# 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.

## Definitions[edit]

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

with codomain *, we define an equivalence relation by if there exists an isomorphism with .
*

Equivalently, 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 *.
*

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, 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[edit]

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 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[edit]

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 also[edit]

## Notes[edit]

**^**Mac Lane, p. 126

## References[edit]

- Mac Lane, Saunders (1998),
*Categories for the Working Mathematician*, Graduate Texts in Mathematics, vol. 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. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.