PlanetMath: subcategory

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subcategory

(Definition)

Let $\mathcal{C}$ be a (small) category. If $\mathcal{S}$ is a collection of both a subset, call it $\operatorname{Ob}(\mathcal{S})$ , of objects of $\mathcal{C}$ and a subset, call it $\operatorname{Mor}(\mathcal{S})$ , of morphisms of $\mathcal{C}$ such that

For each $S\in\operatorname{Ob}(\mathcal{S})$ , the identity morphism of , $id_S\in\operatorname{Mor}(\mathcal{S});$
For each $f\in\operatorname{Mor}(\mathcal{S})$ , $\operatorname{domain}(f)$ and $\operatorname{codomain}(f)\in\operatorname{Ob}(\mathcal{S});$
For every pair $f,g\in\operatorname{Mor}(\mathcal{S})$ such that $f\circ g$ exists, then $f\circ g\in\operatorname{Mor}(\mathcal{S}).$

Then $\mathcal{S}$ is readily seen to be a category. It is called a subcategory of the category $\mathcal{C}.$

Given a category $\mathcal{C}$ and a subcategory $\mathcal{S}$ of $\mathcal{C}$ , a map

$\displaystyle \operatorname{Incl}:\mathcal{S}\hookrightarrow \mathcal{C}$

that sends each object of $\mathcal{S}$ to itself (in $\mathcal{C}$ ), and each morphism of $\mathcal{S}$ to itself (in $\mathcal{C}$ ), is a functor. $\operatorname{Incl}$ is called the inclusion functor, or an embedding. This inclusion functor is a faithful functor. If it is also full, then we call the corresponding subcategory $\mathcal{S}$ a full subcategory of $\mathcal{C}$ . In other words, if $\mathcal{S}$ is a full subcategory of $\mathcal{C}$ , then

$\displaystyle \operatorname{hom_{\mathcal{C}}}(S_1,S_2)=\operatorname{hom_{\mathcal{S}}}(S_1,S_2)$

for pair of $S_1,S_2\in \operatorname{Ob}(\mathcal{S})$ .

Remarks

Let $T:\mathcal{C}\to\mathcal{D}$ be a full and faithful functor. Then $T(\mathcal{C})$ is a full subcategory of $\mathcal{D}$ .
Again, let $T:\mathcal{C}\to\mathcal{D}$ be a full and faithful functor. If $\mathcal{S}$ is a full subcategory of $\mathcal{D}$ , then $T^{-1}(\mathcal{S})$ defined by:
- $\operatorname{Ob}(T^{-1}(\mathcal{S})):=\lbrace C\in\operatorname{Ob}(\mathcal{C})\mid T(C)\in\operatorname{Ob}(\mathcal{S})\rbrace$
- $\operatorname{Mor}(T^{-1}(\mathcal{S})):=\lbrace f\in\operatorname{Mor}(\mathcal{C})\mid T(f)\in\operatorname{Mor}(\mathcal{S})\rbrace$
is a subcategory of $\mathcal{C}$ .

Examples of Subcategories

In Set, the category of finite sets is a full subcategory, and so is the category of -element sets, where is any (possibly infinite) cardinality. If is finite, then every morphism in the subcategory is invertible.
In Top, we have the full subcategories whose objects are Euclidean spaces, compact spaces, or Hausdorff spaces.
In Grp, there is the full subcategory whose objects are abelian groups with additive homomorphisms.
Grp is in fact a subcategory of the category of topological groups, since every group may be viewed as a topological group with the discrete topology.
In Ring, there are the subcategories of commutative rings, matrix rings, or fields. Note that Field is not a full subcategory of Ring, since the ring homomorphism that maps every element to 0 is not a field homomorphism.