Category (mathematics)
In
mathematics,
categories allow one to formalize notions involving abstract structure and processes that preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as
category theory.
For more extensive motivational background and historical notes, see
category theory and the
list of category theory topics.
A
category C consists of
* a
class ob(
C) of
objects:
* a class hom(
C) of
morphisms. Each morphism
f has a unique
source object a and
target object b where
a and
b are in ob(
C). We write
f:
a â†'
b, and we say
"f is a morphism from
a to
b". We write hom(
a,
b) (or hom
C(
a,
b)) to denote the
hom-class of all morphisms from
a to
b. (Some authors write Mor(
a,
b).)
* for every three objects
a,
b and
c, a binary operation hom(
a,
b) × hom(
b,
c) â†' hom(
a,
c) called
composition of morphisms; the composition of
f :
a â†'
b and
g :
b â†'
c is written as
g o
f or
gf (Some authors write
fg.)
such that the following axioms hold:
* (associativity) if
f :
a â†'
b,
g :
b â†'
c and
h :
c â†'
d then
h o (
g o
f) = (
h o
g) o
f, and
* (identity) for every object
x, there exists a morphism 1
x :
x â†'
x called the
identity morphism for x, such that for every morphism
f :
a â†'
b, we have 1
b o
f =
f =
f o 1
a.
From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.
A
small category is a category in which both ob(
C) and hom(
C) are actually
sets and not proper classes. A category that is not small is said to be
large. A
locally small category is a category such that for all objects
a and
b, the hom-class hom(
a,
b) is a set. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small.
The morphisms of a category are sometimes called
arrows due to the influence of
commutative diagrams.
Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.
*The category
Set of all
sets together with
functions between sets, where composition is the usual function composition (The following are
subcategories of
Set, obtained by adding some type of structure onto a set, by requiring that morphisms are functions that respect this added structure, and where morphism composition is simply ordinary function composition.)
**The category
Rel of all
sets with
relations**The category
Ord of all
preordered sets with
monotonic functions
**The category
Mag consisting of all
magmas with their
homomorphisms
**The category
Med consisting of all
medial magmas with their
homomorphisms
**The category
Grp consisting of all
groups with their
group homomorphisms
**The category
Ab consisting of all
abelian groups with their
group homomorphisms
**The category
VectK of all
vector spaces over the
field K (which is held fixed) with their
K-
linear maps
**The category
Top of all
topological spaces with
continuous functions
**The category
Met of all
metric spaces with
short maps
**The category
Uni of all
uniform spaces with
uniformly continuous functions
*The category
Cat of all small categories with functors
*Any
preordered set (
P, ≤) forms a small category, where the objects are the members of
P, the morphisms are arrows pointing from
x to
y when
x ≤
y (The composition law is forced, because there is at most one morphism from any object to another.)
*Any
monoid forms a small category with a single object
x. (Here,
x is any fixed set.) The morphisms from
x to
x are precisely the elements of the monoid, and the categorical composition of morphisms is given by the monoid operation. In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
*Any
directed graph generates a small category: the objects are the
vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the
free category generated by the graph.
*If
I is a
set, the
discrete category on I is the small category that has the elements of
I as objects and only the identity morphisms as morphisms. Again, the composition law is forced.)
*Any category
C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the
dual or opposite category and is denoted
Cop.
*If
C and
D are categories, one can form the
product category C ×
D: the objects are pairs consisting of one object from
C and one from
D, and the morphisms are also pairs, consisting of one morphism in
C and one in
D. Such pairs can be composed componentwise.
A
morphism f :
a â†'
b is called
* a
monomorphism (or
monic) if
fg1 =
fg2 implies
g1 =
g2 for all morphisms
g1,
g2 :
x â†'
a.
* an
epimorphism (or
epic) if
g1f =
g2f implies
g1 =
g2 for all morphisms
g1,
g2 :
b â†'
x.
* a
bimorphism if it is both a monomorphism and an epimorphism.
* a
retraction if it has a right inverse, i.e. if there exists a morphism
g :
b â†'
a with
fg = 1
b.
* a
section if it has a left inverse, i.e. if there exists a morphism
g :
b â†'
a with
gf = 1
a.
* an
isomorphism if it has an inverse, i.e. if there exists a morphism
g :
b â†'
a with
fg = 1
b and
gf = 1
a.
* an
endomorphism if
a =
b. The class of endomorphisms of
a is denoted end(
a).
* an
automorphism if
f is both an endomorphism and an isomorphism. The class of automorphisms of
a is denoted aut(
a).
Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:
*
f is a monomorphism and a retraction;
*
f is an epimorphism and a section;
*
f is an isomorphism.
Relations among morphisms (such as
fg =
h) can most conveniently be represented with
commutative diagrams, where the objects are represented as points and the morphisms as arrows.
* In many categories, the hom-sets hom(
a,
b) are not just sets but actually
abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is
bilinear. Such a category is called
preadditive. If, furthermore, the category has all finite
products and
coproducts, it is called an
additive category. If all morphisms have a
kernel and a
cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an
abelian category. A typical example of an abelian category is the category of abelian groups.
* A category is called
complete if all
limits exist in it. The categories of sets, abelian groups and topological spaces are complete.
* A category is called
cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
* A
topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
* A
groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups,
group actions and
equivalence relations.
* Adámek, JiÅ™Ã, Herrlich, Horst, & Strecker, George E. (1990).
Abstract and Concrete Categories. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
* Asperti, Andrea, & Longo, Giuseppe (1991). [ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf
Categories, Types and Structures]. Originally publ. M.I.T. Press.
* Barr, Michael, & Wells, Charles (2002).
Toposes, Triples and Theories. (revised and corrected free online version of
Grundlehren der mathematischen Wissenschaften (278). Springer-Verlag,1983)
* Borceux, Francis (1994).
Handbook of Categorical Algebra.. Vols. 50-52 of
Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
* Lawvere, William, & Schanuel, Steve. (1997).
Conceptual Mathematics: A First Introduction to Categories. Cambridge: Cambridge University Press.
* Mac Lane, Saunders (1998).
Categories for the Working Mathematician (2nd ed.). Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.
* Jean-Pierre Marquis,
"Category Theory" in
Stanford Encyclopedia of Philosophy, 2006
*
Homepage of the Categories mailing list, with extensive list of resources
*
Category Theory section of Alexandre Stefanov's list of free online mathematics resources
