World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0000540476
Reproduction Date:

Title: Subobject  
Author: World Heritage Encyclopedia
Language: English
Subject: Substructure, Structure (mathematical logic), Subobject classifier, Subquotient, Isomorphism theorem
Publisher: World Heritage Encyclopedia


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 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 which 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, and quotient spaces.


  • Definition 1
  • Examples 2
  • See also 3
  • Notes 4
  • References 5


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

u: SA and
v: TA

with codomain A, say that uv if u factors through v — that is, if there exists w: ST such that u = v \circ w. The binary relation ≡ defined by

uv if and only if uv and vu

is an equivalence relation on the monomorphisms with codomain A, and the corresponding equivalence classes of these monomorphisms are the subobjects of A. If two monomorphisms represent the same subobject of A, then their domains are isomorphic. 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.)

To get the dual concept of quotient object, replace monomorphism by epimorphism above and reverse arrows.


In the category Set, 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. Similar results hold in Grp, 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


  1. ^ Mac Lane, p. 126


  • 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:  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.