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

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

u : SA and v : TA

with codomain A, we write uv if u factors through v—that is, if there exists φ : ST such that ${\displaystyle u=v\circ \varphi }$. The binary relation ≡ defined by

uv if and only if uv and vu

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 uv if and only if there exists an isomorphism φ : ST with ${\displaystyle u=v\circ \varphi }$.)

The relation ≤ induces a 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.

Examples

1. 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.
2. In Grp, the category of groups, the subobjects of A correspond to the subgroups of A.
3. Given a partially ordered class P, we can form a category with the elements of P 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.
4. A subobject of a terminal object is called a subterminal object.