Diagram(2)

Let $G$ and $G ^ { \prime }$ be directed graphs (also called oriented graphs, diagram schemes or pre-categories; cf. also Graph, oriented). A diagram of shape (also called a diagram of type) $G$ in $G ^ { \prime }$ is a morphism of graphs $d : G \rightarrow G ^ { \prime }$; i.e. if $G$ and $G ^ { \prime }$ are given by

\begin{equation*} G : A G \overset{\text{dom}_G}{\underset{\underset{\text{codom}_G}{\rightarrow} }{\rightarrow}} O G, \end{equation*}

(here $O G$ and $A G$ denote, respectively, a set of objects and a set of arrows of $G$), then a morphism $d$ is a pair of mappings

\begin{equation*} d _ { 0 } : O G \rightarrow O G ^ { \prime } , \quad d _ { A } : A G \rightarrow A G ^ { \prime } \end{equation*}

with $\operatorname {dom}_{G^{\prime}} \circ d _ { A } = d _ { 0 } \circ \operatorname {dom}_{G}$, $\operatorname{codom}_{G'} \circ d _ { A } = d _ { 0 } \circ \operatorname{codom}_{G}$.

A diagram is called finite if its shape is a finite graph, i.e. $O G$ and $A G$ are finite sets. A diagram in a category $\mathcal{C}$ is defined as a diagram $G \rightarrow U \mathcal{C}$, where $U \cal C$ denotes the underlying graph of $\mathcal{C}$ (with the same objects and arrows, forgetting which arrows are composites and which are identities).

Every functor $F : \mathcal{C} \rightarrow \mathcal{C} ^ { \prime }$ is also a diagram $U F : U \mathcal C \rightarrow U \mathcal C ^ { \prime }$ between the corresponding graphs. This observation defines the forgetful functor $U : \operatorname{Cat} \rightarrow \operatorname{Graph}$ from small categories to small graphs (cf. also Functor).

Let $d , d ^ { \prime } : G \rightarrow \mathcal{C}$ be two diagrams of the same shape $G$ in the same category $\mathcal{C}$. A morphism between $d$ and $d ^ { \prime }$ is a mapping $\Phi : O G \rightarrow A \mathcal{C}$ that carries each object $g$ of the graph $G$ to an arrow $\Phi g : d g \rightarrow d ^ { \prime } g$, such that for any arrow $a : g \rightarrow g ^ { \prime }$ of $G$ the diagram

commutes.

All diagrams of the shape $G$ in $\mathcal{C}$ and all morphisms between them form a category.

Let $d : G \rightarrow \mathcal C$ be a diagram in the category $\mathcal{C}$ and let $\alpha = ( a _ { 1 } , \dots , a _ { n } )$ be a finite sequence of arrows of the graph $G$ with $\operatorname{dom} a_{i+1}=\operatorname{codom} a_i$, $i = 1 , \dots , n - 1$. Put $d \alpha = d a _ { n } \circ \ldots \circ d a _ { 1 }$. A diagram $d$ is called commutative if $d \alpha = d \alpha'$ for any finite sequence $\alpha ^ { \prime } = ( a ^ { \prime_1 } , \ldots , a ^ { \prime_m } )$ in $G$ with $\operatorname{dom} \alpha_{j+1}^{\prime} = \operatorname { codom } \alpha _ { j } ^ { \prime }$, $j = 1 , \dots , m - 1$, $\operatorname{dom} a_1= \operatorname { dom } a_1'$, $\operatorname{codom}a_n=\operatorname{codom}a_m'$.

A sequence is a diagram $d : G \rightarrow \mathcal C$, where $G$ is of the form

The corresponding diagram is represented by

\begin{equation*} c _ { 1 } \stackrel { \phi _ { 1 } } { \rightarrow } \ldots \stackrel { \phi _ { n - 1 } } { \rightarrow } c _ { n }, \end{equation*}

where $c _ { k } = d ( g _ { k } )$ are objects and $\phi _ { k } = d ( a _ { k } )$ are arrows of $\mathcal{C}$.

A triangle diagram in the category $\mathcal{C}$ is a diagram with shape graph

and is represented as

\begin{equation*} \left. \begin{array} { c c c } { \square } & { c _ { 2 } } & { \square } \\ { \square } & { \square } & { \searrow ^ { \phi _ { 2 } } } \\ { \square ^ { \phi _ { 1 } } \nearrow } & { \vec { \phi _ { 3 } } } &{c_3} \end{array} \right. . \end{equation*}

Commutativity means that $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 }$.

A quadratic diagram (also called a square diagram) in $\mathcal{C}$ corresponds to the graph

and is represented as

Commutativity means $\phi _ { 2 } \circ \phi _ { 1 } = \phi _ { 3 } \circ \phi _ { 4 }$.

References

[a1]	P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)
[a2]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. Ser. II , 9 (1957) pp. 120–221
[a3]	S. Maclane, "Categories for the working mathematician" , Springer (1971)