In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a subobject is, roughly speaking, an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
that sits inside another object in the same
category. The notion is a generalization of concepts such as
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s from
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
,
subgroups from
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
,
[Mac Lane, p. 126] and
subspaces from
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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 . This generalizes concepts such as
quotient sets,
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
s,
quotient spaces,
quotient graph In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some vertex in ''C'' with ...
s, etc.
Definitions
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
:
is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on the monomorphisms with codomain ''
'', and the corresponding
equivalence classes
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of these monomorphisms are the subobjects of ''
''.
The relation ≤ induces a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, 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.''
Interpretation
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.
Examples
In Set, the
category of sets, a subobject of ''A'' corresponds to a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''B'' of ''A'', or rather the collection of all maps from sets
equipotent
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', th ...
to ''B'' with
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
exactly ''B''. The subobject partial order of a set in Set is just its subset
lattice.
In Grp, the
category of groups, the subobjects of ''A'' correspond to the
subgroups
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of ''A''.
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.
A subobject of a
terminal object is called a
subterminal object.
See also
*
Subobject classifier
*
Subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
Notes
References
*
* {{cite book , editor1-last=Pedicchio , editor1-first=Maria Cristina , editor2-last=Tholen , editor2-first=Walter , title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory , series=Encyclopedia of Mathematics and Its Applications , volume=97 , location=Cambridge , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pr ...
, year=2004 , isbn=0-521-83414-7 , zbl=1034.18001
Objects (category theory)