| Version 4 |
Version 3 |
| \begin{defn} |
\begin{defn} |
| Let $\mathcal{C}$ be a category. A \emph{diagram} in $\CC$ is a |
Let $\mathcal{C}$ be a category. A \emph{diagram} in $\CC$ is a |
| directed graph $\GG$ with vertex set $V$ and edge set $E$, (``loops'' |
directed graph $\GG$ with vertex set $V$ and edge set $E$, (``loops'' |
| and ``parallel edges'' are allowed) together with two maps |
and ``parallel edges'' are allowed) together with two maps |
| $o\co V\to\mathrm{Obj}(\CC)$, $m\co E\to \mathrm{Morph}(\CC)$ such that |
$o\co V\to\mathrm{Obj}(\CC)$, $m\co E\to \mathrm{Morph}(\CC)$ such that |
| if $e\in E$ has source $s(e)\in V$ and target $t(e)\in V$ then |
if $e\in E$ has source $s(e)\in V$ and target $t(e)\in V$ then |
| $m(e) \in \text{Hom}_{\CC}\left(s(e),t(e)\right)$. |
$m(e) \in \text{Hom}_{\CC}\left(s(e),t(e)\right)$. |
| \end{defn} |
\end{defn} |
| Usually diagrams are denoted by drawing the corresponding graph |
Usually diagrams are denoted by drawing the corresponding graph |
| and labeling its vertices (respectively edges) with their images under $o$ |
and labeling its vertices (respectively edges) with their images under $o$ |
| (respectively $m$), for example if $f\co A\to B$ is a morphism |
(respectively $m$), for example if $f\co A\to B$ is a morphism |
| $$\xymatrix@1{ {A}\ar[r]^f&{B} }$$ |
$$\xymatrix@1{ {A}\ar[r]^f&{B} }$$ |
| is a diagram. Often (as in the previous example) the vertices themselves are |
is a diagram. Often (as in the previous example) the vertices themselves are |
| not drawn since their position can b deduced by the position of their |
not drawn since their position can b deduced by the position of their |
| labels. |
labels. |
| \begin{defn} |
\begin{defn} |
| Let $D=(\GG,o,m)$ be a diagram in the category $\CC$ and $\Gg=(e_1,\ldots,e_n)$ |
Let $D=(\GG,o,m)$ be a diagram in the category $\CC$ and $\Gg=(e_1,\ldots,e_n)$ |
| be a path in $\GG$. Then the \emph{composition along} $\Gg$ is the following |
be a path in $\GG$. Then the \emph{composition along} $\Gg$ is the following |
| morphism of $\CC$ |
morphism of $\CC$ |
| $$\circ(\Gg):=m(e_n)\circ\cdots\circ m(e_1)\,.$$ |
$$\circ(\Gg):=m(e_n)\circ\cdots\circ m(e_1)\,.$$ |
| We say that $D$ is |
We say that $D$ is |
| \emph{commutative} or that it \emph{commutes} if for any two objects in |
\emph{commutative} or that it \emph{commutes} if for any two objects in |
| the image of $o$, say $A=o(v_1)$ and $B=o(v_2)$, and any two paths $\Gg_1$ |
the image of $o$, say $A=o(v_1)$ and $B=o(v_2)$, and any two paths $\Gg_1$ |
| and $\Gg_2$ that connect $v_1$ to $v_2$ we have |
and $\Gg_2$ that connect $v_1$ to $v_2$ we have |
| $$\circ(\Gg_1)=\circ(\Gg_2)\,.$$ |
$$\circ(\Gg_1)=\circ(\Gg_2)\,.$$ |
| \end{defn} |
\end{defn} |
| For example the commutativity of the triangle |
For example the commutativity of the triangle |
| $$\xymatrix{ |
$$\xymatrix{ |
| {A}\ar[rr]^{f}\ar[dr]_{h}&&{B}\ar[dl]^{g}\\ |
{A}\ar[rr]^{f}\ar[dr]_{h}&&{B}\ar[dl]^{g}\\ |
| &{C}& |
&{C}& |
| translates to $h=g\circ f$, while the commutativity of the square |
translates to $h=g\circ f$, while the commutativity of the square |
| $$\xymatrix{ |
$$\xymatrix{ |
| {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\ |
{A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\ |
| {C}\ar[r]_{h}&{D} |
{C}\ar[r]_{h}&{D} |
| translates to $g\circ f=h\circ k$. |
translates to $g\circ f=h\circ k$. |
