In mathematics, specifically category theory, a subcategory of a category <math>\mathcal{C}</math> is a category <math>\mathcal{S}</math> whose objects are objects in <math>\mathcal{C}</math> and whose morphisms are morphisms in <math>\mathcal{C}</math> with the same identities and composition of morphisms. Intuitively, a subcategory of <math>\mathcal{C}</math> is a category obtained from <math>\mathcal{C}</math> by "removing" some of its objects and arrows.

Formal definition

Let <math>\mathcal{C}</math> be a category. A subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math> is given by

  • a subcollection of objects of <math>\mathcal{C}</math>, denoted <math>\operatorname{ob}(\mathcal{S})</math>,
  • a subcollection of morphisms of <math>\mathcal{C}</math>, denoted <math>\operatorname{mor}(\mathcal{S})</math>.

such that

  • for every <math>X</math> in <math>\operatorname{ob}(\mathcal{S})</math>, the identity morphism id<sub><math>X</math></sub> is in <math>\operatorname{mor}(\mathcal{S})</math>,
  • for every morphism <math>f:X\to Y</math> in <math>\operatorname{mor}(\mathcal{S})</math>, both the source <math>X</math> and the target <math>Y</math> are in <math>\operatorname{ob}(\mathcal{S})</math>,
  • for every pair of morphisms <math>f</math> and <math>g</math> in <math>\operatorname{mor}(\mathcal{S})</math> the composite <math>f\circ g</math> is in <math>\operatorname{mor}(\mathcal{S})</math> whenever it is defined.

These conditions ensure that <math>\mathcal{S}</math> is a category in its own right: its collection of objects is <math>\operatorname{ob}(\mathcal{S})</math>, its collection of morphisms is <math>\operatorname{mor}(\mathcal{S})</math>, and its identities and composition are as in <math>\mathcal{C}</math>. There is an obvious faithful functor <math>I:\mathcal{S}\to\mathcal{C}</math>, called the inclusion functor which takes objects and morphisms to themselves.

Let <math>\mathcal{S}</math> be a subcategory of a category <math>\mathcal{C}</math>. We say that <math>\mathcal{S}</math> is a of <math>\mathcal{C}</math> if for each pair of objects <math>X</math> and <math>Y</math> of <math>\mathcal{S}</math>,

:<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).</math>

A full subcategory is one that includes all morphisms in <math>\mathcal{C}</math> between objects of <math>\mathcal{S}</math>. For any collection of objects <math>A</math> in <math>\mathcal{C}</math>, there is a unique full subcategory of <math>\mathcal{C}</math> whose objects are those in <math>A</math>.

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 <math>K</math>, the category of <math>K</math>-vector spaces forms a full subcategory of the category of (left or right) <math>K</math>-modules.

Embeddings

Given a subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math>, the inclusion functor <math>I:\mathcal{S}\to\mathcal{C}</math> is both a faithful functor and injective on objects. It is full if and only if <math>\mathcal{S}</math> 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 if it is

faithful and

injective on objects.

Equivalently, <math>F</math> is an embedding if it is injective on morphisms. A functor <math>F</math> 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 <math>F:\mathcal{B}\to\mathcal{C}</math> the image of <math>F</math> is a (full) subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math>, and <math>F</math> induces an isomorphism of categories between <math>\mathcal{B}</math> and <math>\mathcal{S}</math>. If <math>F</math> is a full and faithful functor but not necessarily injective on objects, then the image of <math>F</math> is equivalent to <math>\mathcal{B}</math>.

In some categories, one can also speak of morphisms of the category being embeddings.

Types of subcategories

A subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math> is said to be isomorphism-closed or replete if every isomorphism <math>k:X\to Y</math> in <math>\mathcal{C}</math> such that <math>Y</math> is in <math>\mathcal{S}</math> also belongs to <math>\mathcal{S}</math>. An isomorphism-closed full subcategory is said to be strictly full.

A subcategory of <math>\mathcal{C}</math> is wide or lluf (a term first posed by Peter Freyd) if it contains all the objects of <math>\mathcal{C}</math>. 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 <math>\mathcal{S}</math> of an abelian category <math>\mathcal{C}</math> such that for all short exact sequences

:<math>0\to M'\to M\to M\to 0</math>

in <math>\mathcal{C}</math>, <math>M</math> belongs to <math>\mathcal{S}</math> if and only if both <math>M'</math> and <math>M</math> do. This notion arises from Serre's C-theory.

See also

  • Reflective subcategory
  • Exact category, a full subcategory closed under extensions.

References