Subobject

Summary

A subobject of an object $X$ in a category is an equivalence class of monomorphisms into $X$. Two monomorphisms are equivalent when each factors through the other. It is the categorical version of a subset, a subgroup or a subspace, defined only through morphisms. The subobjects of $X$ are partially ordered by factorisation, and in a topos they are classified by the subobject classifier.