concrete categoryConcreteCategory2006-06-30 10:09:162008-10-19 17:48:40Definitionforgetful functorunderlying functorconrete and abstract categories% this is the default PlanetMath preamble. as your knowledge
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& {#6} } }A \emph{concrete category} over a category $\Cat B$ is a category $\Cat A$ together with a
faithful functor $\Func UAB$. (The functor $U$ is sometimes called the \emph{forgetful
functor} or the \emph{underlying functor}.)
A concrete category over $\Set$ is called a \emph{construct}. (Here $\Set$ denotes the category of
sets.)
This means that in a construct objects can be interpreted as sets and morphisms as maps.
{\bf Remarks:}
\begin{enumerate}
\item An alternative meaning of a \emph{concrete category} is that of a category with objects that have elements;
such objects can be classes, semigroups, monoids, groups, groupoids, topological spaces, and so on.
\item Note also the Yoneda-Grothendieck Lemma that relates a category $\mathcal{C}$ to the functor category $\hat{\mathcal{C}}$ of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$, the category of sets.
\end{enumerate}
\begin{thebibliography}{1}
\bibitem{ahs}
J.~Ad\'amek, H.~Herrlich, and G.~Strecker.
\newblock {\em Abstract and Concrete Categories}.
\newblock Wiley, New York, 1990.
\end{thebibliography}