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 $.

References

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

References

[a1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[a2]	S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7