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Viewing Version
7
of
'concrete category'
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| Title of object: |
concrete category |
| Canonical Name: |
ConcreteCategory |
| Type: |
Definition |
| Created on: |
2006-06-30 10:09:16 |
| Modified on: |
2008-09-21 01:14:33 |
| Classification: |
msc:18A05 |
| Defines: |
forgetful functor, underlying functor |
Preamble:
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Content:
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 Remark:}
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.
\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}
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