Subcategory Explained
is a category
whose
objects are objects in
and whose
morphisms are morphisms in
with the same identities and composition of morphisms. Intuitively, a subcategory of
is a category obtained from
by "removing" some of its objects and arrows.
Formal definition
Let
be a category. A
subcategory
of
is given by
- a subcollection of objects of
, denoted
,
- a subcollection of morphisms of
, denoted
.such that
in
, the identity morphism id
is in
,
in
, both the source
and the target
are in
,
- for every pair of morphisms
and
in
the composite
is in
whenever it is defined.
These conditions ensure that
is a category in its own right: its collection of objects is
, its collection of morphisms is
, and its identities and composition are as in
. There is an obvious
faithful functor
, called the
inclusion functor which takes objects and morphisms to themselves.
Let
be a subcategory of a category
. We say that
is a
of
if for each pair of objects
and
of
,
Homl{S}(X,Y)=Homl{C}(X,Y).
A full subcategory is one that includes
all morphisms in
between objects of
. For any collection of objects
in
, there is a unique full subcategory of
whose objects are those in
.
Examples
, the category of
-
vector spaces forms a full subcategory of the category of (left or right)
-
modules.
Embeddings
Given a subcategory
of
, the inclusion functor
is both a faithful functor and
injective on objects. It is
full if and only if
is a full subcategory.
Some authors define an embedding to be a full and faithful functor. Such a functor is necessarily injective on objects up to isomorphism. For instance, the Yoneda embedding is an embedding in this sense.
Some authors define an embedding to be a full and faithful functor that is injective on objects.[1]
Other authors define a functor to be an embedding if it isfaithful andinjective on objects.Equivalently,
is an embedding if it is injective on morphisms. A functor
is then called a
full embedding if it is a full functor and an embedding.
With the definitions of the previous paragraph, for any (full) embedding
the
image of
is a (full) subcategory
of
, and
induces an
isomorphism of categories between
and
. If
is not injective on objects then the image of
is
equivalent to
.
In some categories, one can also speak of morphisms of the category being embeddings.
Types of subcategories
A subcategory
of
is said to be
isomorphism-closed or
replete if every isomorphism
in
such that
is in
also belongs to
. An isomorphism-closed full subcategory is said to be
strictly full.
A subcategory of
is
wide or
lluf (a term first posed by
Peter Freyd[2]) if it contains all the objects of
. A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself.
A Serre subcategory is a non-empty full subcategory
of an
abelian category
such that for all short exact sequences
in
,
belongs to
if and only if both
and
do. This notion arises from Serre's C-theory.
See also
References
- Web site: Jaap van Oosten. Basic category theory.
- Book: Freyd, Peter. Peter J. Freyd
. Peter J. Freyd . 1991. 95–104 . Algebraically complete categories. Lecture Notes in Mathematics . 1488. Springer. Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990). 10.1007/BFb0084215. 978-3-540-54706-8.
