1 Theory of Categories
Category theory can be described as the branch of mathematics concerned with the general,
abstract and universal properties and applications of the fundamental concepts of category, functor
between categories and natural transformations between functors. A category can also be defined as
a mathematical interpretation of the theories of abstract category, or ETAC. A topos is often
considered as a special type of category subject to the topos axioms, and thus based upon a
(commutative) Heyting logic.
1.1 Introduction: Basic concepts
Category theory has developed, and is now being further developed, very rapidly in comparison
with most of the older branches of mathematics, with the notable exception of Topology, certain
aspects of Geometry and Number theory which experienced recently most remarkable advances.
The official birthdate year of Category Theory is 1945, even though an earlier, published report in
1943 utilized categorical concepts.
First of all, a category consists of arrows, called morphisms subject to a very small set of basic
category theory axioms (which is as small as only four axioms in some recent formulations).
Various chains, or geometric forms, composed of such arrows are called (categorical) diagrams.
From a logical point of view, morphisms may also be considered as relations, thus generalizing the
concept of set-theoretical mapping or function. Therefore, any logical relation theory may also be
formalized in terms of morphisms, and any metalogic (metatheory) of relations among relations
may be formalized in terms of categories and functors. One can then also say that “mathematics is
about relations between relations”.
Charles Ehresmann–one of the founders and developers of category theory in Europe– in
evaluating the role played by category theory in modern mathematics pointed out that the
concepts of morphism (arrow, with the specific examples of mapping and mathematical function)
and mathematical structure are the key notions of all modern mathematics, whereas that of a set
or object is relegated to a secondary, less important role. Morphisms of structures such as monoids,
semigroups, groups, rings, modules, vector spaces, groupoids, topological spaces, and so on
‘preserve the basic structure’, thus allowing comparisons to be made between different
mathematical objects with the ‘same’ structure.
1.2 Examples of morphisms and categories
As well-known examples, one considers morphisms of groups defined as group homomorphisms, and
morphisms of topological spaces as homeomorphisms. Interestingly, morphisms between groupoids
are still being called ‘homomorphisms’ even when such objects possess a topological structure as
well. Moreover, groupoids are also regarded as a specific type of categories with all invertible
morphisms, or natural generalizations of groups as a notion of ‘group with many identities’, and
they play fundamental roles in algebraic topology. By analogy one might then expect also that
Barry Mitchell’s concept of a ‘ring with many objects’ , and also its generalizations to algebroids
may play, respectively, important roles in algebraic Geometry and Number Theory.
There are two important differences between groupoids and groups. One is that, unlike
groups, groupoids have a partial multiplication, and the other difference is that the
condition for two elements to be composable is a geometric one (namely the end point of a
groupoid arrow is the starting point of another arrow). Such a partial multiplication law
that one has for groupoid arrows may be thought of as a kind of “group with many
identities”, that reduces to the particular case of a group when there is only one identity. The
other very important difference between groups and groupoids is that the geometry,
or topology, underlying groupoids is that of directed graphs, whereas the geometry
underlying groups is simply that of based sets, that is sets with a chosen base point,
or objects with discrete topology (“set dust”). It is thus quite evident that graphs are
far more interesting than sets, and can reflect more geometry–or by considering their
connectivity properties in the general case– are endowed with a topological structure.
One can also think that the objects of a groupoid allow for the addition of a spatial
component to the simpler objects of group theory that may be considered to have only
algebraic structures. Note, however, that this simplification applies only to groups other
than topological ones; thus one expects a closer connection between a topological group
and a topological groupoid because of their endowment with a topological structure
consistently added to their algebraic structure. Furthermore, one may expect even closer links,
for example, between a locally compact groupoid equipped with an associated Haar
measure and a locally compact group also equipped with a Haar measure; the latter
argument carries over to the relationships between a quantum groupoid and a quantum
group.
Last-but-not-least, the so called ‘trivial’ groupoid structures arise from the presence
of equivalence classes that are important in most fields of mathematics that involve
classification or equivalence relations, and are therefore of fundamental importance in
mathematics.
Second level arrows–those between categories– are called functors, and third level arrows between
functors are called natural transformations, again subject to specific naturality conditions such as
commutativity of diagrams. The third level arrow is the more powerful concept in comparison with
either functors or morphisms, and of course, the functor is a more powerful concept than a
morphism.
On the first conceptual level, mathematical categories provide a most convenient, universal-conceptual
‘language’ founded on the notions of category, functor, natural transformation and functor
category, albeit at such an abstract and universal level that many classical mathematicians chose
to dub it– without any strong justification– as “abstract nonsense”. Nevertheless, there are also
categories with structure, as well as enriched categories, algebraic categories, categories of
categories, 2-categories, double categories, and so on; category theory can be therefore
also also as a kind of metatheory endowed with different structural levels that are all
consistent and natural, in the sense of involving commutativity. Upon imposition of
additional (Ab1 to Ab6) axioms such commutative super-structures become Abelian
categories that generalize or extend the universal properties of categories of Abelian, or
commutative, groups; in fact, half of such axioms are obtained by merely ‘inverting the
arrows’–which is called (categorical) duality. Whereas the last 50 years have been dominated by
developments in (commutative) homology theory (or Homological Algebra) and Abelian
category theory, there is currently occurring a very rapid development of Non-Abelian
Algebraic Topology, (NAAT) in modern mathematics. Such developments have close
connections to recent results in non-Abelian theories in mathematical physics, and also have
potential impact on Topological quantum field theories (TQFT), HQFTs and physical
mathematics.
However, on a different level, when considered as a further sophistication of algebraic topology,
category theory embraces not only algebraic and topological structures, but also geometric and
analytic ones. At still higher levels, category theory provides natural means to define higher
dimensional structures such as higher dimensional algebra (HDA) that open completely
new avenues of mathematical research of recent, substantial interest in mathematical
physics, and especially in the development of quantum gravitation and/or superstring
theories.
On the other hand, at a fundamental level, the universal categorical concepts of adjoint functors
and adjointness semantics may provide a unique foundation of mathematics and algebraic theories,
well beyond set theory with its known limitations and problems.
1.3 Category theory applications
Among the important category theory applications in mathematics itself are:
- Applications in the Foundations of Mathematics
- Algebraic Topology applications
- Algebraic Geometry applications
- Applications to Number theory
- Applications to rings and modules
- Applications to the theory of Abelian groups
- X
- Y
- Z
-
Other category theory applications are in:
1.4 An Index of categories
A partial list of various types of categories.
1.5 A Category theory index
A partial index of the theory of categories.
1.6 An Index of Algebraic Geometry
1.7 An Index of Algebraic Topology
1.8 Bibliography of category theory and its applications
An extensive, but not complete, literature on category theory.
More to come...
