Subcategory

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

Formal definition

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

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

such that

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

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

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

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

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

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

Embeddings

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

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

Types of subcategories

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

A subcategory of ${\mathcal {C}}$ is wide or lluf (a term first posed by Peter Freyd^[2]) if it contains all the objects of ${\mathcal {C}}$ .^[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 ${\mathcal {S}}$ of an abelian category ${\mathcal {C}}$ such that for all short exact sequences

0\to M'\to M\to M''\to 0

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

References

↑ Jaap van Oosten. "Basic category theory" (PDF).
↑ 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.
↑ Wide subcategory at the nLab

[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

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