Subcategory

A particular case of the concept of a substructure of a mathematical structure. A category $\mathfrak{L}$ is called a subcategory of a category $\mathfrak{K}$ if $\mathrm{Ob}(\mathfrak{L} \subseteq \mathrm{Ob}(\mathfrak{L})$, $$ H_{\mathfrak{L}} (A,B) = H_{\mathfrak{K}}(A,B) \cap \mathrm{Mor}(\mathfrak{L}) $$ for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ and if the composite of two morphisms in $\mathfrak{L}$ coincides with their composite in $\mathfrak{K}$. For each subclass $\mathfrak{L}'$ of $\mathrm{Ob}(\mathfrak{K})$ there are smallest and largest subcategories $\mathfrak{L}_1$ and $\mathfrak{L}_2$ of $\mathfrak{K}$ whose classes of objects coincide with $\mathfrak{L}'$; the subcategory $\mathfrak{L}_1$ contains only identity morphisms of objects in $\mathfrak{L}'$ and is called the discrete subcategory generated by $\mathfrak{L}'$; the subcategory $\mathfrak{L}_2$ contains all morphisms in $\mathfrak{K}$ with domain and codomain in $\mathfrak{L}'$ and is called the full subcategory generated by $\mathfrak{L}'$. Any subcategory $\mathfrak{L}$ of $\mathfrak{K}$ for which $H_{\mathfrak{L}}(A,B) = H_{\mathfrak{L}}(A,B)$ for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ is called a full subcategory of $\mathfrak{K}$. The following are full subcategories: the subcategory of non-empty sets in the category of all sets, the subcategory of Abelian groups in the category of all groups, etc. For a small category $\mathfrak{D}$, the full subcategory of the category of all contravariant functors from $\mathfrak{D}$ into the category of sets generated by the hom-functors (morphism functors, $A \mapsto H_{\mathfrak{D}}(A,B)$ is isomorphic to $\mathfrak{D}$ (cf. also Functor). This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.

An arbitrary subcategory of a category $\mathfrak{K}$ need not inherit any of the properties of this category. However, there are important classes of subcategories that inherit many properties of the ambient category, such as reflective subcategories and co-reflective subcategories (cf. Reflective subcategory).

For references see Category; Functor.