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, 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.
In detail, let be an object of some category. Given two monomorphisms
with codomain , we define an equivalence relation by if there exists an isomorphism with .
Equivalently, we write if factors through —that is, if there exists such that . The binary relation defined by
The relation ≤ induces a partial order on the collection of subobjects of .
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, rarely, locally small (this clashes with a different usage of the term locally small, namely that there is a set of morphisms between any two objects).
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.
This definition corresponds to the ordinary understanding of a subobject outside category theory. When the category's objects are sets (possibly with additional structure, such as a group structure) and the morphisms are set functions (preserving the additional structure), one thinks of a monomorphism in terms of its image. An equivalence class of monomorphisms is determined by the image of each monomorphism in the class; that is, two monomorphisms f and g into an object T are equivalent if and only if their images are the same subset (thus, subobject) of T. In that case there is the isomorphism of their domains under which corresponding elements of the domains map by f and g, respectively, to the same element of T; this explains the definition of equivalence.
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.
Given a partially ordered class P = (P, ≤), we can form a category with the elements of P as objects, and a single arrow from p to q iff p ≤ q. 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.
- Mac Lane, p. 126
- Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 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. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.