Diagram(2)
Let and
be directed graphs (also called oriented graphs, diagram schemes or pre-categories; cf. also Graph, oriented). A diagram of shape (also called a diagram of type)
in
is a morphism of graphs
; i.e. if
and
are given by
![]() |
![]() |
(here and
denote, respectively, a set of objects and a set of arrows of
), then a morphism
is a pair of mappings
![]() |
with ,
.
A diagram is called finite if its shape is a finite graph, i.e. and
are finite sets. A diagram in a category
is defined as a diagram
, where
denotes the underlying graph of
(with the same objects and arrows, forgetting which arrows are composites and which are identities).
Every functor is also a diagram
between the corresponding graphs. This observation defines the forgetful functor
from small categories to small graphs (cf. also Functor).
Let be two diagrams of the same shape
in the same category
. A morphism between
and
is a mapping
that carries each object
of the graph
to an arrow
, such that for any arrow
of
the diagram
![]() |
commutes.
All diagrams of the shape in
and all morphisms between them form a category.
Let be a diagram in the category
and let
be a finite sequence of arrows of the graph
with
,
. Put
. A diagram
is called commutative if
for any finite sequence
in
with
,
,
,
.
A sequence is a diagram , where
is of the form
![]() |
The corresponding diagram is represented by
![]() |
where are objects and
are arrows of
.
A triangle diagram in the category is a diagram with shape graph
![]() |
and is represented as
![]() |
Commutativity means that .
A quadratic diagram (also called a square diagram) in corresponds to the graph
![]() |
and is represented as
![]() |
Commutativity means .
References
[a1] | P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967) |
[a2] | A. Grothendieck, "Sur quelques points d'algebre homologique" Tôhoku Math. J. Ser. II , 9 (1957) pp. 120–221 |
[a3] | S. Maclane, "Categories for the working mathematician" , Springer (1971) |
Diagram(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram(2)&oldid=12484