A category with objects X, Y, Z and morphisms
f,
g
In
mathematics,
category
theory deals in an
abstract way
with
mathematical structures
and relationships between them: it abstracts from
set and
function to
objects
linked in diagrams by
morphisms or
arrows.
One of the simplest examples of a category (which is a very
important concept in
topology) is that of
groupoid, defined as a category whose
arrows or morphisms are all invertible.
Categories now appear in most
branches of mathematics, some areas of
theoretical computer science
where they correspond to
types, and
mathematical physics where they can be
used to describe
vector spaces.
Category theory provides both with a unifying notion and
terminology. Categories were first introduced by
Samuel Eilenberg and
Saunders Mac Lane in 1942–45, in
connection with
algebraic
topology.
Category theory has several faces known not just to specialists,
but to other mathematicians. A term dating from the 1940s,
"
general abstract nonsense",
refers to its high level of abstraction, compared to more classical
branches of mathematics.
Homological
algebra is category theory in its aspect of organising and
suggesting manipulations in
abstract
algebra.
Diagram chasing is a
visual method of arguing with abstract "arrows" joined in diagrams.
Note that arrows between categories are called
functors, subject to specific defining commutativity
conditions; moreover, categorical diagrams and sequences can be
defined as functors (viz. Mitchell, 1965). An arrow between two
functors is a
natural
transformation when it is subject to certain naturality or
commutativity conditions. Both functors and natural transformations
are key concepts in category theory, or the "real engines" of
category theory. To paraphrase a famous sentence of the
mathematicians who founded category theory: 'Categories were
introduced to define functors, and functors were introduced to
define natural transformations'.
Topos theory
is a form of abstract
sheaf
theory, with geometric origins, and leads to ideas such as
pointless topology. A topos can
also be considered as a specific type of category with two
additional topos axioms.
Background
The study of
categories is an
attempt to
axiomatically capture what is commonly found in
various classes of related
mathematical structures by
relating them to the
structure-preserving functions
between them. A systematic study of category theory then allows us
to prove general results about any of these types of mathematical
structures from the axioms of a category.
Consider the following example. The
class Grp of
groups consists of all objects having a
"group structure". One can proceed to
prove theorems
about groups by making logical deductions from the set of axioms.
For example, it is immediately proved from the axioms that the
identity element of a group is
unique.
Instead of focusing merely on the individual objects (e.g., groups)
possessing a given structure, category theory emphasizes the
morphisms – the structure-preserving
mappings –
between these objects; by studying these
morphisms, we are able to learn more about the structure of the
objects. In the case of groups, the morphisms are the
group homomorphisms. A group homomorphism
between two groups "preserves the group structure" in a precise
sense – it is a "process" taking one group to another, in a way
that carries along information about the structure of the first
group into the second group. The study of group homomorphisms then
provides a tool for studying general properties of groups and
consequences of the group axioms.
A similar type of investigation occurs in many mathematical
theories, such as the study of
continuous maps (morphisms)
between
topological spaces in
topology (the associated category is called
Top), and the study of
smooth functions (morphisms) in
manifold theory.
If one axiomatizes
relations
instead of
function, one
obtains the theory of
allegories.
Functors
Abstracting again, a category is
itself a type of
mathematical structure, so we can look for "processes" which
preserve this structure in some sense; such a process is called a
functor. A functor associates to every
object of one category an object of another category, and to every
morphism in the first category a morphism in the second.
In fact, what we have done is define a category
of categories
and functors – the objects are categories, and the morphisms
(between categories) are functors.
By studying categories and functors, we are not just studying a
class of mathematical structures and the morphisms between them; we
are studying the
relationships between various classes of
mathematical structures. This is a fundamental idea, which
first surfaced in
algebraic
topology. Difficult
topological questions can be
translated into
algebraic questions which are often easier
to solve. Basic constructions, such as the
fundamental group or
fundamental groupoid of a
topological space, can be expressed as
fundamental functors to the category of
groupoids in this way, and the concept is
pervasive in algebra and its applications.
Natural transformation
Abstracting yet again, constructions are often "naturally related"
– a vague notion, at first sight. This leads to the clarifying
concept of
natural
transformation, a way to "map" one functor to another. Many
important constructions in mathematics can be studied in this
context. "Naturality" is a principle, like
general covariance in physics, that cuts
deeper than is initially apparent.
Historical notes
In 1942–45,
Samuel Eilenberg and
Saunders Mac Lane introduced
categories, functors, and natural transformations as part of their
work in topology, especially
algebraic topology. Their work was an
important part of the transition from intuitive and geometric
homology to
axiomatic homology
theory. Eilenberg and Mac Lane later wrote that their goal was
to understand natural transformations; in order to do that,
functors had to be defined, which required categories.
Stanislaw Ulam, and some writing on
his behalf, have claimed that related ideas were current in the
late 1930s in Poland. Eilenberg was Polish, and studied mathematics
in Poland in the 1930s. Category theory is also, in some sense, a
continuation of the work of
Emmy
Noether (one of Mac Lane's teachers) in formalizing abstract
processes; Noether realized that in order to understand a type of
mathematical structure, one needs to understand the processes
preserving that structure. In order to achieve this understanding,
Eilenberg and Mac Lane proposed an axiomatic formalization of the
relation between structures and the processes preserving
them.
The subsequent development of category theory was powered first by
the computational needs of
homological algebra, and later by the
axiomatic needs of
algebraic
geometry, the field most resistant to being grounded in either
axiomatic set theory or the
Russell-Whitehead view of united
foundations. General category theory, an extension of
universal algebra having many new features
allowing for
semantic flexibility and
higher-order logic, came later;
it is now applied throughout mathematics.
Certain categories called
topoi (singular
topos) can even serve as an alternative to
axiomatic set theory as a foundation of
mathematics. These foundational applications of category theory
have been worked out in fair detail as a basis for, and
justification of,
constructive mathematics. More
recent efforts to introduce undergraduates to categories as a
foundation for mathematics include
Lawvere and Rosebrugh (2003) and Lawvere and
Schanuel (1997).
Categorical logic is now a
well-defined field based on
type theory
for
intuitionistic logics, with
applications in
functional
programming and
domain theory,
where a
cartesian closed
category is taken as a non-syntactic description of a
lambda calculus. At the very least, category
theoretic language clarifies what exactly these related areas have
in common (in some
abstract
sense).
Categories, objects and morphisms
A
category C consists of the following three
mathematical entities:
- A class ob(C), whose
elements are called objects;
- A class hom(C), whose elements are called morphisms or maps
or arrows. Each morphism f has a unique
source object a and target object b. We write
f: a → b, and we say "f is a
morphism from a to b". We write hom(a,
b) (or Hom(a, b), or
hom_{C}(a, b), or
Mor(a, b), or C(a, b))
to denote the hom-class of all morphisms from a
to b.
- A binary operation \circ,
called composition of morphisms, such that for any three
objects a, b, and c, we have
hom(a, b) × hom(b, c) →
hom(a, c). The composition of f:
a → b and g: b → c is
written as g\circ f or gf Some authors compose in the
opposite order, writing fg or f\circ g for g\circ f.
Computer scientists using category theory very commonly write
f;g for g\circ f, governed by two axioms:
- * Associativity: If f :
a → b, g : b → c and
h : c → d then h\circ(g\circ f)=(h\circ
g)\circ f, and
- * Identity: For every
object x, there exists a morphism 1_{x} :
x → x called the identity morphism for x, such that
for every morphism f : a → b, we have
{\rm 1}_b\circ f=f=f\circ{\rm 1}_a.
From these axioms, it can be proved that there is exactly one
identity morphism for every
object. Some authors deviate from the definition just given by
identifying each object with its identity morphism.
Relations among morphisms (such as
fg =
h) are
often depicted using
commutative
diagrams, with "points" (corners) representing objects and
"arrows" representing morphisms.
Properties of morphisms
Some morphisms have important properties. A morphism
f :
a →
b is:
- a monomorphism (or monic)
if fog_{1} =
fog_{2} implies
g_{1} = g_{2} for all morphisms
g_{1}, g_{2} : x →
a.
- an epimorphism (or epic) if
g_{1}of =
g_{2}of implies
g_{1} = g_{2} for all morphisms
g_{1}, g_{2} : b →
x.
- an isomorphism if there exists a
morphism g : b → a with
fog = 1_{b} and
gof = 1_{a}.
- an endomorphism if a =
b. end(a) denotes the class of endomorphisms of
a.
- an automorphism if f is
both an endomorphism and an isomorphism. aut(a) denotes
the class of automorphisms of a.
Functors
Functors are structure-preserving maps
between categories. They can be thought of as morphisms in the
category of all (small) categories.
A (
covariant) functor
F from a category
C to a category
D, written
F:
C
→
D, consists of:
- for each object x in C, an object
F(x) in D; and
- for each morphism f : x → y in
C, a morphism F(f) :
F(x) → F(y),
such that the following two properties hold:
- For every object x in C,
F(1_{x}) =
1_{F(x)};
- For all morphisms f : x → y and
g : y → z, F(g\circ f)=F(g)\circ
F(f).
A
contravariant functor
F:
C →
D, is like a covariant functor, except that it "turns
morphisms around" ("reverses all the arrows"). More specifically,
every morphism
f :
x →
y in
C
must be assigned to a morphism
F(
f) :
F(
y) →
F(
x) in
D. In
other words, a contravariant functor is a covariant functor from
the
opposite category
C^{op} to
D.
Natural transformations and isomorphisms
A
natural transformation is a relation between two
functors. Functors often describe "natural constructions" and
natural transformations then describe "natural homomorphisms"
between two such constructions. Sometimes two quite different
constructions yield "the same" result; this is expressed by a
natural isomorphism between the two functors.
If
F and
G are (covariant) functors between the
categories
C and
D, then a natural transformation
from
F to
G associates to every object
x
in
C a morphism η
_{x} :
F(
x) →
G(
x) in
D such
that for every morphism
f :
x →
y in
C, we have η
_{y} o
F(
f) =
G(
f)
o
η
_{x}; this means that the following diagram is
commutative:
The two functors
F and
G are called
naturally
isomorphic if there exists a natural transformation from
F to
G such that η
_{x} is an
isomorphism for every object
x in
C.
Universal constructions, limits, and colimits
Using the language of category theory, many areas of mathematical
study can be cast into appropriate categories, such as the
categories of all sets, groups, topologies, and so on. These
categories surely have some objects that are "special" in a certain
way, such as the
empty set or the
product of two topologies, yet in the
definition of a category, objects are considered to be atomic,
i.e., we
do not know whether an object
A is a
set, a topology, or any other abstract concept – hence, the
challenge is to define special objects without referring to the
internal structure of those objects. But how can we define the
empty set without referring to elements, or the product topology
without referring to open sets?
The solution is to characterize these objects in terms of their
relations to other objects, as given by the morphisms of the
respective categories. Thus, the task is to find
universal properties that uniquely
determine the objects of interest. Indeed, it turns out that
numerous important constructions can be described in a purely
categorical way. The central concept which is needed for this
purpose is called categorical
limit, and can be dualized to
yield the notion of a
colimit.
Equivalent categories
It is a natural question to ask: under which conditions can two
categories be considered to be "essentially the same", in the sense
that theorems about one category can readily be transformed into
theorems about the other category? The major tool one employs to
describe such a situation is called
equivalence of
categories, which is given by appropriate functors between two
categories. Categorical equivalence has found numerous applications
in mathematics.
Further concepts and results
The definitions of categories and functors provide only the very
basics of categorical algebra; additional important topics are
listed below. Although there are strong interrelations between all
of these topics, the given order can be considered as a guideline
for further reading.
- The functor category
D^{C} has as objects the functors from
C to D and as morphisms the natural
transformations of such functors. The Yoneda lemma is one of the most famous basic
results of category theory; it describes representable functors in
functor categories.
- Duality: Every statement,
theorem, or definition in category theory has a dual which
is essentially obtained by "reversing all the arrows". If one
statement is true in a category C then its dual will be
true in the dual category C^{op}. This duality,
which is transparent at the level of category theory, is often
obscured in applications and can lead to surprising
relationships.
- Adjoint functors: A functor can
be left (or right) adjoint to another functor that maps in the
opposite direction. Such a pair of adjoint functors typically
arises from a construction defined by a universal property; this
can be seen as a more abstract and powerful view on universal
properties.
Higher-dimensional categories
Many of the above concepts, especially equivalence of categories,
adjoint functor pairs, and functor categories, can be situated into
the context of
higher-dimensional categories. Briefly, if
we consider a morphism between two objects as a "process taking us
from one object to another", then higher-dimensional categories
allow us to profitably generalize this by considering
"higher-dimensional processes".
For example, a (strict)
2-category is a
category together with "morphisms between morphisms", i.e.,
processes which allow us to transform one morphism into another. We
can then "compose" these "bimorphisms" both horizontally and
vertically, and we require a 2-dimensional "exchange law" to hold,
relating the two composition laws. In this context, the standard
example is
Cat, the 2-category of all (small)
categories, and in this example, bimorphisms of morphisms are
simply
natural
transformations of morphisms in the usual sense. Another basic
example is to consider a 2-category with a single object; these are
essentially
monoidal categories.
Bicategories are a weaker notion of
2-dimensional categories in which the composition of morphisms is
not strictly associative, but only associative "up to" an
isomorphism.
This process can be extended for all
natural numbers n, and these are
called
n-categories. There is
even a notion of
ω-category
corresponding to the
ordinal number
ω.
Higher-dimensional categories are part of the broader mathematical
field of
higher-dimensional
algebra,a concept introduced by
Ronald Brown. For a
conversational introduction to these ideas, see
John
Baez, 'A Tale of n-categories' (1996).
See also
Notes
- Note that a morphism that is both epic and monic is not
necessarily an isomorphism! For example, in the category consisting
of two objects A and B, the identity morphisms,
and a single morphism f from A to B,
f is both epic and monic but is not an isomorphism.
References
Freely available online:
- Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990)
Abstract and concrete categories. John Wiley
& Sons. ISBN 0-471-60922-6.
- Freyd, Peter J. (1964) Abelian Categories. New York: Harper and
Row.
- Michael Barr and Charles Wells (1999) Category Theory Lecture Notes. Based on their
book Category Theory for Computing Science.
- ——— (2002) Toposes, triples and theories. Revised and
corrected translation of Grundlehren der mathematischen
Wissenschaften (Springer-Verlag, 1983).
- Leinster, Tom (2004) Higher
operads, higher categories (London Math. Society Lecture
Note Series 298). Cambridge Univ. Press.
- Schalk, A. and Simmons, H. (2005) An introduction to Category Theory in four easy
movements. Notes for a course offered as part of the
MSc. in Mathematical Logic,
Manchester
University.
- Turi, Daniele (1996–2001) Category Theory Lecture Notes. Based on Mac Lane
(1998).
- Goldblatt, R (1984) Topoi: the Categorial Analyis of Logic
A clear introduction to categories, with particular emphasis on the
recent applications to logic.
- A. Martini, H. Ehrig, and D. Nunes (1996) Elements of Basic Category Theory (Technical
Report 96-5, Technical University Berlin)
Other:
- Awodey, Steven (2006). Category Theory (Oxford Logic
Guides 49). Oxford University Press.
- Borceux, Francis (1994). Handbook of categorical
algebra (Encyclopedia of Mathematics and its Applications
50-52). Cambridge Univ. Press.
- Freyd, Peter J. & Scedrov,
Andre, (1990). Categories, allegories (North Holland
Mathematical Library 39). North Holland.
- Hatcher, William S. (1982). The Logical Foundations of
Mathematics, 2nd ed. Pergamon. Chpt. 8 is an idiosyncratic
introduction to category theory, presented as a first order theory.
- Lawvere, William, &
Rosebrugh, Robert (2003). Sets for mathematics. Cambridge
University Press.
- Lawvere, William, &
Schanuel, Steve (1997). Conceptual mathematics: a first
introduction to categories. Cambridge University Press.
- Mac Lane, Saunders (1998).
Categories
for the Working Mathematician. 2nd ed. (Graduate Texts in
Mathematics 5). Springer-Verlag.
- ——— and Garrett Birkhoff
(1967). Algebra. 1999 reprint of the 2nd ed., Chelsea.
ISBN 0-8218-1646-2. An introduction to the subject making judicious
use of category theoretic concepts, especially commutative diagrams.
- May, Peter (1999). A Concise Course in Algebraic
Topology. University of Chicago Press, ISBN
0-226-51183-9.
- Pedicchio, Maria Cristina & Tholen, Walter (2004).
Categorical foundations (Encyclopedia of Mathematics and
its Applications 97). Cambridge University Press.
- Taylor, Paul (1999). Practical Foundations of
Mathematics. Cambridge University Press. An introduction to
the connection between category theory and constructive mathematics.
- Pierce, Benjamin (1991). Basic Category Theory for Computer
Scientists. MIT Press.
External links
- Chris Hillman, Categorical primer, formal introduction to
Category Theory.
- J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats
- Stanford
Encyclopedia of Philosophy: " Category Theory" -- by Jean-Pierre Marquis. Extensive
bibliography.
- Homepage of the Categories mailing list, with
extensive resource list.
- Baez, John, 1996," The
Tale of n-categories." An informal introduction to
higher order categories.
- The catsters" a Youtube channel about category
theory.
- Categories, Logic and the Foundations of Physics,
Webpage dedicated to the use of Categories and Logic in the
Foundations of Physics.
- Interactive Web page which generates examples
of categorical constructions in the category of finite sets.
Written by Jocelyn Paine