Subobject
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In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. 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 o ...
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 concer ...
,
subgroup 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 ...
s from
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,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 ho ...
. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
that describes how one object sits inside another, rather than relying on the use of elements. The
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
concept to a subobject is a . This generalizes concepts such as
quotient set 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 ...
s,
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 ''A'' be an object of some category. Given two monomorphisms :u: S \to A \ \text \ v: T\to A with
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
''A'', we define an equivalence relation by u \equiv v if there exists an isomorphism \phi: S \to T with u = v \circ \phi. Equivalently, we write u \leq v if u factors through ''v''—that is, if there exists \phi: S \to T such that u = v \circ \phi. The binary relation \equiv defined by :u \equiv v \iff u \leq v \ \text \ v\leq u 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 ''A'', 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 a ...
of these monomorphisms are the subobjects of ''A''. 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 A. The collection of subobjects of an object may in fact be a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
; 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.''


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 g^ \circ f 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 In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
, 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 o ...
''B'' of ''A'', or rather the collection of all maps from sets equipotent 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 In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories T ...
, 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 In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...
is called a subterminal object.


See also

*
Subobject classifier In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "tru ...
* Subquotient


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 Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, year=2004 , isbn=0-521-83414-7 , zbl=1034.18001 Objects (category theory)