Maintained by: Mara Alagic
Curriculum and Instruction Department
Wichita State University
Wichita, KS 67260-0028
Updated March 17, 2000
Category Theory
Category theory studies structural aspects of mathematics that are common to many fields
of mathematics: e.g., algebra, topology, functional analysis, logic, and computer science.
Thus, category theorists tend to have many diverse interests. My research interests have
included: relational categories, categories of some abstract structures and categorical
semantics of programming languages.
Category theory is a mathematical language which arose in the study of limits for
universal coefficient theorems in Cech cohomology by Eilenberg and Mac Lane (1942);
so the topic has its origins in some sophisticated topology.
However, soon category theory became a field in itself. The reason for this is that it
provides a unifying mathematical modeling language. It lends itself very well to
extracting and generalizing elementary and essential notions and constructions from many
mathematical disciplines. Thanks to its general nature, the language of category theory
enables one to "transport" problems from one area of mathematics, via suitable
"functors", to another area, where the solution may be easier to find.
Categories have successfully been applied in formulating and solving
problems in topology, algebra, geometry and functional analysis. Moreover, in the sixties
Lawvere started a project aiming at a purely categorical foundation of all mathematics,
beginning with an appropriate axiomatization of the category of sets. This has led to a
huge interest in and development of sheaf and topos theory.
More recently, computer science discovered category theory. It quickly found applications in the field of algebraic semantics and the theory of (functional) programming languages; it is still unclear what kind of applications might be expected in the scope of cognitive science.
The basic notions of the language are actually quite simple.
A category consists of a set of OBJECTS and a set of MORPHISMS.
Every morphism has a SOURCE object and a TARGET object.
Given objects X and Y, we write Hom(X,Y) for the set of morphisms from X to Y (i.e.,
having X as source and Y as target).
The axioms for a category are that it consist of a set of objects and for any 2
objects X and Y a set Hom(X,Y) of morphisms from X to Y, and
(i) Given a morphism g in Hom(X,Y) and a morphism f in Hom(Y,Z), there is morphism which
we call f*g in Hom(X,Z). (This binary operation * is called COMPOSITION.)
(ii) Composition is associative: (f*g)*h = f*(g*h).
(iii) For each object X there is a morphism id|X from X to X, called the IDENTITY ON X.
(iv) Given any f in Hom(X,Y), f*id|X = f and id|Y*f = f.
The classic example is Set, the category with sets as objects
and functions as morphisms, and the usual composition as composition! Or else
Vect --- vector spaces as objects, linear maps as morphisms
Group --- groups as objects, homomorphisms as morphisms
Top --- topological spaces as objects, continuous functions as morphisms
Diff --- smooth manifolds as objects, smooth maps as morphisms
Ring --- rings as objects, ring homomorphisms as morphisms
Note that in all these cases the morphisms are actually a special sort of functions. That need not be the case in general! For example, an ordered set is a category with its elements as objects and one morphism in each Hom(X,Y) if X is less than or equal to Y, but none otherwise.
Many nice things in mathematics are functors. A functor is a kind of
map between categories. A FUNCTOR F from a category C to a category D is a map from the
set of objects of C to the set of objects of D together with a map from the set Hom(X,Y)
for any objects X,Y of C to Hom(F(X),F(Y)). That is, objects go to objects and morphisms
go to morphisms.
Category theory is popular among algebraic topologists. Typically an algebraic topologist will try to assign algebraic invariants to topological structures. The golden rule of such invariants is that they should be FUNCTORIAL. That is, they should be functors! For example, the fundamental group is functorial. Topologists know how to cook up a group called the fundamental group from any space. (The group keeps track of how many holes the space has.) But ALSO, any map between spaces determines a homomorphism of the fundamental groups. So the fundamental group is really a functor from the category Top to the category Group.
This allows us to completely transpose any situation involving spaces
and continuous maps between them to a parallel situation involving groups and
homomorphisms, and thus reduce some topology problems to algebra problems!
The category version of the definition of a group: A group is a category with one object in which all the morphisms are isomorphisms.
A representation of a group, if we think of a group as a category as
is just a functor from that category to the category Vect of vector spaces. So we can
define a representation of a category to be a functor from that category to the category
of vector spaces.
There are NATURAL TRANSFORMATIONS between functors. Suppose we have two functors F and G
from the category C to the category D. A natural transformation n from F to G consists of:
(i) for each object X in C, a morphism n(X) from F(X) to G(X), such that
(ii) the following diagram commutes:
n(Y)*F(f) = G(f)*n(X): F(X) --------> G(Y)
An example would be "abelianization", which maps a
group H to the abelian group H/[H,H]. If F were the fundamental group and G were the first
homology group, we could say that abelianization is a natural transformation from F to G.
If you want to get into deeper waters, think about this question:
It turns out to be, not just a category, but a 2-category. That means
that in addition to objects and morphisms, it has "2-morphisms", that is,
morphisms between morphisms. To see how this goes, let's call the 2-category of all
categories "Cat". Then the objects of Cat are categories, the morphisms of Cat
are functors, and the 2-morphisms are natural transformations!
Let me just say a bit about where things go from here. First of all, it turns out that we
can keep playing this game ad infinitum. We can define a notion of "n-category"
having objects, morphisms between objects, 2-morphisms between morphisms, and so on up to
n-morphisms... and it turns out that "category of all n-categories" is really an
(n+1)-category.
Just as we can talk about a representation of a category, that is, a functor to Vect, we
can talk about a representation of an n-category. This is the same as an "n-functor
to nVect", where nVect is the n-category of "n-vector spaces". While this
doubtless may seem like insane generalization for its own sake,
Apart from the possibility of studying properties of given structures
by formulating them within such a category, one can look at them in respect to other types
of structures. This is were the concept of functor comes in: a functor is nothing but a
mapping between two categories, again fulfilling some elementary properties concerning its
behavior on the objects and arrows of the "source" category.
On top of the framework given by the two basic notions of category and functor, a whole
theory of typical constructions has been built, including natural transformations, adjoint
functors, and limits. While it is beyond the scope of this introduction to explain these
in detail, it should be stressed here that all of those abound in all fields of
mathematics, sometimes obviously so, sometimes quite heavily disguised. It is precisely
in discovering such disguises where category theory has proven most useful, since this
leads to more insight in the studied structures.
For example, It is interesting to note why associativity without
commutativity is studied so much more than commutativity without associativity. Basically,
because most of our examples of binary operations can be interpreted as composition of
functions. For example, if write simply x for the operation of adding x to a real number
(where x is a real number), then x + y is just x composed with y. Composition is always
associative so the + operation is associative!
If we try to generalize the heck out of the concept of a group, keeping associativity as a sacred property, we get the notion of a category. Categories are some of the most basic structures in mathematics.
TOPOLOGY and CATEGORY THEORY SITES
Applied and Computational Category Theory at RISC-Linz
Topology Atlas: Questions in Topology
Ask a Topologist Bulletin Board Topology Atlas: Commentary
Monty Harper Song Lyrics - Topologically Speaking
Categories Home Page (at Mount Allison University)
Theory and Applications of Categories
Centre de Recherche en Théorie des Catégories -- Montréal
Links to ftp and WWW Sites of Categorical Interest
Electronic Journal on Theory and Applications of Categories
Information on the Xy-pic package for LaTeX documents
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Updated March 17, 2000 |