Subcategory

Summary

A subcategory of a category $\mathcal{C}$ is a choice of some objects and some morphisms of $\mathcal{C}$. The choice must be closed under composition and contain the identities of its objects. It is a category in its own right, and its inclusion into $\mathcal{C}$ is a faithful functor. Important special cases are the full subcategories, which keep all morphisms between the chosen objects, and the wide subcategories, which keep all objects. The category of abelian groups inside the category of groups is a standard example.