# 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,^{[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.

## DefinitionsEdit

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

with codomain * *, we write if factors through * *—that is, if there exists such that . The binary relation defined by

is an equivalence relation on the monomorphisms with codomain * *, and the corresponding equivalence classes of these monomorphisms are the **subobjects** of * *. (Equivalently, one can define the equivalence relation by if and only if there exists an isomorphism with .)

The relation ≤ induces a partial order on the collection of subobjects of .

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 sometimes *locally small*.

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.*

## ExamplesEdit

- 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*p*≤*q*. 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.

## See alsoEdit

## NotesEdit

**^**Mac Lane, p. 126

## ReferencesEdit

- Mac Lane, Saunders (1998),
*Categories for the Working Mathematician*, Graduate Texts in Mathematics,**5**(2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001 - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.