# Difference between revisions of "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 $D ( u) \in \mathop{\rm Hom} ( D ( i) , D ( j) )$ 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 $D ( \phi ) : D ( i) \rightarrow D ( j)$ 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

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

The corresponding diagram is represented as follows:

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

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

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

and is represented as follows:

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

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

A square diagram corresponds to the graph

$$\begin{array}{lcl} i _ {1} &\cdot \rightarrow ^ { {\phi _ 1} } \cdot &i _ {2} \\ size - 3 {\phi _ {4} } \downarrow &{} &\downarrow size - 3 {\phi _ {2} } \\ i _ {4} &\cdot \mathop \rightarrow \limits _ { {\phi _ {3} }} \cdot &i _ {3} \\ \end{array}$$

and is represented as follows:

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

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 $\nu _ {i} : D ( i) \rightarrow D _ {1} ( i)$, 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 $C$.

How to Cite This Entry:
Diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram&oldid=14728
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article