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.


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


with codomain  , we write   if   factors through  —that is, if there exists   such that  . The binary relation   defined by


is an equivalence relation on the monomorphisms with codomain  , and the corresponding equivalence classes of these monomorphisms are the subobjects of  . (Equivalently, one can define the equivalence relation by   if and only if there exists an isomorphism   with  .)

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 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.


  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 = (P, ≤), we can form a category with the elements of P as objects, and a single arrow from p to q iff pq. 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.

See alsoEdit


  1. ^ Mac Lane, p. 126


  • Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 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. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.