category theoryCategoryTheory2009-06-09 20:18:412009-06-10 02:07:23Topicmorphismsobjectdiagramscategory theory applicationsfunctorfunctor categorygroupmonoidsemigroupAbelian groupcommutativitynaturality conditionAbelian category$Ab$-axiomscategorical dualitydualitycategories with structureenriched categorymetatheoryNon-Abelian Algebraic TopologyNAATcategorycategory theorytoposfunctornatural transformationsmetagraphmetacategoryalgebraic topologyhomology theoryETACunit lawassociativity axiomsidentitycompositionoperationsgraph\section{Theory of Categories}
{\em 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 \PMlinkname{category}{Category}, functor between categories and natural transformations between functors. A {\em category} can also be defined as a mathematical interpretation of the elementary theory of abstract categories, or ETAC. A {\em topos} is often considered also as a special type of category subject to the topos axioms.
\subsection{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 is 1945 even though an earlier, published report in 1943 utilized categorical concepts.
First of all, a {\em category} consists of arrows, called {\em 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) {\em diagrams}.
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 {\em structure} are the key notions of all modern mathematics, whereas that of a set or {\em 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. As well-known examples, one consider morphisms of groups defined as
{\em group homomorphisms), and morphisms of topological spaces as homeomorphisms. Interestingly, morphisms between groupoids are still being called `homomorphisms' even when such objects posses 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 objects', and they play fundamental roles in Algebraic Topology. By analogy one might then expect also that Barry Mitchell's concept of `{\em rings with many objects}' , and also their generalizations to {\em algebroids} may play, respectively, important roles in Algebraic Geometry and Number Theory.
Second level arrows--those between categories-- are called {\em functors}, and third level arrows between functors (second level arrows) are called {\em natural transformations} again subject to specific natural conditions such as commutativity of diagrams. The third level arrow is the more powerful concept in comparison either with either functors or morphisms, and of course, a functors 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 {\em categories with structure}, as well as {\em 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 {\em 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) {\em 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 \PMlinkexternal{Non-Abelian Algebraic Topology (NAAT)}{http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e040609.pdf} in modern mathematics. Such developments have close connections to recent results in
non-Abelian theories in mathematical physic, and also have potential impact
on Topological Quantum Field Theories (TQFT) and HQFTs.
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.
\subsection{Category theory applications}
Among the important category theory applications in mathematics itself are:
\begin{itemize}
\item Applications in the Foundations of Mathematics
\item Algebraic Topology applications
\item Algebraic Geometry applications
\item Applications to Number theory
\item Applications to rings and modules
\item Applications to the theory of Abelian groups
\item X
\item Y
\item Z
\item
\item
\end{itemize}
Other category theory applications are in:
\begin{itemize}
\item Theoretical and Mathematical Physics
\item Mathematical Biophysics, Complex Systems Biophysics and Theoretical Biology
\item Theory of Logic Algebras
\item Categorical logic
\item Algebraic logic
\item Computer theory
\item Quantum Operator Algebra (QOA)
\item Theoretical Ecology and Environmental sciences
\item Genomics and Interactomics
\end{itemize}
\subsection{An Index of categories}
A partial list of various types of categories.
\subsection{A Category theory index}
A partial \PMlinkname{index of the theory of categories}{IndexOfCategoryTheory}.
\subsection{An Index of Algebraic Geometry}
\subsection{An Index of Algebraic Topology}
\subsection{Bibliography of category theory and its applications}
An extensive, but not complete, literature on category theory.
\textbf{More to come...}