categorical algebraCategoricalAlgebra2009-01-31 20:10:592009-02-14 03:05:48Topicdirectlyalgebraic representationfunctor representationrepresentable functorcategory of algebraic structuresalgebraic categorycategory of logic algebrasextensions of categorical algebraalgebraic category of -logic algebrasnon-Abelian structuresabelian categorysupplemental axioms for an Abelian categoryhigher dimensional generalized Van Kampen theorems (HD-VKT)axiomatic theory of supercategories and metacategoriescategorical quantum logics as quantum LM-algebraic logicnon-commuting graphnon-Abelian structurestopic entry on foundations of mathematicstopic on algebra classificationalgebraic categories and classes of algebrasrepresentable functorR-supercategorieshomotopy groupoids and crossed complexes: non-commutative structures in higher dimensional algebra (HDA)index of categories% this is the default PlanetPhysics preamble.
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\def\D{\mathsf{D}}\subsection{An Outline of Categorical Algebra}
This topic entry provides an outline of an important mathematical field called {\em categorical algebra}; although specific definitions are in use for various applications of categorical algebra to specific algebraic structures, they do not cover the entire field. In the most general sense, \emph{categorical algebras}-- as introduced by Mac Lane in 1965 -- can be described as the study of representations of algebraic structures, either concrete or abstract, in terms of categories, functors and natural transformations.
\subsection{Extensions of Categorical Algebra}
\begin{itemize}
\item Thus, ultimately, since categories are interpretations of the \emph{axiomatic elementary theory of abstract categories (ETAC)}, so are categorical algebras.
The general definition of representation introduced above can be still further extended by introducing \emph{supercategorical algebras as interpretations of ETAS}, as explained next.
\item Mac Lane (1976) wrote in his {\em Bull. AMS} review cited here as a verbatim quotation:
\emph{``On some occasions I have been tempted to try to define what algebra is,
can, or should be - most recently in concluding a survey [72] on Recent
advances in algebra. But no such formal definitions hold valid for long, since
algebra and its various subfields steadily change under the influence of ideas
and problems coming not just from logic and geometry, but from analysis,
other parts of mathematics, and extra mathematical sources. The progress of
mathematics does indeed depend on many interlocking, unexpected and
multiform developments.''}
\end{itemize}
\subsection{Basic Definitions}
An \emph{algebraic representation} is generally defined as a \emph{morphism $\rho$ from an abstract algebraic structure $\mathcal{A}_S$ to a concrete algebraic structure $A_c$}, a Hilbert space $\mathcal{H}$, or a family of linear operator spaces.
The key notion of \PMlinkname{representable functor}{RepresentableFunctor} was first reported by Alexander Grothendieck in 1960.
\begin{definition}
Thus, when the latter concept is extended to categorical algebra, one has a \emph{representable} functor $S$ from an arbitrary category $\mathcal{C}$ to the category of sets $Set$ if $S$ admits a \emph{functor representation} defined as follows. A \emph{functor representation of $S$} is defined as a pair, $({R}, \phi)$, which consists of an object $R$ of $\mathcal{C}$ and a family $\phi$ of equivalences $\phi (C): \Hom_{\mathcal{C}}(R,C) \cong S(C)$, which is natural in C, with C being any object in $\mathcal{C}$. When the functor $S$ has such a representation, it is also said to be \emph{represented by the object $R$} of $\mathcal{C}$. For each object $R$ of $\mathbf{C}$
one writes $h_{R}: \mathcal{C} \lra Set$ for the covariant $\Hom$--functor $h_{R}(C)\cong \Hom_{\mathcal{C}}(R,C)$. A \emph{representation} $(R, \phi)$ of ${S}$ is therefore \emph{a natural equivalence of functors}:
\begin{equation}
\phi: h_{R} \cong {S}~.
\end{equation}
\end{definition}
\begin{remark}
The equivalence classes of such functor representations (defined as natural equivalences) obviously determine an algebraic (\emph{groupoid}) structure.
\end{remark}
\begin{thebibliography}{9}
\bibitem{SML65}
Saunders Mac Lane: Categorical algebra., {\em Bull. AMS}, \textbf{71} (1965), 40-106., Zbl 0161.01601, MR 0171826,
\bibitem{SML76}
Saunders Mac Lane: Topology and Logic as a Source of Algebras., {\em Bull. AMS}, \textbf{82}, Number 1, 1-36,
January 1, 1976.
\end{thebibliography}