The notion of a category was introduced in
1942
by
S. Eilenberg
and
S. MacLane
[a1],
and has since found numerous applications in algebra,
topology and the foundations of mathematics. The intuitive idea is
that a category consists of all the objects in some
"universe of mathematical discourse"
together with all the mappings between
them. By identifying an object with its identity morphism it is possible
to define the notion of a category in terms of morphisms alone.
It is intuitively clearer and also more customary to use both
objects and morphisms. The two approaches are mixed-up to some extent
in the article above. The object and morphism definition is as follows. A category
consists of
A1) a class
whose elements are called
objects
of
,
and a class
whose elements are called
morphisms
or
arrows
of
;
A2) operations assigning to each morphism
of
a pair of objects
,
called the
domain
and
codomain
(or
source
and
target)
of
.
One writes
"a: A
B"
or
"AaB"
to mean
"a is a morphism with domain A and codomain B" ;
this rephrases 1) above;
A3) an operation assigning to each object
of
a morphism
,
called the
identity morphism
on
;
this is the precise meaning of part of 5) above;
A4) a partial (binary) product operation for morphisms, the product
(called the
composite
of
and
)
being defined if and only if
,
and satisfying
and
whenever it is defined; this rephrases 3);
these data being subject to the axioms
A5) composition is associative, i.e.
whenever both sides are defined; this rephrases 4) above;
A6) identity morphisms are units for composition, i.e.
and
whenever the composites are defined; this and A3) rephrase 5).
The classes
and
are not required to be sets, and in many of the leading examples (see,
e.g. the main text above and the examples
(A1)–(A4),
(A7)
below) they
are not sets. However, most examples have the property that, for each pair of objects
,
the collection of morphisms
with
and
forms a set (usually denoted by
or
);
such categories are sometimes called
locally small,
although other writers include this condition as part
of the definition of a category. A category
is said to be
small
if
and
are sets (cf.
Small category);
it turns out that many of the fundamental
mathematical structures may be regarded as small categories (e.g.
(A6)
and
(A7)
below).
It follows from the definition that each object in a category has
a unique identity morphism; thus it is possible to identify
objects with their identity morphisms, leading to an axiomatization of
categories in which
"morphism"
and
"composite"
are the only primitive notions (see
[9]).
Examples of categories.
(A1)
See the main article. The category
of sets is more often denoted by
.
(A3)
Next to the category
one may consider the categories
of Abelian groups,
of (right) modules over a fixed ring
,
etc.
(A4)
The category of binary relations between sets is usually denoted by
.
(A5)
A semi-group with identity
is also called a
monoid.
It defines a category with one object
,
the elements of
being interpreted as morphisms
.
(A6)
A
partially ordered set
defines a category whose objects are the elements of
,
and whose morphisms are the instances of the
order-relation: that is, there is just one morphism
if
,
and none otherwise.
(A7)
Similarly, an oriented
graph
(or
diagram scheme)
can be interpreted as a category of which the
objects are the vertices and the morphism from
to
are the oriented paths from
to
including the trivial identity paths from
to
for all vertices. Inversely a category can be seen as
a (very large) oriented graph with loops and multiple edges
together with an equivalence relation identifying certain paths, cf.
[11],
Section
II.7.
(A8)
The
homotopy category
or
has the same objects as
,
but its morphisms are homotopy classes of
continuous mappings. This category can be proved to be nonconcrete.
In the study of categories, functors (morphisms of categories) play an essential role. A
functor
consists of two functions, one assigning to each object
of
an object
of
,
and the other assigning to each morphism
of
a morphism
of
,
in such a way that the categorical structure is preserved:
,
and
whenever
is defined. There is also a third level of structure: if
and
are both functors
,
a
natural transformation
(or
functorial morphism
is a function assigning to each object
of
a morphism
in
,
such that for every
in
the diagram
commutes (i.e.
).
Functors may be composed (and every category has
an identity functor); thus there is a category
of (small) categories and functors between them. Natural
transformations may be composed; thus, given two categories
and
,
there is a category
(or
)
of functors
and natural transformations between them. This is one
important way in which new categories are constructed
from existing ones. The resulting categories are called
categories of functors
or
categories of diagrams.
The latter name is especially understandable if
is the category corresponding to a diagram
scheme (oriented graph). Indeed, then a functor
"is"
a diagram in
.
Other important constructions to obtain new categories are: taking quotients (cf.
Quotient category),
taking localizations (cf.
Localization in categories)
and constructing relative and comma categories. The important notion of a
derived category
involves several of these constructions. If
is a category and
an object of
,
then the
relative category
of objects over
has as objects all morphisms
of
into
and a morphism in
of
to
is a morphism
such that
.
Dually there is the notion of the relative category of objects in
under a given object. Intuitively an object
of
is a family of objects of
parametrized by
or a
fibre object
(fibred object).
The systematic consideration of these relative objects, i.e.
fibre objects (and their duals) combined with
base change
and
deformation
ideas has become a most important technique in many parts
of mathematics, especially in algebra (notably homological algebra), algebraic
and differential geometry, topology, and differential and algebraic topology.
It is especially important to find the right fibrewise
versions of definitions, theorems and concepts. (A second additional
not unrelated major trend involves finding the right equivariant versions in
those case in which there is a group of symmetries present (as well).)
The idea of a
comma category
generalizes that of
categories of objects over or under a given object.
Let there be given three categories
,
,
and two functors
arranged as follows
Then the
comma category
or
has as objects all triples
consisting of an object
,
an object
and a morphism
in
.
A morphism
from
to
consists of a pair of morphisms
and
such that
.
Examples of functors.
ITEM {(A9}
The
forgetful functor
sends each topological space to its underlying set, and each
continuous function to itself
( "forgetting"
the
continuity). Similarly, one has forgetful functors
,
,
etc.
(A10)
If
is a locally small category, then for each
there is a functor
sending
to the set
and
to the function which sends
to
.
Such functors (or functors isomorphic to them in
)
are called
representable
(cf.
Representable functor).
(A11)
There is a functor
sending a set
to the free group generated by
,
and a function
to the unique homomorphism
sending the generator
of
to
,
for each
.
(A12)
The (singular) homology groups of spaces (cf.
Homology group)
define functors
(one for each dimension
).
(A13)
A functor between
monoids,
considered as categories, is just a monoid homomorphism.
(A14)
A functor between partially ordered sets, considered
as categories, is just an order-preserving mapping.
(A15)
If
is a
group,
considered as a category, a functor
(respectively,
)
is a permutation (respectively, an
-linear
representation) of
(cf.
Representation of a group).
A functor
is said to be
faithful
(cf.
Faithful functor)
if it is
"injective on Hom-sets" ;
i.e. if, given two morphisms
and
with the same domain and codomain in
,
implies
.
is said to be
full
if it is
"surjective on Hom-sets"
in a similar sense. Forgetful functors (as in
(A9)
above) are always faithful. The property that a category
be
concrete
can now be rephrased as: There is a faithful functor
.
Given a category
,
one can form its
opposite
or
dual category
by keeping the same objects as
and reversing all the morphisms. The category
is isomorphic to its opposite, though most familiar categories are not. A functor
is sometimes called a
contravariant functor
from
to
;
for emphasis, functors
are then called
covariant.
(For example, if
is locally small, one may define a contravariant functor
from
to
,
by analogy with the covariant functor
of example
(A10)
above.) The
duality principle
for categories is essentially the assertion that something which is
true for all categories is true for the duals of all categories.
S. MacLane
[a4]
introduced the idea that Cartesian products can be characterized
in categorical terms, by a universal property; this
gave rise to the general categorical notion of
limit
(and the dual notion of
colimit),
which includes products as a special case (cf.
Limit
and
Universal problems).
The key notion of
adjunction
came latter
[a5]:
given functors
and
,
one says that
is
left adjoint
to
(written
)
if there is a bijection between morphisms
and morphisms
which is natural in
and
;
this is equivalent to the existence of natural transformations
and
satisfying certain identities
[11]
(cf.
Adjoint functor).
For example, the free group functor (example
(A11)
above) is left adjoint to the forgetful functor
;
Galois connections (cf.
Galois correspondence)
are examples of (contravariant) adjunctions between partially ordered sets. A functor
which has a left adjoint preserves all limits; the
converse implication is valid under suitable
"smallness conditions"
(the
adjoint functor theorem,
see
[9]).
Given an adjunction
as above, the composite functor
is equipped with natural transformations
and
satisfying certain identities; these data define the notion of a
monad
or
triple
on a category, which played a central role in
much categorical research in the
1960's and later years.
The identities which a triple
on a category
is required to satisfy are the following:
,
,
.
An
algebra for the triple
,
or
-algebra,
is an object
of
together with a morphism
such that the following identities hold:
,
.
It is a good idea to write out these requirements
in terms of commutative diagrams. They are
reminiscent of associativity and unit requirements.
Dually, i.e. reversing all arrows, there is the notion of a
cotriple
and the corresponding notion of a
co-algebra
over such a cotriple. An important example of a cotriple in the category
of commutative rings with unit element is the functor
of the
big Witt vectors
together with the structure of a
special
-ring
on
.
The co-algebras for this cotriple are precisely the special
-algebras
(cf.
Witt vector
and
-ring).
Important examples of triples arise from adjunctions
involving forgetful functors. For example, let
be the forgetful functor from the category of commutative rings with unit
element to the category of sets. This one has an adjoint
which assigns to a set
the
free commutative ring
with generator
,
i.e. the ring
of commutative polynomials over
in the variables
,
.
The
freeness property
of
,
i.e. the property that for every ring
and every collection of elements
of
there is precisely one homomorphism of rings
such that
for all
,
precisely expresses the fact that
and
are adjoint functors:
.
The corresponding natural transformation
is given by
(cf. also
Adjoint functor).
Every monad and comonad can be induced by an
adjunction; in fact there is a
"best possible"
such adjunction, in which
is taken to be the
category of
(Eilenberg–Moore)
algebras
for the monad,
[a7].
A general adjunction
is said to be
monadic
(or
is said to be monadic over
)
if
is (canonically) equivalent to the category of algebras for the induced monad on
.
The adjunction between the
and
,
mentioned above, is monadic; more generally, the categories which are monadic over
can be characterized
[a8]
as those which arise from varieties of universal algebras
(provided one allows infinitary as well as finitary algebraic operations;
the finitary case can also be characterized in categorical terms
[9],
using the notion of
algebraic theory).
See also
Variety of universal algebras.
Another phrase that is used to denote a triple
is
algebraic theory
(in monad form) over the category
.
It is so to speak the theory of the category of
-algebras.
There are, at least, two more equivalent ways in
which this notion is approached. One is as follows
[a6].
An
algebraic theory in clone form
consists of an
"object assignment function"
(
-terms
with variables in
)
for all objects
,
an
"insertion of variables mapping"
for all
and a
"clone-composition function"
for each ordered triple
of objects of
.
For each
in
let
be the composite
.
Then the data
are supposed to satisfy the following axioms. For all
,
,
and
,
This defines a new category
,
the
Kleishi category
of
.
The objects of
are the objects of
,
,
composition is given by
,
and the identity morphisms are the
in
.
A simple example of an algebraic theory in clone form is as follows. Let
be a ring with unit. For a set
let
be the vector space
.
A matrix with columns indexed by
and rows indexed by
is a mapping
,
i.e. a morphism in
;
is the
-th
column of the matrix. Given an
matrix
and a
matrix
,
define their composite
by the usual matrix product, i.e.
is the
-vector
with components
The insertion of variables assignment
is defined by
where
denotes the Kronecker delta (cf.
Kronecker symbol).
It is easily checked that the axioms above are satisfied.
Let
be an algebraic theory in clone form. For
in
define
as the composite
.
It follows readily that
is then a functor and that
is a natural transformation. Further define
as the composite
.
Then
is also a natural transformation and
is a triple. Moreover, this construction yielding a triple for each algebraic theory
in clone form is a bijection. For a discussion of the algebraic
theories (in clone and monad form) coming from a universal algebra
and a third categorical way of viewing universal algebras see
Universal algebra.
The language of categories and functors was originally introduced to
meet the needs of algebraic topology and homological algebra
[a1],
[a4].
In the
1950's and early
1960's much attention was focused on Abelian categories (cf.
Abelian category),
which may be defined as categories satisfying all the elementary properties of
;
it was shown in
[2]
that they provide an adequate foundation for
the development of homological algebra, and in
[a11]
that every small Abelian category admits a full
imbedding, preserving finite limits and colimits, into
for some
.
In an Abelian category
,
the
"Hom-sets"
have a natural
Abelian group
structure; this observation provided one of the incentives for developing the theory of
enriched
(or
relative)
categories
[a12],
that is, categories whose
"Hom-sets"
are objects of some
"base category"
.
Categories enriched over themselves (such as
and
)
are called
closed categories
[a13]
(cf.
Closed category);
an important class of closed categories (including
but not
)
consists of those where the closed structure (the
"internal Hom" )
is related by an adjunction to the categorical
product structure — such categories are called
Cartesian closed.
The notion of a Cartesian closed category played an important
role in
F.W. Lawvere's
axiomatization of the category
of small categories as a foundation for mathematics
[a14],
and in his latter development with
M. Tierney
of the notion of an
elementary
topos,
which has dominated much of categorical research in the
1970's and
1980's (see
[a15]).
Cartesian closed categories are also of importance in
logic, since they provide models for the (typed)
-calculus
(see
[a16]).
Categories enriched over
(commonly called
-categories)
have also received a good deal of attention in recent
years. They are distinguished from the general run of
enriched categories by the possibility of considering diagrams within them
which commute
"up to isomorphism"
but not exactly; the weaker notion of a
bicategory
[a17]
is a further expression of this idea.
-categories and higher-dimensional categories have also been studied, and have
proved to be of importance in the algebraic study of homotopy types
[a18].
In these areas of category theory
coherence theorems
play an important part: these are theorems which allow
one to deduce the commutativity of a large class
of diagrams from that of certain particular diagrams (see
[a19],
for example).
Further areas of category theory in which much work has
been done in recent years include the theory of
fibred categories
[a2]
(which, together with enriched category theory, is an expression of the idea that
can be replaced by some more general base category
as a foundation for much of mathematics), and the theory of
topological categories
[a3]
(which is concerned with the study of concrete categories whose forgetful functors to
have good infinitary properties, similar to those of the forgetful functor
,
see also
Topologized category).
In addition to the books
[9]
and
[11],
[12]
and
[13]
may also be recommended as general accounts of category theory.