Subcategory

https://en.wikipedia.org/wiki/Subcategory

A subcategory 𝒟 of a category 𝒞 is defined by restricting the collection of objects in 𝒞 to a subcollection of objects in 𝒟 and the collection of morphisms in 𝒞 to a subcollection of morphisms in 𝒟 subject to the requirement that 𝒟 contains:

• domain and codomain of any morphism in 𝒟,

• the identity morphism of any object in 𝒟, and

• the composite of any composable pair of morphisms in 𝒟.

For example, there is a subcategory CRing ⊂ Ring of commutative unital rings. Both of these form subcategories of the category Rng of not-necessarily unital rings and homomorphisms that need not preserve the multiplicative unit.

Any category 𝒞 contains a maximal groupoid, the subcategory containing all of the objects and only those morphisms that are isomorphisms.

A subcategory S of a category C is a category 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 objects and arrows.

A category C whose objects and arrows are subclasses of those of a category A, and whose source, target, identities and compositions are those of A is said to be a subcategory of A.

A subcategory C of a category A is said to be a full subcategory when, for all pairs of objects A and B in C, if f : A → B is an arrow in A then it is also an arrow in C. A full subcategory of a category A is determined by objects alone.