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Subobject

Summary

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,[1] 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, quotient spaces, quotient graphs, etc.

Definitions

In detail, let ${\displaystyle A}$  be an object of some category. Given two monomorphisms

${\displaystyle u:S\to A\ {\text{and}}\ v:T\to A}$

with codomain ${\displaystyle A}$ , we define an equivalence relation by ${\displaystyle u\equiv v}$  if there exists an isomorphism ${\displaystyle \phi :S\to T}$  with ${\displaystyle u=v\circ \phi }$ .

Equivalently, we write ${\displaystyle u\leq v}$  if ${\displaystyle u}$  factors through ${\displaystyle v}$ —that is, if there exists ${\displaystyle \phi :S\to T}$  such that ${\displaystyle u=v\circ \phi }$ . The binary relation ${\displaystyle \equiv }$  defined by

${\displaystyle u\equiv v\iff u\leq v\ {\text{and}}\ v\leq u}$

is an equivalence relation on the monomorphisms with codomain ${\displaystyle A}$ , and the corresponding equivalence classes of these monomorphisms are the subobjects of ${\displaystyle A}$ .

The relation ≤ induces a partial order on the collection of subobjects of ${\displaystyle A}$ .

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 called well-powered or, rarely, locally small (this clashes with a different usage of the term locally small, namely that there is a set of morphisms between any two objects).

To get the dual concept of quotient object, replace "monomorphism" by "epimorphism" above and reverse arrows. A quotient object of A is then an equivalence class of epimorphisms with domain A.

Interpretation

This definition corresponds to the ordinary understanding of a subobject outside category theory. When the category's objects are sets (possibly with additional structure, such as a group structure) and the morphisms are set functions (preserving the additional structure), one thinks of a monomorphism in terms of its image. An equivalence class of monomorphisms is determined by the image of each monomorphism in the class; that is, two monomorphisms f and g into an object T are equivalent if and only if their images are the same subset (thus, subobject) of T. In that case there is the isomorphism ${\displaystyle g^{-1}\circ f}$  of their domains under which corresponding elements of the domains map by f and g, respectively, to the same element of T; this explains the definition of equivalence.

Examples

In Set, the category of sets, a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Set is just its subset lattice.

In Grp, the category of groups, the subobjects of A correspond to the subgroups of A.

Given a partially ordered class P = (P, ≤), we can form a category with the elements of P as objects, and a single arrow from p to q iff pq. If P has a greatest element, the subobject partial order of this greatest element will be P itself. This is in part because all arrows in such a category will be monomorphisms.

A subobject of a terminal object is called a subterminal object.