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arrows p1 : P в€’ A and p2 : P в€’ B with the property that for any elements x of
в†’ в†’
A and y of B based on T there is a unique element (x, y) of A Г— 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 Г— 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, AГ—B) and Hom(T, A)Г—Hom(T, B)
(this is merely a rewording of the statement that A Г— B represents {(x, y): x в€€
A, y в€€ B}).
Note that (x, x ) and (x , x) are distinct elements of A Г— A if x and x are
distinct, because p1 (x, x ) = x, whereas p1 (x , x) = x . In fact, (x, x ) = (p2 , p1 ) в—¦
(x, x ).
1.7 Limits 39
If f : A в€’ C and g: B в€’ D, then we deп¬Ѓne
в†’ в†’

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

Thus for elements x of A and y of B deп¬Ѓned on the same object, (f Г—g)(x, y) =
(f (x), g(y)).
It should be noted that the notation A Г— B carries with it the information
about the arrows p1 and p2 . Nevertheless, one often uses the notation A Г— 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 Г— 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 deп¬Ѓning ordered pairs.
In a poset regarded as a category, the product of two elements is their inп¬Ѓmum,
if it exists. In a group regarded as a category, products donвЂ™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 deп¬Ѓned analo-
gously as the limit of the diagram D: I в€’ C where I is the index set considered
в†’
as the objects of a graph with no arrows and D is the indexing function. This
product is denoted iв€€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, (AГ—B)Г—C is isomorphic to AГ—(B Г—C),
and both are isomorphic to A Г— B Г— 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 п¬Ѓnite products if any п¬Ѓnite indexed set of
objects has a product. By an inductive argument (Exercise 2), it is suп¬ѓcient for
п¬Ѓnite 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 п¬Ѓnite limits or is left exact if every diagram D: I в€’ C in в†’
which I is a п¬Ѓnite graph, has a limit. A functor is left exact if it preserves п¬Ѓnite
limits; it is continuous if it preserves limits of all small diagrams. A category has
designated п¬Ѓnite limits if it has the additional structure of an operation that
40 1 Categories
takes each п¬Ѓnite diagram to a speciп¬Ѓc limit cone over that diagram. One deп¬Ѓnes
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 deп¬Ѓne certain notions of abstract algebra
in a category. Thus a binary operation on an object A of a category is an arrow
m: A Г— A в€’ A (so of course, the product must exist). For elements x, y of A
в†’
deп¬Ѓned on T , we write xy for m(x, y) just as in sets. Observe that the expression
xy is deп¬Ѓned only if x and y are elements of A deп¬Ѓned on the same object. We
will use inп¬Ѓx 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 deп¬Ѓned 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 deп¬Ѓned 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 в€’ G, and a global element
в†’
e of G with the properties that e()x = xe() = x and xi(x) = i(x)x = e for all
x в€€ G. (In notation such as вЂњe()xвЂќ, the element () is assumed to have the same
domain as x.) Abelian groups, rings, R-modules, monoids, and so on can all be
deп¬Ѓned in this way.

Equalizers

The equalizer of two arrows f, g: A в€’ B (such arrows are said to be paral-
в†’
lel) is the object [x в€€ A | f (x) = g(x)]. As such this does not describe a commu-
tative cone, but the equivalent expression [(x, y) | x в€€ A, y в€€ 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
Aв€’в†’Bв€’
в€’в†’
в€’
g
We will also call it Eq(f, g). In Set, the equalizer of f and g is of course the
set {x в€€ A | f (x) = g(x)}. In Grp, the kernel of a homomorphism f : G в€’ H is
в†’
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 deп¬Ѓned there. In a category C that has п¬Ѓnite limits, an equivalence
relation on an object A is a subobject (u, v): E в€’ A Г— A which is reп¬‚exive,
в†’
symmetric and transitive: for any elements x, y, z of A based on T , the following
must be true:
1. (x, x) в€€T E.
2. If (x, y) в€€T E then so is (y, x).
3. If (x, y) and (y, x) are both in E then so is (x, z).
These deп¬Ѓnitions can be translated into statements about diagrams (see Ex-
ercise 19).
The two projections
Eв€’ A
в†’
в€’в†’
of an equivalence relation E )в€’ A Г— A are also called the equivalence relation.
в†’
Exercise 18 describes conditions on a parallel pair of arrows which make it an
equivalence relation, thus giving a deп¬Ѓnition which works in categories without
products.
Related to this is the concept of kernel pair. If f : A в€’ B is any arrow of
в†’
C , a parallel pair of arrows h: K в€’ A, k: K в€’ A is a kernel pair for f if
в†’ в†’
f в—¦ h = f в—¦ k and whenever s, t: L в€’ A is a pair of arrows for which f в—¦ s = f в—¦ t,
в†’
then there is a unique arrow j: L в€’ K for which s = h в—¦ j and t = k в—¦ j. K is the
в†’
pullback of f along itself and h and k are the projections (Exercise 20). Thus
K = [(x, x ) | f в—¦ x = f в—¦ 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 п¬Ѓnite 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

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 п¬Ѓnite
products. Given a diagram D: I в€’ C , with I a non-empty п¬Ѓnite graph, we
в†’
let A = iв€€ObI Di and B = О±в€€ArI cod О±. We deп¬Ѓne two arrows f, g: A в€’ B в†’
by pО± в—¦ f = О± в—¦ pdom О± and pО± в—¦ g = pcod О± . This means that the following diagrams
commute.
gE
fE
Di cod О± iв€€ObI Di cod О±
iв€€ObI О±в€€ArI О±в€€ArI
d В
pdom О± pО± pcod О±d В pО±
d В
dВ
E D cod О±
D dom О± D cod О±
О±
h
If E в€’ в†’ Di is an equalizer of f and g, then f в—¦ h = g в—¦ h expresses the fact
в€’
that h: E в€’ D is a cone, while the universal mapping property into the equalizer
в†’
expresses the universality of that cone. As for the empty cone, its limit is the
terminal object.
By suitable modiп¬Ѓcations of this argument, we can show that a functor pre-
serves п¬Ѓnite limits if and only if it preserves binary products, the terminal object
and equalizers.

Preservation of limits

Let D: I в€’ C be a diagram and F : C в€’ B be a functor. Let d: lim D в€’ D
в†’ в†’ в†’
be a universal element of D. We say that F preserves lim D if F d: F (lim D) в€’в†’
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 в€’ F D and there is an isomorphism g: F (lim D) в€’ lim F D with
в†’ в†’
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
В‚
Hom(T, A) Hom(F T, F d) Hom(F T, lim F D)
В
В Hom(F T, d )
В
c cВ
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 в€ј 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 п¬Ѓnite limits) if it preserves the limit of every diagram (respectively every
п¬Ѓnite diagram). F preserves products (respectively п¬Ѓnite products) if F preserves
the limit of every discrete diagram (respectively every п¬Ѓnite discrete diagram).
To preserve п¬Ѓnite products it is suп¬ѓcient to preserve terminal objects and each
product of two objects.
A functor which preserves п¬Ѓnite 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 в€’ B creates limits of a given type if whenever D: I в€’ C
в†’ в†’
is a diagram of that type and d: lim F D в€’ F D is a universal element of F D,
в†’
then there is a unique element u: X в€’ D for which F u = d and moreover u is a
в†’
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 GГ—H (really
on U G Г— U H) making it the product in Grp.
F reп¬‚ects limits of a given type if whenever D: I в€’ C is a diagram of that
в†’
type, d: lim F D в€’ F D is a universal element of F D and c is a cone to D for
в†’
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 Г— B) Г— C) and
A Г— (B Г— C) can both be regarded as the product of A, B and C (by п¬Ѓnding
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 п¬Ѓnite 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 Г— 1 exists
for any object A and is (up to isomorphism) just A itself equipped with the
projections id: A в€’ A and (): A в€’ 1.
в†’ в†’

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

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