Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that 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 that 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.

In detail, let A be an object of some category. Given two monomorphisms

with codomain A, say that u ≤ v if u factors through v — that is, if there exists w: S → T such that ${\displaystyle u=v\circ w}$. The binary relation ≡ defined by

is an equivalence relation on the monomorphisms with codomain A, and the corresponding equivalence classes of these monomorphisms are the subobjects of A. If two monomorphisms represent the same subobject of A, then their domains are isomorphic. The collection of monomorphisms with codomain A under the relation ≤ forms a preorder, but the definition of a subobject ensures that the collection of subobjects of A is a partial order. (The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a set, the category is well-powered.)

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