----
> [!definition] Definition. ([[slice category]])
> Let $\mathsf{C}$ be a [[category]] and $A$ be an object of $\mathsf{C}$. We define a [[category]] $\mathsf{C}_{A}$ whose objects are certain *morphisms* in $\mathsf{C}$ and whose morphisms are certain *[[diagram|diagrams]]* of $\mathsf{C}$.
>
$\text{Obj}(\mathsf{C}_{A})$ is defined to be all morphisms from any object of $\mathsf{C}$ to $A$, thus, an object of $\mathsf{C}_{A}$ is a morphism $f \in \text{Hom}(Z,A)$ from some object $Z$ of $\mathsf{C}$.
> ```tikz
\begin{document}
\begin{tikzpicture}
% Nodes
\node (Z) at (0,0) {$Z$};
\node (A) at (0,-2) {$A$};
>
% Arrows
\draw[->] (Z) -- (A) node[midway, left] {$f$};
\end{tikzpicture}
\end{document}
>```
There is really only one sensible way to define the morphisms of $\mathsf{C}_A$. Given two objects of $\mathsf{C}_{A}$:
>```tikz
\begin{document}
\begin{tikzpicture}
% Nodes for the left part of the diagram
\node (Z) at (0,0) {$Z_1$};
\node (A) at (0,-2) {$A$};
>
% Arrow for the left part of the diagram
\draw[->] (Z) -- (A) node[midway, left] {$f_1$};
>
% Nodes for the right part of the diagram
\node (Y) at (2,0) {$Z_2$};
\node (B) at (2,-2) {$A$};
>
% Arrow for the right part of the diagram
\draw[->] (Y) -- (B) node[midway, right] {$f_2$};
\end{tikzpicture}
\end{document}
>```
>
we look for $\sigma$ in $\text{Hom}_{\mathsf{C}}(Z_{1}, Z_{2})$ such that the [[diagram]] commutes:
>```tikz
>\begin{document}
>\begin{tikzpicture}
% Nodes for the left part of the diagram
\node (Z) at (0,0) {$Z_1$};
\node (A) at (1,-1.5) {$A$};
>
% Arrow for the left part of the diagram
\draw[->] (Z) -- (A) node[midway, left] {$f_1$};
>
% Nodes for the right part of the diagram
\node (Y) at (2,0) {$Z_2$};
\draw[->] (Z) -- (Y) node[midway, above] {$\sigma$};
>
% Arrow for the right part of the diagram
\draw[->] (Y) -- (A) node[midway, right] {$f_2$};
\end{tikzpicture}
\end{document}
>```
that is, $\sigma$ such that $f_{1}=f_{2}\sigma$. Then we define $\text{Hom}_{\mathsf{C}_{A}}(f_{1}, f_{2})$ to equal the set of all such diagrams; there is one such morphism per such $\sigma$.
^definition
> [!justification]
> Time to check the [[category]] axioms. The identity $1_{f}$ corresponds to the diagram
>
>```tikz
\begin{document}
\begin{tikzpicture}
% Nodes
\node (Z1) at (0,0) {$Z$};
\node (Z2) at (2,0) {$Z$};
\node (A) at (1,-1.5) {$A$};
>
% Arrows
\draw[->] (Z1) -- (Z2) node[midway, above] {$1_Z$};
\draw[->] (Z1) -- (A) node[midway, left] {$f$};
\draw[->] (Z2) -- (A) node[midway, right] {$f$};
\end{tikzpicture}
\end{document}
>```
>
which commutes since $\mathsf{C}$ is a [[category]]. The notion of composition is obtained by first 'concatenating' two commutative diagrams:
>
>```tikz
\begin{document}
\begin{tikzpicture}
% Nodes
\node (Z1) at (0,0) {$Z_1$};
\node (Z2) at (2,0) {$Z_2$};
\node (Z3) at (4,0) {$Z_3$};
\node (A) at (2,-2) {$A$};
>
% Arrows
\draw[->] (Z1) -- (Z2) node[midway, above] {$\sigma$};
\draw[->] (Z2) -- (Z3) node[midway, above] {$\tau$};
\draw[->] (Z1) -- (A) node[midway, left] {$f_1$};
\draw[->] (Z2) -- (A) node[midway, left] {$f_2$};
\draw[->] (Z3) -- (A) node[midway, right] {$f_3$};
\end{tikzpicture}
\end{document}
>```
and then removing the central arrow:
>
>```tikz
\begin{document}
\begin{tikzpicture}
% Nodes
\node (Z1) at (0,0) {$Z_1$};
\node (Z3) at (4,0) {$Z_3$};
\node (A) at (2,-2) {$A$};
>
% Arrows
\draw[->] (Z1) -- (Z3) node[midway, above] {$\tau \sigma$};
\draw[->] (Z1) -- (A) node[midway, left] {$f_1$};
\draw[->] (Z3) -- (A) node[midway, right] {$f_3$};
\end{tikzpicture}
\end{document}
>```
the resultant diagram commutes because $\mathsf{C}$ is a [[category]]. Associativity is rather immediate.
^justification
> [!basicexample]
> Consider the [[category]] constructed [[category#^basic-example-2|here]] with $S=\mathbb{Z}$ and $\sim$ the relation $\leq$. The category $\mathsf{C}_{3}$ has as objects the $\mathsf{C}$-morphisms $(n,3) \in \mathbb{Z} \times \mathbb{Z}$ for $n \leq 3$. There is a morphism $(m,3) \to (n,3)$
if and only if $m \leq n$.
^basic-example
----
####
----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```