Diagram
in a category
A mapping of an oriented graph
with set of vertices
and set of edges
into the category
for which
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and if the edge
has source (origin)
and target (end)
. The concept of a diagram in
may also be defined as the image of the mapping
, in order to obtain a better visualization of diagrams.
Let be an oriented chain of the graph
with source
and target
, 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
denote the composition of morphisms
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The diagram is said to be commutative if
for any two oriented chains
and
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
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The corresponding diagram is represented as follows:
![]() |
where the are objects in the category
, while the
are morphisms in this category.
A triangular diagram in a category corresponds to the graph
![]() |
and is represented as follows:
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Commutativity of this diagram means that .
A square diagram corresponds to the graph
![]() |
and is represented as follows:
![]() |
Commutativity of this diagram means that .
The class of diagrams with a given graph forms a category. A morphism of a diagram
into a diagram
is taken to be a family of morphisms
, where
runs through the set of vertices of
, so that for any edge
with source
and target
the condition
is met. In particular, one may speak of isomorphic diagrams. The graph
is sometimes referred to as the scheme of a diagram in
.
References
[1] | A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221 |
Comments
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 |
Diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram&oldid=46644