In category theory, a subobject is, roughly speaking, an object that is contained in another category object. The definition generalizes the older concepts of subsets in set theory and subgroups in group theory. [1] Since the “real” structure of objects is not considered in category theory, the definition is based on the use of morphisms, rather than “elements”.
Content
- 1 Definition
- 2 Examples
- 3 notes
- 4 Literature
Definition
Let A be an object of some category. Having two monomorphisms :
- u : S → A and
- v : T → A
with the general image of A , we say that u ≤ v if u “passes through” v , that is, if there exists a morphism w : S → T such that u = v ∘ w . We define the following binary relation:
- u ≡ v if and only if u ≤ v and v ≤ u .
This is an equivalence relation on monomorphisms with the image of A ; we call its equivalence classes subobjects of A. Monomorphisms with the image A and the relation ≤ form a preorder , but the definition of a subobject guarantees that the subobjects A form a partially ordered set .
The dual concept to a subobject is a factor object; that is, in order to obtain a definition of a factor object, it is necessary to replace “monomorphism” with “epimorphism” in the definition above and change the direction of all arrows.
Examples
In the category of sets, the subobjects of A correspond to the subsets of A , or, more precisely, to the class of all embeddings of sets that are equipotent to the given subset. The same is true in the group category and in some other categories.
Notes
- ↑ Mac Lane, p. 126
Literature
- Mac Lane, Saunders. Categories for the Working Mathematician. - 2nd. - New York, NY: Springer-Verlag , 1998 .-- Vol. 5. - ISBN 0-387-98403-8 .