<<

. 9
( 60 .)



>>

arrows p1 : P ’ A and p2 : P ’ B with the property that for any elements x of
’ ’
A and y of B based on T there is a unique element (x, y) of A — B based on T
such that p1 (x, y) = x and p2 (x, y) = y. These arrows are conventionally called
the projections, even though they need not be epimorphisms. Conversely, any
element h of A — B based on T must be of the form (x, y) for some elements of A
and B respectively based on T : namely, x = p1 (h) and y = p2 (h). In other words,
there is a canonical bijection between Hom(T, A—B) and Hom(T, A)—Hom(T, B)
(this is merely a rewording of the statement that A — B represents {(x, y): x ∈
A, y ∈ B}).
Note that (x, x ) and (x , x) are distinct elements of A — A if x and x are
distinct, because p1 (x, x ) = x, whereas p1 (x , x) = x . In fact, (x, x ) = (p2 , p1 ) —¦
(x, x ).
1.7 Limits 39
If f : A ’ C and g: B ’ D, then we de¬ne
’ ’

f — g = (f —¦ p1 , g —¦ p2 ): A — B ’ C — D


Thus for elements x of A and y of B de¬ned on the same object, (f —g)(x, y) =
(f (x), g(y)).
It should be noted that the notation A — B carries with it the information
about the arrows p1 and p2 . Nevertheless, one often uses the notation A — B to
denote the object P ; the assumption then is that there is a well-understood pair
of arrows which make it the genuine product. We point out that in general there
may be no canonical choice of which object to take be X — Y , or which arrows as
projections. There is apparently such a canonical choice in Set but that requires
one to choose a canonical way of de¬ning ordered pairs.
In a poset regarded as a category, the product of two elements is their in¬mum,
if it exists. In a group regarded as a category, products don™t exist unless the
group has only one element. The direct product of two groups is the product in
Grp and the product of two topological spaces with the product topology is the
product in T There are similar constructions in a great many categories of sets
op.
with structure.
The product of any indexed collection of objects in a category is de¬ned analo-
gously as the limit of the diagram D: I ’ C where I is the index set considered

as the objects of a graph with no arrows and D is the indexing function. This
product is denoted i∈I Di, although explicit mention of the index set is often
omitted. Also, the index is often subscripted as Di if that is more convenient.
One particular case of a product is the product over the empty index set; this is
necessarily a terminal object (Exercise 1).
There is a general associative law for products which holds up to isomorphism;
for example, for any objects A, B and C, (A—B)—C is isomorphic to A—(B —C),
and both are isomorphic to A — B — C, meaning the product over a three-element
index set with D1 = A, D2 = B and D3 = C.
There is certainly no reason to expect two objects in an arbitrary category
to have a product. A category has products if any indexed set of objects in
the category has a product. It has ¬nite products if any ¬nite indexed set of
objects has a product. By an inductive argument (Exercise 2), it is su¬cient for
¬nite products to assume an empty product and that any pair of objects has a
product. Similar terminology is used for other types of limits; in particular, a
category C has ¬nite limits or is left exact if every diagram D: I ’ C in ’
which I is a ¬nite graph, has a limit. A functor is left exact if it preserves ¬nite
limits; it is continuous if it preserves limits of all small diagrams. A category has
designated ¬nite limits if it has the additional structure of an operation that
40 1 Categories
takes each ¬nite diagram to a speci¬c limit cone over that diagram. One de¬nes
categories with designated products, designated limits, designated pullbacks, and
so on, in the same manner.

Algebraic structures in a category

The concept of product allows us to de¬ne certain notions of abstract algebra
in a category. Thus a binary operation on an object A of a category is an arrow
m: A — A ’ A (so of course, the product must exist). For elements x, y of A

de¬ned on T , we write xy for m(x, y) just as in sets. Observe that the expression
xy is de¬ned only if x and y are elements of A de¬ned on the same object. We
will use in¬x notation for symbols for binary operations such as +.
The operation m is commutative if xy = yx for all elements x and y of A;
spelled out, m(x, y) = m(y, x) for all elements x and y of A de¬ned on the same
object. The operation is associative if (xy)z = m(m(x, y), z) = m(x, m(y, z)) =
x(yz) for all elements x, y, and z de¬ned on the same object.
Thus a group in a category is an object G of the category together with an
associative binary operation on G, a function i: G ’ G, and a global element

e of G with the properties that e()x = xe() = x and xi(x) = i(x)x = e for all
x ∈ G. (In notation such as “e()x”, the element () is assumed to have the same
domain as x.) Abelian groups, rings, R-modules, monoids, and so on can all be
de¬ned in this way.

Equalizers

The equalizer of two arrows f, g: A ’ B (such arrows are said to be paral-

lel) is the object [x ∈ A | f (x) = g(x)]. As such this does not describe a commu-
tative cone, but the equivalent expression [(x, y) | x ∈ A, y ∈ B, f (x) = g(x) = y]
does describe a commutative cone, so the equalizer of f and g is the limit of the
diagram
f
A’’B’
’’

g
We will also call it Eq(f, g). In Set, the equalizer of f and g is of course the
set {x ∈ A | f (x) = g(x)}. In Grp, the kernel of a homomorphism f : G ’ H is

the equalizer of f and the constant map at the group identity.
1.7 Limits 41
Equivalence relations and kernel pairs

In Set, an equivalence relation E on a set A gives rise to a quotient set A/E,
the set of equivalence classes. In this section, we will explore the two concepts
(equivalence relations and kernel pairs) in an arbitrary category. In exercises here
and in Section 1.8 we explore their connection with the concept of coequalizer,
which is de¬ned there. In a category C that has ¬nite limits, an equivalence
relation on an object A is a subobject (u, v): E ’ A — A which is re¬‚exive,

symmetric and transitive: for any elements x, y, z of A based on T , the following
must be true:
1. (x, x) ∈T E.
2. If (x, y) ∈T E then so is (y, x).
3. If (x, y) and (y, x) are both in E then so is (x, z).
These de¬nitions can be translated into statements about diagrams (see Ex-
ercise 19).
The two projections
E’ A

’’
of an equivalence relation E )’ A — A are also called the equivalence relation.

Exercise 18 describes conditions on a parallel pair of arrows which make it an
equivalence relation, thus giving a de¬nition which works in categories without
products.
Related to this is the concept of kernel pair. If f : A ’ B is any arrow of

C , a parallel pair of arrows h: K ’ A, k: K ’ A is a kernel pair for f if
’ ’
f —¦ h = f —¦ k and whenever s, t: L ’ A is a pair of arrows for which f —¦ s = f —¦ t,

then there is a unique arrow j: L ’ K for which s = h —¦ j and t = k —¦ j. K is the

pullback of f along itself and h and k are the projections (Exercise 20). Thus
K = [(x, x ) | f —¦ x = f —¦ x ]. In Set, an equivalence relation (u, v) is the kernel
pair of its class map.

Existence of limits

The existence of some limits sometimes implies the existence of others. We
state a theorem giving the most useful variations on this theme.
Proposition 2.
(a) In any category C , the following are equivalent:
(i) C has all ¬nite limits.
42 1 Categories
(ii) C has a terminal object, all equalizers of parallel pairs, and all binary
products.
(iii) C has a terminal object and all pullbacks.

(b) A category C has all limits if and only if it has all equalizers of parallel
pairs and all products.

Proof. For (a), that (i) implies (iii) is trivial, and that (iii) implies (ii) follows
from Exercise 4.
The construction that shows (ii) implies (i) and the construction for the hard
half of (b) are essentially the same.
With a terminal object and binary products, we get, by induction, all ¬nite
products. Given a diagram D: I ’ C , with I a non-empty ¬nite graph, we

let A = i∈ObI Di and B = ±∈ArI cod ±. We de¬ne two arrows f, g: A ’ B ’
by p± —¦ f = ± —¦ pdom ± and p± —¦ g = pcod ± . This means that the following diagrams
commute.
gE
fE
Di cod ± i∈ObI Di cod ±
i∈ObI ±∈ArI ±∈ArI
d  
pdom ± p± pcod ±d  p±
d  
c c ‚©

E D cod ±
D dom ± D cod ±
±
h
If E ’ ’ Di is an equalizer of f and g, then f —¦ h = g —¦ h expresses the fact

that h: E ’ D is a cone, while the universal mapping property into the equalizer

expresses the universality of that cone. As for the empty cone, its limit is the
terminal object.
By suitable modi¬cations of this argument, we can show that a functor pre-
serves ¬nite limits if and only if it preserves binary products, the terminal object
and equalizers.

Preservation of limits

Let D: I ’ C be a diagram and F : C ’ B be a functor. Let d: lim D ’ D
’ ’ ’
be a universal element of D. We say that F preserves lim D if F d: F (lim D) ’’
F D is a universal element of F D. The following proposition gives an equivalent
condition for preserving a limit.
1.7 Limits 43
Proposition 3. F preserves the limit of D if and only if F D has a limit
d : lim F D ’ F D and there is an isomorphism g: F (lim D) ’ lim F D with
’ ’
the property that for any object T ,
F E Hom(F T, F (lim D))
Hom(T, lim D)
d
d Hom(F T, g)
d
d

Hom(T, A) Hom(F T, F d) Hom(F T, lim F D)
 
 Hom(F T, d )
 
c c 
©
E Cone(F T, F D)
Cone(T, D)
F
commutes.
The proof is trivial, but we include this diagram because it is analogous to a
later diagram (Diagram (3), Section 5.3) which is not so trivial.
Requiring that lim F D ∼ F (lim D) is not enough for preservation of limits
=
(see Exercise 16).
Given any arbitrary class of diagrams each of which has a limit in C , the
functor F preserves that class of limits if it preserves the limit of each diagram
in that class. We say, for example, that F preserves all limits (respectively
all ¬nite limits) if it preserves the limit of every diagram (respectively every
¬nite diagram). F preserves products (respectively ¬nite products) if F preserves
the limit of every discrete diagram (respectively every ¬nite discrete diagram).
To preserve ¬nite products it is su¬cient to preserve terminal objects and each
product of two objects.
A functor which preserves ¬nite limits is called left exact. This coincides
with the concept with the same name when the functor goes from a category of
R-modules to Ab.
A functor F : C ’ B creates limits of a given type if whenever D: I ’ C
’ ’
is a diagram of that type and d: lim F D ’ F D is a universal element of F D,

then there is a unique element u: X ’ D for which F u = d and moreover u is a

universal element of D. The underlying set functor from Grp to Set creates limits.
For example, that it creates products is another way of stating the familiar fact
that given two groups G and H there is a unique group structure on G—H (really
on U G — U H) making it the product in Grp.
F re¬‚ects limits of a given type if whenever D: I ’ C is a diagram of that

type, d: lim F D ’ F D is a universal element of F D and c is a cone to D for

which F c = d, then c is a universal element of D.
44 1 Categories
Exercises 1.7

1. Show that an object T is the terminal object if and only if it is the product
of the empty set of objects.

2. (a) Let A, B and C be objects in a category. Show how (A — B) — C) and
A — (B — C) can both be regarded as the product of A, B and C (by ¬nding
appropriate projection maps), so that they are isomorphic.
(b) Let C be category with an empty product and with the property that
any two objects have a product. Show that C has all ¬nite products.
(c) If you really care, state and prove a general associative law saying that
any way of meaningfully parenthesizing a sequence of objects of a category with
products gives a product which is isomorphic to that given by any other way of
parenthesizing the same sequence.

3. Show that in a category with a terminal object 1, the product A — 1 exists
for any object A and is (up to isomorphism) just A itself equipped with the
projections id: A ’ A and (): A ’ 1.
’ ’

4. Prove that in a category with ¬nite products, the equalizer of

<<

. 9
( 60 .)



>>