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%%% Primary Title: algebraic categories and representations of classes of algebras
%%% Primary Category Code: 02.
%%% Filename: AlgebraicCategoriesAndRepresentationsOfClassesOfAlgebras.tex
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%%% Author(s): bci1
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\begin{document}
\subsection{Introduction}
Classes of algebras can be categorized at least in two \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html}: either classes of \emph{specific
algebras}, such as: \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} algebras, K-algebras, \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} algebras, logic algebras, and so on,
or \emph{general} ones, such as general classes of: \htmladdnormallink{categorical algebras}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html}, \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}), supercategorical algebras, universal algebras, and so on.
\subsection{Basic concepts and definitions}
\begin{itemize}
\item {\bf Class of algebras}
\begin{definition}
A \emph{class of algebras} is defined in a precise sense as an \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in the
\emph{\htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}}.
\end{definition}
\item {\bf Monad on a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{C}$, and a T-algebra in $\mathcal{C}$}
\begin{definition}
Let us consider a category $\mathcal{C}$, two \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/Functor.html}: $T: \mathcal{C} \to \mathcal{C}$ (called the \emph{monad functor}) and $T^2: \mathcal{C} \to \mathcal{C} = T \circ T$, and two \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}:
$\eta: 1_ \mathcal{C} \to T$ and $\mu: T^2 \to T$. The triplet $(\mathcal{C},\eta,\mu)$
is called a {\em monad on the category $\mathcal{C}$}. Then, a {\em T-algebra} $(Y,h)$ is defined as an object $Y$ of a category $\mathcal{C}$ together with an arrow $h: TY \to Y $ called the {\em structure map} in $\mathcal{C}$ such that:
\begin{enumerate}
\item $$Th: T^2 \to TY,$$
\item $$h \circ Th = h \circ \mu_Y,$$
where: $\mu_Y: T^2 Y \to TY;$ and
\item $$ h \circ \eta_Y = 1_Y.$$
\end{enumerate}
\end{definition}
\item {\bf Category of Eilenberg-Moore algebras of a monad $T$}
An important definition related to abstract classes of algebras and universal algebras is that of the category of Eilenberg-Moore algebras of a monad $T$:
\begin{definition}
The category $\mathcal{C}^T$ of $T$-algebras and their \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is called the {\em Eilenberg-Moore category} or {\em category of Eilenberg-Moore algebras} of the monad T.
\end{definition}
\end{itemize}
\subsection{Pertinent remarks:}
\begin{itemize}
\item {\bf a. Algebraic category definition}
\begin{remark}
With the above definition, one can also define a \emph{category of classes of algebras and their
associated \htmladdnormallink{groupoid homomorphisms}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism.html}} which is then an algebraic category.
Another example of algebraic category is that of the \htmladdnormallink{category of C*-algebras}{http://planetphysics.us/encyclopedia/Homomorphisms.html}.
Generally, a category $\mathcal{A}_C$ is called \emph{algebraic} if it is \htmladdnormallink{monadic}{http://planetphysics.us/encyclopedia/CoIntersections.html} over the category of sets and set-theoretical mappings, $Set$; thus, a functor $G: \mathcal{D} to \mathcal{C}$ is called \emph{monadic} if it has a left adjoint
$F: \mathcal{C}\to \mathcal{D}$ forming a {\em monadic adjunction} $(F,G,\eta,\epsilon)$ with $G$ and $\eta, \epsilon$
being, respectively, the unit and counit; such a {\em monadic adjunction} between categories
$\mathcal{C}$ and $\mathcal{D}$ is defined by the condition that category $\mathcal{D}$ is equivalent to the to the Eilenberg-Moore category $\mathcal{C} ^T$ for the monad
$$T = GF.$$
\end{remark}
\item b. Equivalence classes
\begin{remark}
Although all classes can be regarded as equivalence, weak equivalence, etc., classes of
algebras (either specific or general ones), do not define identical, or even isomorphic structures, as the notion of `equivalence' can have more than one meaning even in the algebraic case.
\end{remark}
\end{itemize}
\subsection{Algebraic representations}
\begin{itemize}
\item \htmladdnormallink{group representations}{http://planetphysics.us/encyclopedia/GroupRepresentations.html} \item \htmladdnormallink{groupoid representations}{http://planetphysics.us/encyclopedia/GroupRepresentations.html} \item Convolution C*-algebra groupoid representations
\item Functorial representations and \htmladdnormallink{representable functors}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html}
\item Categorical group representations
\item Algebroid representations
\item Quantum Algebroid (QA) representations
\item Double groupoid representations
\item Double Algebroid representations
\item Grassman-Hopf representations
\end{itemize}
\end{document}
