
that
121
are
equalizers.
(4.10)
PROPOSITION.
L et A, BE 21 and :
sentation. Then
is
a
coequalizer.
PROOF. The
identity
and hence lift
to a
e
is
a
reflexive
Now the
are
rows
single
coequalizers,
diagonal
equalizers
( 4.11 )
of
.
same
a
is
true
a
C 0) --+ A
Thus
of
standard
a
5-presentation
of
B,
diagram
chase
to see
that the
fixed object and consider the resultant diagram
sets. )
PROPOSITION.
coequalized by I
of
from which it is
column is. ( Hom it into
of
map
coequalizer.The
and the
are
maps
For
P ROOF. Let
122
be
is
a
of
Al.
Then
coequalizer. From ( 4.8 )
a
are
6-presentation
equalizers,
(4.12)
and ( 4.9 ),
from which the
PROPOSITION.
we
have that
isomorphism follows.
There is
1-1
a
correspondence
between :
PROOF. Let
6-presentation.
be
a
is
an
equalizer.
equalizer.
(4.13)
Since it is
a
coequalizer, it
That the third is follows from
THEOREM.
equipped with - 0 -
Given the
and
a
applying hom (1, -)
to
Section 1, the category
closed monoidal category.
hypotheses of
(-, -) is
follows that the first line of
an
r 21,
5. CARTESIAN CLOSED CATEGORIES.
( 5.1 ) Suppose the functor -O - :5 x --+ E is the cartesian product, denoted
as usual by - X - . We would like to know when the induced tensor on ;7 21 is
123
the cartesian
Of
product.
course we
may well be different from those
purposes of
know
by example
in 21 . As above,
we
products in 17 21
replace 17 a by U for
that
exposition.
( 5 .2 ) Let A E 21 and let
be
6 -pre s en tation . Th en for
a
C
E5 ,
coequalizer.The projections Cfu X C -+ C give a map Z ( Cw X C) --+ C,
which coequalizes the maps from I
( CY XC) and hence induces a map from
is
a
A O C
also
to
C. The map
coequalizes
AO C
to
the
A . This
C’c T ,
the map
factors
through
whose
projections
map
have
two
gives
any
on
A e C --+ A X C. If C’ -+ A X C is
and determines
A and C
a
a
map from
a
map with
map
that
are
the
5-presentation
is
the
map
and thus induces
CY X C
given ones. This implies that the
AO C --+ A x C is a regular epimorphism since otherwise there would
to be a map C’ -+ A X C which doesn’t factor through the image.
( 5.3 ) Thus for
so
a
Cw --+ A
some
from I
maps
a
regular epimorphism. By adjointness
Z(CwXC ) --+ (ZCw ) X C
is
a
regular epimorphism.
Let
us now
add
following
H YPOTHESIS. For any set of
onical
objects
map Z(Cw X A ) --+ (ZCw ) X A
I C w}
is
124
a
of Z and
any AE 21 ,
monomorphism.
the
can-
( 5.4 ) This hypothesis allows
Let Ao =
Y- C60
5-sieve
terminal
a
hence
a
AoXC --+ A X C
coequalizer,
( 5 .5 )
that
so
XA Ao
is
a
that
{CY --+ Ao XA Ao}
are
regular epimorphism
and
The sieve
presentation.
ZCY--+ Ao
immediately
is
so
Since also
is
in the above
conclude
us to
Now
ing (4.10)
a
it follows that
regular epimorphism,
from which A O C = A X C is immediate.
simply
to
is
repeat the argument with
object B
an
in
place
of
C,
us-
initiate the argument used in ( 5.2 ).
( 5 .6 ) One easy way in which the hypothesis of ( 5.3 ) may be satisfied is if
5 has finite sums, if any map from an object of Z to a sum in 5 factors
through
a
finite
and
sum
in 6 the injections
to
a
sum
are
monomorphic.
First consider the map
If this fails
with the
w
=
to
same
be
monomorphism,
composite in (ZCw )
a
1, ... , n , suffice
so
there
XC.
are two
maps C’
A finite number of
are
injection into the sum a monomorphism,
equal. An analysis of the argument used in (5.4)
show A 0 C = A
gument
( 5.7 )
to
derive the
THEOREM.
C for C E 5 which
X
indices, say
that the maps from C’’ factor
and with the
to
==++Z ( Cw X C)
hypothesis
Given the
for
can
right
This
hypotheses of Section
two
maps
shows that this suffices
be fed
arbitrary A .
125
it follows the
can
back into this
be summarized
1 with the tensor
on
aras
(Z
product, then the resultant category
hypothesis of (5.3) is satisfied.
the cartesian
provided
the
77
U is cartesian closed
6.EXAMPLES.
(6.1 ) The first example leads to the category of compactly generated spaces.
Let 21 be the category of Hausdorff spaces and 5 the full subcategory of
compact ones.The
tensor
is the cartesian
product
and the hom
equips the function space with the uniform convergence topology. That the
hypotheses of Section 1 are satisfied is standard ( Kelley [1955] ), and that
these of (5.6) are is trivial. The construction gives the category of compactly generated spaces with internal hom given by the compact/open topo-
logy.
(6.2) With I as above, any full subcategory of compact spaces may be taken as 6 provided it is closed under finite products. In fact, even that restriction may be
in Z
are
dropped provided
5-generated.
we can
show that finite
For then the classes of spaces
products
of
objects
generated by T
and
generated by its finite product closure are the same. Since the full
subcategory generated by the one is cartesian closed - the hypothesis of
(5.3) being satisfied in any case - it follows that the full subcategory ge-
these
nerated
buy 6
is also.
(6.3) For example, take for 5 the category consisting of the space
with the usual
spaces
topology
and its
sequentially generated.
endomorphisms.
We call the
6-generated
It is clear that first countable spaces - in
particular all Cn - are sequentially generated and so are their quotients
(see Kelley, problem 3-R where it is shown that the euclidean plane with
one axis shrunk to a point is a quotient of a first countable space that isn’t
first countable ) . Conversely, any sequentially generated space is a quotient
of a union of copies of C and hence a quotient of a first countable space.
126
Thus
sequentially generated
means
a
of
quotient
first countable space.
a
(6.4) For another example let 5 be the category consisting of the unit interval J and its endomorphisms. Let X be any space such that each point has
a
countable
decreasing
basis of
first countable.Then the
Given such
ces.
x .
By refining
and xn
connected
is determined
by
{
Un}
be
the sequence, if necessary,
we
topology
sequence {xn }--+ x,
a
Then map J --+ X
xn +1
pathwise
let
the interval
Un .
Call it
pathwise
base
neighborhood
a
may suppose that
[ 1 / n+1, 1/ n]
by letting
entirely inside
which lies
sets.
the convergent sequen-
go
to a
This describes
path
a
at
xn E Un .
between
continuous
map J --+ X and it is clear that such maps determine the
topology. Since each
Jn is pathwise first countable the 5-generated spaces - usually called simplicially generated - form a cartesian closed category.They are the same as
the
pathwise first countable
These
spaces.
topological examples
have also been considered, from
ly different points of view - by Day [1972]
substantially the same results.
(6.5) We
can
also
play
and
slightWyler [1973] - who obtain
this game with the category of
separated
uniform
spaces. A direct limit in the category of uniform spaces of compact Haus-
uniformity. For any function which is continuous
is continuous, hence uniformly continuous, on every compact subspace,
and hence uniformly continuous on the whole space. The same argument
shows it is compactly generated in the category of completely regular spa-
dorff spaces has the fine
ces.
The
converse
pletely regular
is also
true so
spaces which
are
the resultant category consists of the
compactly generated
com-
in that category.
(6.6) We may vary the examples of ( 6.2 ) - ( 6.4 ) by considering pointed
Hausdorff spaces
for 21 .
The
is
the smash
product ( the
cartesian product with the naturally embedded sum shrunk to a point) and
the internal hom of C to X is the subspace of base point preserving maps
with the same compact/open topology. The fact that the m ap
tensor
product
127
now
consists in the identification of
proof that the inverse image
vation, the equivalence
of
single
a
a
compact
compact
set
set to a
point
allows the
is compact. From that obser-
follows
readily from the analogous result for non-pointed spaces.
This example may also be varied by considering different possibilities for 5 to get pointed sequentially generated spaces, pointed simplicially generated spaces, etc...
topology) is meant a real or complex topological space equipped with an auxiliary norm. Morphisms are required to be
linear, continuous in the topology and norm reducing. Various conditions
may be imposed on the spaces but the one most useful here is the supposition that the norm and the topology are those induced by mappings to Banach
spaces. This means the space is a subspace, both in norm and topology, of
a product of Banach spaces. If complete, it is a closed subspace. VUe let
21 be the category of such complete MT spaces. Let B be the full subcategory of the Banach spaces ( i. e. the topology is that of the norm ) and
the full subcategory of those whose unit ball is compact. It is known ( see
Semadeni [1960] or Barr [1976] ) that B and 6 are dual by functors
(6.7) By
an
MT (for mixed
( describable
the space of linear functionals
topologized, respectively,
topologies). If B E B , C e 6 , ( C , B ) is defined
to be the space of morphisms C - B normed by the sup norm. It is a Banach
space. If A E 21 , let A C II Bw , then ( C, A ) is the space of morphisms given
the topology and norm induced by ( C, A ) C II ( C, Bw ). Finally, for Cj, C2
in 5 , we define
as
with the weak and the
norm
which lies in 6 . The
completeness
Barr [1976 a] .
( which
from
proof that
uses
the conditions of section 1
of the spaces,
128
by
the
way)
can
are
satisfied
be worked
out
REFERENCES
M. BARR,
M.
Duality
of Banach spaces, Cahiers
B ARR, Closed categories and Banach
B. DAY, A reflection theorem for closed
1- 11.
Topo. et
Géo.
Diff.
spaces, Idem XVII-4
( 1976), 15.
( 1976 a), 135.
categories, J. Pure Appl. Algebra
S. EILENBERG and G.M. KELLY, Closed categories, Proc.
P.
XVII- 1
Algebra (La Jolla, 1965), Springer, 1966.
GABRIEL and M. ZISMAN, Calculus of fractions
and
Conf.
on
2
( 1972),
Categorical
homotopy theory, Springer,
1967.
J.
L.
KELLEY, General Topology,
Z. SEMADENI,
Van
Projectivity, injectivity
O. WYLER, Convenient
Nostrand, 1955.
and
duality, Rozprawy
categories for Topology, General Topology and Applications
3 (1973), 225-242.
Forschungsinstitut
Mat. 35 ( 1963).
fur Mathematik
E. T. H. ZURICH
and
Mathematics Department
Mc Gill University
MONTREAL, Qu6bec
CANADA
129