This post is part of a series on category theory. See Overview of Blog Posts for all posts.
Introduction
Many objects or contstructions can be characterized by a so-called universal property that describes either the morphisms to them or the morphisms from them. By the virtue of the Yoneda lemma, we can forget about any concrete representation of the object and can only work with its universal property. One can think of a universal property as a “universal” solution to a problem in the sense that it is uniquely related to all other solutions. For example, consider the problem of finding a commutative ring and two elements
such that
. The universal solution to this problem is given by the elements
and
in
; this solution is universal in the sense that for every other commutative ring
and elements
such that
, there’s a unique ring homomorphism
such that
and
. The general definition is a straightforward abstraction from this example. Universal properties are closely related to representable functors and this relation is established by the Yoneda lemma. Understanding objects in terms of universal properties is the epitome of categorial thinking.
Representability and Universal Elements
Throughout this section, let be a locally small category and let
be a functor.
Definition C5.1 is called representable if there is natural isomorphism
for some object
. We say that
is represented by
and call
a representing object. The pair
is called a representation of
.
Definition C5.2 For an object , we say that an element
is universal if the following property holds: For any object
and
, there’s a unique morphism
with
. In this case, this property is called the universal property of
.
Lemma C5.3 For an object , the representations
are in bijection to universal elements of the form
via
Proof Via the Yoneda lemma, we know that natural transformations correspond to elements in
via
. We need to check that
is an isomorphism if and only if
is an universal element. For every object
, we have
, but by naturality of
, we have for any morphism
,
(cf. the proof of the Yoneda lemma). Thus
is a natural isomorphism if and only if for every object
, the map
is a bijection, which is just a restatement of saying that
is a universal element.
Example C5.4 Let be a commutative ring and consider the forgetful functor
from the category of commutative
-algebras
to the category of sets
, that sends an
-algebra to its underlying set. Then
is the represented by the polynomial ring
. Concretely, we have a natural isomorphism
.
is the corresponding universal element. This holds, because any
-algebra homomorphism
is given by evaluation at a uniquely determined element, namely
.
Example C5.5 Let be a group with a presentation
with
being the set of generators and
the set of relations. Then for if we have for a group
for every generator
an element
such that the relations in
are satisfied by those corresponding elements, this uniquely determines a group homomorphism
. Consequently, for the functor
sending a group
to the families of elements in
indexed by
such that the corresponding relations are satisfied, we see that the family
in
is an universal element and
is represented by
.
Example C5.6 Let be a ring and
, then consider the functor
, i.e. from the category of left
-modules with
-linear maps to the category of sets that sends a module
to the set
. Then this functor is represented by
and
is a universal element.
Example C5.7 Let be the functor from the category of small categories to the category of sets that sends a small category
to the set of all morphisms in
. Then
is represented by the arrow category from definition C3.6. The unique morphism
in the arrow category is a universal element.
Example C5.8 Let be a small category, let
be an object and consider the functor from the functor category
to
sending a functor
to
and a natural transformation
to
. Then the Yoneda lemma in this context can be stated by saying that this functor is representable by
and the identity
is a universal element.
Example C5.9 Let be a group, not necessarily abelian. Then consider the functor from the category of all abelian groups to the category of sets
given by taking all group homomorphism to an abelian group:
(this is a hom-functor from a larger category restricted to a subcategory). Then this functor is representable by the abelianization
with the quotient map
as a universal element in
: This is the universal property of the abelianization: Every map from
to an abelian group factorizes over the quotient map
.
Example C5.10 To give an example of a contravariant representable functor, consider the functor from the opposite category of topological spaces to the category of sets that sends a topological space to the set consisting of all of its open subsets. A continuous map
is sent to the map
. This functor is represented by the Sierpinski space
whose points are
with the topology that has as open sets
. (For all algebraic geometers: this arises as the spectrum of a discrete valuation ring.) The open subset
is a universal element. This reason this holds is that for any map
, we always have
and
, so the only interesting part of being continuous is requiring that
is open. Since
has only two points, knowing
completely determines
. Putting these things together, we get that the map
is a bijection, proving the claim.
Example C5.11 Not every functor is representable. As a boring example, one can take the constant functor that sends every object to the empty set. (Exercise: why will this never be representable, provided that
is non-empty?) For a more interesting example, consider the functor from the category of all groups to the category of sets that sends a group to the set of all of its torsion elements, i.e. elements of finite order.
. (This defines a functor because group homomorphisms send torsion elements to torsion elements.) Suppose that
is represented by a group
and let
be a universal element. Then for every
, we have an element
. By universality of
, there is a group homomorphism
that sends
to
. But this implies that
divides the order of
. As
was arbitrary, this is impossible, as the order of
is finite.
Universal Elements as Initial Objects
One can ask for a characterization for representability of a functor that doesn’t refer to a representing . In this section, we shall introduce the notion of an initial object to obtain such a criterion, although the criterion will be a mere reformulation of the definition. This reformulation will be put to use in the future to prove a more useful criterion, for which we lack the tools at the moment. We shall return to the notion of initial objects later when we generalize them vastly to colimits.
Definition C5.12 Let be a one-point set. An object
in a category
is called initial if it represents the constant functor
, given by sending each object in
to
and every morphism to
.
Unfolding the definition, this means that for an object is initial if for any object
, there is exactly one morphism
We prove an elementary, but useful property of initial objects:
Lemma C5.13 If a category has an initial object, it is unique up to unique isomorphism.
Proof Let and
be initial objects in
, then there are unique morphisms
and
. Then both
and
are morphisms
, so they must agree, thus
. By symmetry,
proving that
and
are mutually inverse isomorphisms.
is necessarily unique as
is initial.
Definition C5.14 Let be a functor where
is a locally small category. Define the category of elements
as follows:
- Objects are pairs
consisting of an element
and an element
.
- Morphisms
are morphisms
in
such that that
.
Lemma C5.15 satisfies the universal property for
if and only if
is an initial object in
.
Proof Immediate from the definitions.
Corollary C5.16 Object characterized by a universal property are unique up to unique isomorphism in the following sense: if and
both satisfy the universal property for a functor
, then there exists a unique isomorphism
such that
.
Proof By lemma C5.15, both and
are initial objects in
. Now apply lemma C5.13.
This concludes this post on representability and universal properties. Next time, we will see an important class of examples for those concepts, given by (co)limits.