Subobject classifier: Difference between revisions

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== 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 '' : ''A'' → ''S'' we can assign the function ''χ_A'' from ''S'' to Ω that maps precisely the elements of ''A'' 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 ''S'' to Ω arises in this fashion from precisely one subset ''A''.

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

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.pngsvg|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 Ω of '''C''' is a ''subobject classifier'' for '''C''' if there exists a morphism

:1 → Ω

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.pngsvg|center|frameless|class=skin-invert]]

:is a [[pullback diagram]]—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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| year = 1983

| isbn = 0-444-85207-7

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

*{{cite book

| last = Johnstone

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| year = 1992

| isbn = 0-19-853392-6}}

* {{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

| last = Taylor