Subcategory

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

Formal definition

Let ${\displaystyle {\mathcal {C}}}$ be a category. A subcategory ${\displaystyle {\mathcal {S}}}$ of ${\displaystyle {\mathcal {C}}}$ is given by

• a subcollection of objects of ${\displaystyle {\mathcal {C}}}$, denoted ${\displaystyle \operatorname {ob} ({\mathcal {S}})}$,
• a subcollection of morphisms of ${\displaystyle {\mathcal {C}}}$, denoted ${\displaystyle \operatorname {mor} ({\mathcal {S}})}$.

such that

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

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

Let ${\displaystyle {\mathcal {S}}}$ be a subcategory of a category ${\displaystyle {\mathcal {C}}}$. We say that ${\displaystyle {\mathcal {S}}}$ is a full subcategory of ${\displaystyle {\mathcal {C}}}$ if for each pair of objects ${\displaystyle X}$ and ${\displaystyle Y}$ of ${\displaystyle {\mathcal {S}}}$,

${\displaystyle \mathrm {Hom} _{\mathcal {S}}(X,Y)=\mathrm {Hom} _{\mathcal {C}}(X,Y).}$

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

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.^[1]

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^[2]) if it contains all the objects of ${\displaystyle {\mathcal {C}}}$.^[3] 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

${\displaystyle 0\to M'\to M\to M''\to 0}$

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

See also

Look up subcategory in Wiktionary, the free dictionary.

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

References