In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a subobject is, roughly speaking, an object which sits inside another object in the same category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

. The notion is a generalization of the older concepts of subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 from set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 and subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 from group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

. Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 which describes how one object sits inside another, rather than relying on the use of elements.

Definition

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

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s

u: S → A and

v: T → A

with codomain A, say that u ≤ v if u factors through v. The binary relation ≡ defined by

u ≡ v if and only if u ≤ v and v ≤ u

is an equivalence relation

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 on the monomorphisms with codomain A, and the corresponding equivalence classes of these monomorphisms are the subobjects of A. The collection of monomorphisms with codomain A under the relation ≤ forms a preorder

Preorder

In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...

, 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.)

The dual

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

concept to a subobject is a quotient object; that is, to define quotient object replace monomorphism by epimorphism above and reverse arrows.

Examples

In the category 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 Sets is just its subset lattice. Similar results hold in Groups, 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.