In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory.[1] Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a morphism which describes how one object sits inside another, rather than relying on the use of elements.
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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. 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. 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.)
The dual 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.
See also
Notes
- ^ Mac Lane, p. 126
References
- Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, ISBN 0-387-98403-8
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