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comma category
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(Definition)
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Let
be a category and two subcategories and of
. We form a category as follows:
- The objects of
are of the form , where is an object in , is an object in , and is a morphism (in
) from to :
- The morphisms of
are of the form
, where and are morphisms, such that
is a commutative diagram: .
It is easy to check that is indeed a category. For example, given object , is the corresponding identity morphism. Furthermore, if we have the following two commutative diagrams
we may combine them and form the following commutative diagram
which shows that
, so that
is the composition of and .
Definition. , so constructed, is called a comma category (the comma between and ), or a slice category.
Examples. In the following examples, identity morphisms and compositions of morphisms are implicitly assumed to be included in the subcategories.
and each consists of a single object of
, say , then is just . The objects of are just morphisms , and the morphism of is . is a discrete category.
consists of an object in
and
. Then the comma category, written
or
, may be visualized as a “cone” with apex and base
.
- Similarly, we can form an “inverted cone”
or
.
- Taking
, then the comma category
is the arrow category of
whose objects are morphisms of
and morphisms can be identified with commutative squares in
.
The diagrams , can be joined to the original category
via the inclusion functors :
which suggests that a comma category may be more generally defined in terms of a pair of categories
, and a pair of functors into a certain given category
. Specifically, let
and
be categories and
and
be functors into a specific category
. A comma category of the diagram
written or
, consists of the following:
- objects have the form
, where
is an object of
,
is an object of
, and
-
is a morphism in
;
- morphisms from
to have the form , where
is a morphism of
,
is a morphism of
, such that
- the following diagram
is commutative:
.
- morphism composition in
is given by
, where
and
.
It is an easy exercise to verify that indeed,
is a category. For example, the identity morphism on is provided by the morphism .
Remark. If
and
happen to be subcategories of
and are the inclusion functors, then we may write as
or
.
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"comma category" is owned by CWoo.
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(view preamble | get metadata)
| Other names: |
slice category |
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Cross-references: functors, terms, inclusion functors, diagrams, squares, commutative, arrow category, base, apex, discrete category, composition, identity, commutative diagram, morphism, objects, subcategories, category
There are 6 references to this entry.
This is version 9 of comma category, born on 2006-09-13, modified 2008-10-05.
Object id is 8347, canonical name is CommaCategory.
Accessed 1390 times total.
Classification:
| AMS MSC: | 18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories) |
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Pending Errata and Addenda
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