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

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