subcategorySubcategory2004-05-11 03:20:402007-11-22 01:37:24Definitionfull subcategoryinclusion functor% this is the default PlanetMath preamble. as your knowledge
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Let $\mathcal{C}$ be a (small) category. If $\mathcal{S}$ is a collection of both a subset, call it $\operatorname{Ob}(\mathcal{S})$, of objects of $\mathcal{C}$ and a subset, call it $\operatorname{Mor}(\mathcal{S})$, of morphisms of $\mathcal{C}$ such that
\begin{enumerate}
\item
For each $S\in\operatorname{Ob}(\mathcal{S})$, the identity morphism of $S$, $id_S\in\operatorname{Mor}(\mathcal{S});$
\item
For each $f\in\operatorname{Mor}(\mathcal{S})$, $\operatorname{domain}(f)$ and
$\operatorname{codomain}(f)\in\operatorname{Ob}(\mathcal{S});$
\item
For every pair $f,g\in\operatorname{Mor}(\mathcal{S})$ such that $f\circ g$ exists, then $f\circ g\in\operatorname{Mor}(\mathcal{S}).$
\end{enumerate}
Then $\mathcal{S}$ is readily seen to be a category. It is called a \emph{subcategory} of the category $\mathcal{C}.$
Given a category $\mathcal{C}$ and a subcategory $\mathcal{S}$ of $\mathcal{C}$, a map $$\operatorname{Incl}:\mathcal{S}\hookrightarrow \mathcal{C}$$ that sends each object of $\mathcal{S}$ to itself (in $\mathcal{C}$), and each morphism of $\mathcal{S}$ to itself (in $\mathcal{C}$), is a functor. $\operatorname{Incl}$ is called the \emph{inclusion functor}, or an \emph{embedding}. This inclusion functor is a faithful functor. If it is also \PMlinkname{full}{fullfunctor}, then we call the corresponding subcategory $\mathcal{S}$ a \emph{full subcategory} of $\mathcal{C}$. In other words, if $\mathcal{S}$ is a full subcategory of $\mathcal{C}$, then $$\operatorname{hom_{\mathcal{C}}}(S_1,S_2)=\operatorname{hom_{\mathcal{S}}}(S_1,S_2)$$ for pair of $S_1,S_2\in \operatorname{Ob}(\mathcal{S})$.
\textbf{Remarks}
\begin{enumerate}
\item
Let $T:\mathcal{C}\to\mathcal{D}$ be a full and faithful functor. Then $T(\mathcal{C})$ is a full subcategory of $\mathcal{D}$.
\item
Again, let $T:\mathcal{C}\to\mathcal{D}$ be a full and faithful functor. If $\mathcal{S}$ is a full subcategory of $\mathcal{D}$, then $T^{-1}(\mathcal{S})$ defined by:
\begin{itemize}
\item
$\operatorname{Ob}(T^{-1}(\mathcal{S})):=\lbrace C\in\operatorname{Ob}(\mathcal{C})\mid T(C)\in\operatorname{Ob}(\mathcal{S})\rbrace$
\item
$\operatorname{Mor}(T^{-1}(\mathcal{S})):=\lbrace f\in\operatorname{Mor}(\mathcal{C})\mid T(f)\in\operatorname{Mor}(\mathcal{S})\rbrace$
\end{itemize}
is a subcategory of $\mathcal{C}$.
\end{enumerate}
\textbf{Examples of Subcategories}
\begin{enumerate}
\item
In \textbf{Set}, the category of finite sets is a full subcategory, and so is the category of $k$-element sets, where $k$ is any (possibly infinite) cardinality. If $k$ is finite, then every morphism in the subcategory is invertible.
\item
In \textbf{Top}, we have the full subcategories whose objects are Euclidean spaces, compact spaces, or Hausdorff spaces.
\item
In \textbf{Grp}, there is the full subcategory whose objects are abelian groups with additive homomorphisms.
\item
\textbf{Grp} is in fact a subcategory of the category of topological groups, since every group may be viewed as a topological group with the discrete topology.
\item
In \PMlinkescapetext{\textbf{Ring}}, there are the subcategories of commutative rings, matrix rings, or fields. Note that \textbf{Field} is not a full subcategory of \textbf{Ring}, since the ring homomorphism that maps every element to $0$ is not a field homomorphism.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{Ma}S. Mac Lane, \emph{Categories for the Working Mathematician} (2nd edition), Springer-Verlag, 1997.
\end{thebibliography}