# 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, and quotient spaces.

## Definition

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

*u*:*S*→*A*and*v*:*T*→*A*

with codomain *A*, say that *u* ≤ *v* if *u* factors through *v* — that is, if there exists *w*: *S* → *T* such that . The binary relation ≡ defined by

*u*≡*v*if and only if*u*≤*v*and*v*≤*u*

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*.)

To get the dual concept of quotient object, replace *monomorphism* by *epimorphism* above and reverse arrows.

## Examples

In the category **Set**, 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. Similar results hold in **Grp**, and some other categories.

Given a partially ordered class **P**, we can form a category with **P'**s elements as objects and a single arrow going from one object (element) to another if the first is less than or equal to the second. 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

## Notes

- ↑ Mac Lane, p. 126

## References

- 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.