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%%% Primary Title: categorical diagrams defined as functors
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%%% Filename: CategoricalDiagramsDefinedByFunctors.tex
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\begin{document}
\subsection{Categorical Diagrams Defined by Functors}
Any {\em categorical diagram} can be defined \emph{via} a corresponding \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} (associated with a \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} as shown by Mitchell, 1965, in ref. \cite{BMitchell65}). Such functors associated with diagrams are very useful in the categorical theory of \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} as in the case of \htmladdnormallink{categorical algebra}{http://planetphysics.us/encyclopedia/CategoricalAlgebra.html}. As a particuarly useful example in (commutative) homological algebra let us consider the case of an exact \htmladdnormallink{categorical sequence}{http://planetphysics.us/encyclopedia/HomologicalSequence2.html} that has a correspondingly defined \emph{exact functor} introduced for example in \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} theory.
\subsection{Examples}
Consider a scheme $\Sigma$ as defined in ref. \cite{BMitchell65}. Then one has the following short list of important examples of diagrams and functors:
\begin{enumerate}
\item Diagrams of adjoint situations: \htmladdnormallink{adjoint functors}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}
\item Equivalence of \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}
\item \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} diagrams
\item Diagrams of \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
\item Category of diagrams and 2-functors
\item \htmladdnormallink{monad}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} on a category
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{BMitchell65}
Barry Mitchell., \emph{Theory of Categories.}, Academic Press: New York and London (1965), pp.65-70.
\end{thebibliography}
\end{document}
