Wikipedia / Subcategory
Wikipedia / Subcategory
In mathematics, specifically category theory, a subcategory of a category whose objects and whose morphisms by "removing" some of its objects and arrows.
Formal definition
be a category. A subcategory is given by
,
.
such that
, the identity morphism id,
,
whenever it is defined.
. There is an obvious faithful functor, called the inclusion functor which takes objects and morphisms to themselves.
is a of,
A full subcategory is one that includes all.
Examples
The category of finite sets forms a full subcategory of the category of sets.
The category whose objects are sets and whose morphisms are bijections forms a non-full subcategory of the category of sets.
The category of abelian groups forms a full subcategory of the category of groups.
The category of rings (whose morphisms are unit-preserving ring homomorphisms) forms a non-full subcategory of the category of rngs.
For a field-vector spaces-modules.
Embeddings
is both a faithful functor and injective on objects. It is full 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.
Other authors define a functor to be an embedding is then called a full embedding if it is a full functor and an embedding.
the image induces an isomorphism of categories is equivalent.
In some categories, one can also speak of morphisms of the category being embeddings.
Types of subcategories
is said to be isomorphism-closed or replete. An isomorphism-closed full subcategory is said to be strictly full.
is wide or lluf (a term first posed by Peter Freyd. A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself.
A Serre subcategory of an abelian category such that for all short exact sequences
do. This notion arises from Serre's C-theory.
See also
Exact category, a full subcategory closed under extensions.
References