# Diagram

in a category $C$

A mapping $D$ of an oriented graph $\Gamma$ with set of vertices $I$ and set of edges $U$ into the category $C$ for which

$$D (I) \subset \mathop{\rm Ob} (C) ,\ D (U) \subset \mathop{\rm Mor} (C) ,$$

and $D (u) \in {\mathop{\rm Hom}\nolimits} ( D (i) , D (j) )$ if the edge $u \in U$ has source (origin) $i$ and target (end) $j$. The concept of a diagram in $C$ may also be defined as the image of the mapping $D$, in order to obtain a better visualization of diagrams.

Let $\phi = ( u _ {1} \dots u _ {n} )$ be an oriented chain of the graph $\Gamma$ with source $i$ and target $j$, i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let $D ( \phi ) : D (i) \mathop \rightarrow \limits D (j)$ denote the composition of morphisms

$$D (u _ {n} ) \circ \dots \circ D (u _ {1} ) .$$

The diagram $D$ is said to be commutative if $D ( \phi ) = D ( \phi ^ \prime )$ for any two oriented chains $\phi$ and $\phi ^ \prime$ with identical source and target.

The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form

$$\mathop \cdot \limits _ {i _ {1}} \mathop \rightarrow \limits ^ {u _ {1}} \mathop \cdot \limits _ {i _ {2}} \mathop \rightarrow \limits \dots \mathop \rightarrow \limits \mathop \cdot \limits _ {i _ {n-1}} \mathop \rightarrow \limits ^ {u _ {n-1}} \mathop \cdot \limits _ {i _ {n}} .$$

The corresponding diagram is represented as follows:

$$A _ {1} \mathop \rightarrow \limits ^ {f _ {1}} A _ {2} \mathop \rightarrow \limits \dots \mathop \rightarrow \limits A _ {n-1} \mathop \rightarrow \limits ^ {f _ {n-1}} A _ {n} ,$$

where the $A _ {k} = D ( i _ {k} )$ are objects in the category $C$, while the $f _ {k} = D ( u _ {k} )$ are morphisms in this category.

A triangular diagram in a category $C$ corresponds to the graph

$$\begin{array}{lcr} {i _ {1}} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}} &\mathop \cdot \limits {i _ {2}} \\ {} _ {u _ {3}} \searrow &{} &\swarrow _ {u _ {2}} \\ {} &\mathop \cdot \limits _ {i _ {3}} &{} \\ \end{array}$$

and is represented as follows:

$$\begin{array}{lcr} A _ {1} &\mathop \rightarrow \limits ^ {f _ {1}} &A _ {2} \\ {} _ {f _ {3}} \searrow &{} &\swarrow _ {f _ {2}} \\ {} &A _ {3} &{} \\ \end{array}$$

Commutativity of this diagram means that $f _ {3} = f _ {2} \circ f _ {1}$.

A square diagram corresponds to the graph

$$\begin{array}{rcl} i _ {1} \mathop \cdot \limits &\mathop \rightarrow \limits ^ {u _ {1}} &\mathop \cdot \limits i _ {2} \\ \scriptsize {u _ {4}} \downarrow &{} &\downarrow \scriptsize {u _ {2}} \\ i _ {4} \mathop \cdot \limits &\mathop \rightarrow \limits _ {u _ {3}} &\mathop \cdot \limits i _ {3} \\ \end{array}$$

and is represented as follows:

$$\begin{array}{lcl} A _ {1} &\mathop \rightarrow \limits ^ {f _ {1}} &A _ {2} \\ \scriptsize {f _ {4}} \downarrow &{} &\downarrow \scriptsize {f _ {2}} \\ A _ {4} &\mathop \rightarrow \limits _ {f _ {3}} &A _ {3} \\ \end{array}$$

Commutativity of this diagram means that $f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4}$.

The class of diagrams with a given graph $\Gamma$ forms a category. A morphism of a diagram $D$ into a diagram $D _ {1}$ is taken to be a family of morphisms $\nu _ {i} : D (i) \mathop \rightarrow \limits D _ {1} (i)$, where $i$ runs through the set of vertices of $\Gamma$, so that for any edge $u$ with source $i$ and target $j$ the condition $D _ {1} (u) \circ \nu _ {i} = \nu _ {j} \circ D (u)$ is met. In particular, one may speak of isomorphic diagrams. The graph $\Gamma$ is sometimes referred to as the scheme of a diagram in $C$.

#### References

 [1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221