Subcategory

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 ${\displaystyle K}$, the category of ${\displaystyle K}$-vector spaces forms a full subcategory of the category of (left or right) ${\displaystyle K}$-modules.

Embeddings

Given a subcategory ${\displaystyle \mathcal{S}}$ of ${\displaystyle \mathcal{C}}$, the inclusion functor ${\displaystyle I:\mathcal{S}\to \mathcal{C}}$ is both a faithful functor and injective on objects. It is full if and only if ${\displaystyle \mathcal{S}}$ 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, ${\displaystyle F}$ is an embedding if it is injective on morphisms. A functor ${\displaystyle F}$ 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 ${\displaystyle F:\mathcal{B}\to \mathcal{C}}$ the image of ${\displaystyle F}$ is a (full) subcategory ${\displaystyle \mathcal{S}}$ of ${\displaystyle \mathcal{C}}$, and ${\displaystyle F}$ induces an isomorphism of categories between ${\displaystyle \mathcal{B}}$ and ${\displaystyle \mathcal{S}}$. If ${\displaystyle F}$ is a full and faithful functor but not necessarily injective on objects, then the image of ${\displaystyle F}$ is equivalent to ${\displaystyle \mathcal{B}}$.

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

Types of subcategories

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

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

${\displaystyle 0\to M^{\prime }\to M\to M^{″}\to 0}$

in ${\displaystyle \mathcal{C}}$, ${\displaystyle M}$ belongs to ${\displaystyle \mathcal{S}}$ if and only if both ${\displaystyle M^{\prime }}$ and ${\displaystyle M^{″}}$ do. This notion arises from Serre's C-theory.