What is "subobject"

n. (context category theory English) Given an object ''B'', a ''subobject'' of it is an equivalence class of objects $A\_i$ which relate to ''B'' through monomorphisms $m\_i\; :\; A\_i\; rightarrow\; B$. (If a pair of monomorphisms with codomain ''B'' http://en.wikipedia.org/wiki/Mathematical_jargon%23factor_through each other, then their domains are isomorphic and thus belong to an equivalence class which defines a ''subobject'' of ''B''). The ''subobject'' generalizes its interpretation in category '''Set''' as a set which is a subset (though an http://en.wikipedia.org/wiki/inclusion%20map, which is a monomorphism) of another set.

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism which describes how one object sits inside another, rather than relying on the use of elements.

The dual concept to a subobject is a quotient object. This generalizes concepts such as quotient sets, quotient groups, and quotient spaces.