Subcategory

For subcategories on Wikipedia, see Wikipedia:Subcategories.

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 not 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

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

References

1. Jaap van Oosten. "Basic category theory" (PDF).
2. Freyd, Peter (1991). "Algebraically complete categories". Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990). Lecture Notes in Mathematics. Vol. 1488. Springer. pp. 95–104. doi:10.1007/BFb0084215. ISBN   978-3-540-54706-8.
3. Wide subcategory at the nLab