category theoryCategoryTheory2009-06-09 20:18:412009-06-09 20:42:31Topiccategory theory applicationsfunctorfunctor categorygroupmonoidsemigroupcategorycategory theorytoposfunctornatural transformationsmetagraphmetacategoryalgebraic topologyhomology theoryETACunit lawassociativity axiomsidentitycompositionoperationsgraph\section{The Theory of Categories}
{\em Category theory} can be described as the branch of mathematics concerned
with the 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 as a special type of category subject to the
topos axioms.
\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...}