Subobject classifier


Subobject classifier

In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects 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 referred 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 toelements belonging to the subobject in question, and false to elements not belongingto the subobject. This is why the subobject classifier is widely used in the categorical description of logic.

Introductory example

As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset "j":"U" → "X" we can assign the function "χj" from "X" to Ω that maps precisely the elements of "U" to 1 (see characteristic function). Every function from "X" to Ω arises in this fashion from precisely one subset "U".

To render this example more clear let us 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"→{0,1}, which is defined as follows:

:chi_A(x) = egin{cases} 0, & mbox{if }x otin A \ 1, & mbox{if }xin Aend{cases}

(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 certain subset. Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow: true: {0} → {0, 1} which maps 0 to 1. Given this definition it can be easily seen that the subset "A" can be uniquely defined through the characteristic function "A"=χ"A"-1(1). Therefore the diagramis a pullback.

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

Axiom: Given a category C, then there exists an isomorphism,:y: SubC("X") ≅ HomC(X, Ω) ∀ "X" ∈ C

In Set this axiom can be restated as follows:

Axiom: The collection of all subsets of S denoted by mathcal{P}(S), and the collection of all maps from S to the set {0, 1}=2 denoted by 2"S" are isomorphic i.e. the function y:mathcal{P}(S) ightarrow2^S, which in terms of single elements of mathcal{P}(S) 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 subobject 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:

:for each monomorphism "j": "U" → "X" there is a unique morphism "χ j": "X" → Ω such that the following commutative diagram:is a pullback diagram — that is, "U" is the limit of the diagram:

The morphism "χ j" is then called the classifying morphism for the subobject represented by "j".

Further examples

Every topos has a subobject classifier. For the topos of sheaves of sets on a topological space "X", it can be described in these terms: take the disjoint union Ω of all the open sets "U" of "X", and its natural mapping π to "X" coming from all the inclusion maps. Then π is a local homeomorphism, and the corresponding sheaf is the required subobject classifier (in other words the construction of Ω is by means of its espace étalé). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation between points "x" and open sets "U" of "X". For a small category C, the subobject classifer in the topos of presheaves mathcal{S}^{C^{op is given as follows. For any c in C, Omega(c) is the set of sieves on c.

References

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last = Artin
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authorlink = Michael Artin
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last = Barr
first = Michael
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publisher = Springer-Verlag
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isbn = 0-387-96115-1

*cite book
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first = John
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publisher = Oxford University Press
location = Oxford
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*cite book
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first = Robert
title = Topoi: The Categorial Analysis of Logic
publisher = North-Holland, Reprinted by Dover Publications, Inc (2006)
year = 1983
isbn = 0444852077
url = http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3

*cite book
last = Johnstone
first = Peter
title = Sketches of an Elephant: A Topos Theory Compendium
publisher = Oxford University Press
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*cite book
last = Johnstone
first = Peter
title = Topos Theory
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isbn = 0123878500

*cite book
last = Mac Lane
first = Saunders
authorlink = Saunders Mac Lane
coauthors = Ieke Moerdijk
title = Sheaves in Geometry and Logic: a First Introduction to Topos Theory
publisher = Springer-Verlag
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isbn = 0387977104

*cite book
last = McLarty
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title = Elementary Categories, Elementary Toposes
publisher = Oxford University Press
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*cite book
last = Taylor
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title = Practical Foundations of Mathematics
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*"Topos-physics": An explanation of Topos theory and its implementation in Physics: [http://topos-physics.org/ Topos-physics, Where Geometry meets Dynamics]


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