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.

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

*u*:*S*->*A*and*v*:*T*->*A*

with codomain *A*, we write *u* v if *u* factors through *v*--that is, if there exists *?* : *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*. (Equivalently, one can define the equivalence relation by *u* ? *v* if and only if there exists an isomorphism *?* : *S* -> *T* with .)

The relation partial order on the collection of subobjects of 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 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.*

- 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**, 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.

**^**Mac Lane, p. 126

- 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.

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