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In mathematics, especially in [[category theory]], a '''subobject classifier''' is a special object Ω of a category; such that, intuitively, the [[subobject]]s of anany object ''X'' in the category correspond to the morphisms from ''X'' to Ω. Intuitively,In astypical the name suggestsexamples, whatthat amorphism subobjectassigns classifier does is"true" to identify/classifythe subobjectselements of athe givensubobject objectand according to which elements belong"false" to the subobjectother in question. Becauseelements of this''X.'' roleTherefore, thea subobject classifier is also referred toknown as thea "truth value object". In factand the wayconcept inis whichwidely theused subobjectin classifierthe classifiescategorical subobjectsdescription of alogic. givenNote object,however is by assigning the values true to elements belonging to thethat subobject inclassifiers question,are andoften falsemuch tomore elementscomplicated not belonging tothan the subobject.simple Thisbinary islogic whytruth thevalues subobject{true, classifier is widely used in the categorical description of logicfalse}.

== Introductory example ==

As an example, the set Ω = {0,1} is a subobject classifier in the [[category of sets]] and functions: to every subset ''A'' of ''S'' defined by the inclusion function  '' j '' : '' U A'' → '' X S'' we can assign the function ''χ_jA'' from '' X S'' to Ω that maps precisely the elements of ''UA'' to 1, and the elements outside ''A'' to 0 (seein other words, ''χ_A'' is the [[indicator function|characteristic function]] of ''A''). EveryConversely, every function from ''XS'' to Ω arises in this fashion from precisely one subset ''UA''.

To be clearer, consider a [[subset]] ''A'' of ''S'' (''A'' ⊆ ''S''), where ''S'' is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χ_''A'' : S → {0,1}, which is defined as follows:

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\end{cases}</math>

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong or not to a certainthe subset. Since in any category subobjects are identified as [[monomorphism|monic arrows]], we identify the value true with the arrow: ''A'true''': {0} → {0, 1} which maps 0 to 1. GivenIn this definitionfact, the subset χ_''A'' canis betrue uniquelyprecisely defined throughon the characteristicelements functionof ''A'' = χ_''A''⁻¹(1). Therefore the diagram

[[Image:SubobjectClassifier-01.png|center]]

is a [[pullback (category theory)|pullback]].

In this way, the collection of all subsets of ''S'' and the collection of all maps from ''S'' to Ω = {0,1} are [[isomorphic]].

The above example of subobject classifier in '''Set''' is very useful because it enables us to easily prove the following axiom:

To categorize this notion, recall that, in category theory, a subobject is actually represented by a pair consisting of an object ''A'' and a [[monomorphism|monic arrow]] ''A → S'' (interpreted as the inclusion into another object ''S''). Accordingly, '''true''' refers to the element 1, which is selected by the arrow: '''true''': {0} → {0, 1} that maps 0 to 1. The subset ''A'' of ''S'' can now be defined as the [[pullback (category theory)|pullback]] of '''true''' along the characteristic function χ_''A'', shown on the following diagram:

'''Axiom''': Given a category '''C''', then there exists an [[isomorphism]],

[[Image:SubobjectClassifier-01.pngsvg|center|frameless|class=skin-invert]]

:y: Sub_'''C'''(''X'') ≅ Hom_'''C'''(X, Ω) ∀ ''X'' ∈ '''C'''

Defined that way, χ is a morphism ''Sub''_C(''S'') → Hom_C(S, Ω). By definition, Ω is a '''subobject classifier''' if this morphism χ is an isomorphism.

In '''Set''' this axiom can be restated as follows:

'''Axiom''': The collection of all subsets of S denoted by <math>\mathcal{P}(S)</math>, and the collection of all maps from S to the set {0, 1} = 2 denoted by 2^''S'' are [[isomorphic]] i.e. the function <math>y:\mathcal{P}(S)\rightarrow2^S</math>, which in terms of single elements of <math>\mathcal{P}(S)</math> is ''A'' → χ_''A'', is a [[bijection]].

The above axiom implies the alternative definition of a subobject classifier:

'''Definition''': Ω is a '''subobject classifier''' iff there is a one to one correspondence between subobjects of ''X'' and [[morphisms]] from ''X'' to Ω.

== Definition ==

For the general definition, we start with a category '''C''' that has a [[terminal object]], which we denote by 1. The object Ω of '''C''' is a ''subobject classifier'' for '''C''' if there exists a morphism

:1 → Ω

with the following property:

:forFor each [[monomorphism]] ''j'': ''U'' → ''X'' there is a unique morphism ''χ_j'': ''X'' → Ω such that the following [[commutative diagram]]

[[Image:SubobjectClassifier-02.pngsvg|center|frameless|class=skin-invert]]

:is a [[pullback diagram]] — that—that is, ''U'' is the [[limit (category theory)|limit]] of the diagram:

[[Image:SubobjectClassifier-03.pngsvg|center|frameless|class=skin-invert]]

The morphism ''χ_j'' is then called the '''classifying morphism''' for the subobject represented by ''j''.

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== Further examples ==

=== Sheaves of sets ===

Every [[elementary topos]], defined as a category with finite [[Limit (category theory)|limits]] and [[power object]]s, necessarily has a subobject classifier.<ref>Pedicchio & Tholen (2004) p.8</ref> For the topos of [[sheaf (mathematics)|sheaves]] of sets on a [[topological space]] ''X'', it can be described in these terms: For any [[open set]] ''U'' of ''X'', <math>\Omega(U)</math> is the set of all open subsets of ''U''. Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset ''U'' is the open subset of ''U'' where the assertion is true.

The category of [[sheaf (mathematics)|sheaves]] of sets on a [[topological space]] ''X'' has a subobject classifier Ω which can be described as follows: For any [[open set]] ''U'' of ''X'', Ω(''U'') is the set of all open subsets of ''U''. The terminal object is the sheaf 1 which assigns the [[Singleton (mathematics)|singleton]] {*} to every open set ''U'' of ''X.'' The morphism η:1 → Ω is given by the family of maps η_''U'' : 1(''U'') → Ω(''U'') defined by η_''U''(*)=''U'' for every open set ''U'' of ''X''. Given a sheaf ''F'' on ''X'' and a sub-sheaf ''j'': ''G'' → ''F'', the classifying morphism ''χ_j'' : ''F'' → Ω is given by the family of maps ''χ_j,U'' : ''F''(''U'') → Ω(''U''), where ''χ_j,U''(''x'') is the union of all open sets ''V'' of ''U'' such that the restriction of ''x'' to ''V'' (in the sense of sheaves) is contained in ''j_V''(''G''(''V'')).

For a small category <math>C</math>, the '''subobject classifer''' in the '''topos of presheaves''' <math>\mathrm{Set}^{C^{op}}</math> is given as follows. For any <math>c \in C</math>, <math>\Omega(c)</math> is the set of [[Sieve (category theory)|sieves]] on <math>c</math>.

Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset ''U'' is the open subset of ''U'' where the assertion is true.

== References ==

=== Presheaves ===

<references />

Given a small category <math>C</math>, the category of [[presheaves]] <math>\mathrm{Set}^{C^{op}}</math> (i.e. the [[functor category]] consisting of all contravariant functors from <math>C</math> to <math>\mathrm{Set}</math>) has a subobject classifer given by the functor sending any <math>c \in C</math> to the set of [[Sieve (category theory)|sieves]] on <math>c</math>. The classifying morphisms are constructed quite similarly to the ones in the sheaves-of-sets example above.

=== Elementary topoi ===

Both examples above are subsumed by the following general fact: every [[elementary topos]], defined as a category with finite [[Limit (category theory)|limits]] and [[power object]]s, necessarily has a subobject classifier.<ref>Pedicchio & Tholen (2004) p.8</ref> The two examples above are [[Topos|Grothendieck topoi]], and every Grothendieck topos is an elementary topos.

== Related concepts ==

A [[quasitopos]] has an object that is almost a subobject classifier; it only classifies strong subobjects.

== Notes ==

{{Reflist}}

== References ==

*{{cite book

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| author-link = Michael Artin |author2=Alexander Grothendieck |author2-link=Alexander Grothendieck |author3=Jean-Louis Verdier |author3-link=Jean-Louis Verdier | title = Séminaire de Géometrie Algébrique IV

| authorlink = Michael Artin

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| title = Séminaire de Géometrie Algébrique IV

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*{{cite book

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| url = httphttps://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3}}

*{{cite book

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*{{cite book

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* {{cite book | last=Mac Lane | first=Saunders | authorlinkauthor-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | ___location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 }}

*{{cite book

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| title = Sheaves in Geometry and Logic: a First Introduction to Topos Theory

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| last = McLarty

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| title = Elementary Categories, Elementary Toposes

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Line 119 ⟶ 122:

| year = 1992

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* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina|editor1-link=M. Cristina Pedicchio | 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]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}

*{{cite book

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| year = 1999

| isbn = 0-521-63107-6}}

*''Topos-physics'': An explanation of Topos theory and its implementation in Physics

:[http://topos-physics.org/ Topos-physics, Where Geometry meets Dynamics]

[[Category:Topos theory]]