Subcategory

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

Formal definition

Let $𝒞$ be a category. A subcategory $𝒮$ of $𝒞$ is given by

a subcollection of objects of $𝒞$ , denoted $ob (𝒮)$ ,
a subcollection of morphisms of $𝒞$ , denoted $mor (𝒮)$ .

such that

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

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

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

{H o m}_{𝒮} (X, Y) = {H o m}_{𝒞} (X, Y) .

A full subcategory is one that includes all morphisms in $𝒞$ between objects of $𝒮$ . For any collection of objects $A$ in $𝒞$ , there is a unique full subcategory of $𝒞$ 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 $𝒮$ of $𝒞$ , the inclusion functor $I : 𝒮 \to 𝒞$ is both a faithful functor and injective on objects. It is full if and only if $𝒮$ 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, $F$ 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 $F : ℬ \to 𝒞$ the image of $F$ is a (full) subcategory $𝒮$ of $𝒞$ , and $F$ induces an isomorphism of categories between $ℬ$ and $𝒮$ . If $F$ is a full and faithful functor but not necessarily injective on objects, then the image of $F$ is equivalent to $ℬ$ .

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

Types of subcategories

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

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

0 \to M^{'} \to M \to M^{″} \to 0

in $𝒞$ , $M$ belongs to $𝒮$ 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