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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 any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of ''X.'' Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.

{{context}}

{{references}}

In [[category theory]], a '''subobject classifier''' is a special object Ω of a category; intuitively, the [[subobject]]s of an object ''X'' correspond to the morphisms from ''X'' to Ω. As the name suggests, what a '''subobject classifier''' does is to identify/classify subobjects of a given object according to which elements belong to the subobject in question. Because of this role, the subobject classifier is also refered to as the '''truth value object'''. In fact the way in which the '''subobject classifier''' classifies subobjects of a given object, is by assigning the values true to

elements belonging to the subobject in question, and false to elements not belonging

to the subobject. This is way the '''subobject classifier''' is widely used in the categorical description of logic.

== Introductory example ==

As an example, the set ΩΩ = {0,1} is a subobject classifier in the [[category of sets]] and functions: to every subset ''jA'': of ''US'' defined by the inclusion function&rarrnbsp;'' j ''X : ''A'' → ''S'' we can assign the function ''χχ_jA'' from ''XS'' to ΩΩ 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 ΩΩ 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:

:<math>\chi_A(x) =

\begin{cases}

0, & \mbox{if }x\notin A \\

1, & \mbox{if }x\in A

\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 to the subset ''A''. In fact, χ_''A'' is true precisely on the elements of ''A''.

The above example of subobject classifier in Sets is very usefull because it enables us to easliy prove the following axiom:

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

'''Axiom''': Given a category '''C''', then there exists an [[Isomorphisms]], <math>y:Sub_C(X)\cong Hom_C(X,\Omega)</math> <math>\forall X\in C</math>

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 <math>\{0,1\}=2</math> denoted by <math>2^S</math> 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 <math>A\rightarrow\chi_A</math>, it is a [[bijection]].

The above axiom implyes the alternative definition of a subobject calssifier:

'''Definition''': <math>\Omega</math> is a '''Subobject classifier''' iff there is a ``one to one" correspondence between subobject of X and [[morphisms]]

from X to <math>\Omega</math>.

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:

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

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.

== Definition ==

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

:1 → Ω

:1 &rarr; &Omega;

with the following property:

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

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

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

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

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

:for each [[monomorphism]] ''j'': ''U'' &rarr; ''X'' there is a unique morphism ''&chi;_j'': ''X'' -> &Omega; such that the following [[commutative diagram]]

== Further examples ==

''U'' -> 1

| |

v v

''X'' -> &Omega;

=== Sheaves of sets ===

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

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'')).

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.

<i>

1

|

v

''&chi;_j'': ''X'' -> &Omega;

</i>

=== Presheaves ===

The morphism ''&chi;_j'' is then called the '''classifying morphism''' for the subobject represented by ''j''.

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.

== FurtherRelated examplesconcepts ==

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

== Notes ==

Every [[topos]] has a subobject classifier. For the topos of [[sheaf (mathematics)|sheaves]] of sets on a [[topological space]] ''X'', it can be described in these terms: take the [[disjoint union]] &Omega; of all the [[open set]]s ''U'' of ''X'', and its natural mapping &pi; to ''X'' coming from all the [[inclusion map]]s. Then &pi; is a [[local homeomorphism]], and the corresponding sheaf is the required subobject classifier (in other words the construction of &Omega; is by means of its [[espace étalé]]). One can also consider &Omega; to be, in a (tautological) sense, the graph of the membership relation between points ''x'' and open sets ''U'' of ''X''.

{{Reflist}}

Let us consider an example of a '''subobject classifer''' in the '''Topos of presheafs''' <math>\mathcal{S}^{C^{op}}</math>. The formal definition goes as follows

'''Definition''':

A '''Subobject Classifier''' <math>\Omega</math> is a [[presheaf]]

<math>\Omega:C\rightarrow\mathcal{S}^{C^{op}}</math> such that to each object <math>A\in C</math>

there corresponds an object <math>\Omega(A)\in\mathcal{S}^{C^{op}}</math> which represents the set

of all sieves (see [[sieve]]).

== References ==

*{{cite book

* Robert Goldblatt: ''Topoi, the Categorial Analysis of Logic''. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

| last = Artin

: This book has been reprinted by Dover Publications, Inc (2006). The book can also be accessed freely on [http://www.mcs.vuw.ac.nz/~rob/ Robert Goldblatt's homepage]: [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3 Topoi, the Categorial Analysis of Logic].

| first = Michael

*Cecilia-Flori: ''Topos-physics'', [http://topos-physics.org/]

| 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

An explanation of Topos theory and its implementation in Physics

| publisher = [[Springer-Verlag]]

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}}

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* {{cite book | last=Mac Lane | first=Saunders | author-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 }}

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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 }}

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[[Category:Topos theory]]

[[category:Topos theory]]

[[Category:Objects (category theory)]]

[[es:Clasificador de subobjetos]]