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\begin{document}
\subsection{An Outline of Categorical Algebra}
This topic entry provides an outline of an important mathematical \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} called {\em categorical algebra}; although specific definitions are in use for various applications of categorical algebra to specific \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, 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 \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of algebraic structures, either concrete or abstract, in terms of \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}, \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}.
In a narrow sense, a \emph{categorical algebra} is an associative algebra, defined for any locally finite category and a \htmladdnormallink{commutative ring}{http://planetphysics.us/encyclopedia/PAdicMeasure.html} with unity. This notion may be considered as a generalization of both the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} algebra and that of an incidence algebra, much as the concept of category generalizes the notions of group and partially ordered set.
\subsection{Extensions of Categorical Algebra}
\begin{itemize}
\item Thus, ultimately, since categories are interpretations of the \emph{axiomatic \htmladdnormallink{theories of abstract category}{http://planetphysics.us/encyclopedia/Formula.html} (\htmladdnormallink{ETAC}{http://planetphysics.us/encyclopedia/Formula.html})}, so are categorical algebras.
The general definition of representation introduced above can be still further extended by introducing \emph{supercategorical algebras as interpretations of \htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html}}, 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{\htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\rho$ from an abstract algebraic structure $\mathcal{A}_S$ to a concrete algebraic structure $A_c$}, a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\mathcal{H}$, or a family of \htmladdnormallink{linear operator}{http://planetphysics.us/encyclopedia/Commutator.html} spaces.
The key notion of \htmladdnormallink{representable functor}{http://planetphysics.us/encyclopedia/RepresentableFunctor.html} was first reported by \htmladdnormallink{Alexander Grothendieck}{http://planetphysics.us/encyclopedia/AlexanderGrothendieck.html} 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 \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $R$ 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 \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} 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) directly determine an \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} (\emph{\htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}}) 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}
\end{document}
