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Subobject Classifier as a Generalized Bit of Information

Updated on July 1, 2012

Motivating Example

In this hub, I would like to explore a rabbit trail, so to speak, motivated by the aside mentioned in http://conflicttheory.hubpages.com/hub/Mathematical-Conflict-Theory-Development-A-Topos-Theory-Perspective.

In that hub, we mentioned that a working definition of classical bit is any two-valued set [1].

Since a two-valued set is the subobject classifier of the topos Sets [2], it made sense to ask the question whether the quantum bit is also a subobject in some suitable topos.

In this hub, we shall begin to explore some properties of a subobject classifier in a topos Δ. The hopes will be to find out in what way the subobject classifier of the topos Δ behaves like a bit of information for that topos

The first property I wish to point out is that the subobject classifier, call it Ω, in the topos Δ is that the subobject classifier Ω differs from being a terminal object of the topos Δ only in so much that the unique arrow δ:Ω→1 from Ω to the terminal object 1 in Δ is not monic.

Consider the diagram the arrows: truth:1→Ω and δ:Ω→1. Then δ o truth=id1. If truth o δ= idΩ

then Ω would be isomorphic to 1. Thus the only thing preventing Ω from being the terminal object Ω in Δ (or equivalently, the onlything preventing the terminal object from being the subobject classifier in Ω in Δ) is the failure of the morphism truth o δ from being equal to the identity morphism on Ω in Δ. This, in turn, is logically equivalent to the failure of the unique morphism from δ:Ω→1 from being a monic in Δ. If Δ is a locally small topos, δ:Ω→1 is monic if and only if HomΔ(A,Ω) is a singleton set for every object A in Δ. Thus, for Δ locally small, the failure of δ:Ω→1 to be a monic is logically equivalent to the existence of at least one object A in Δ such that the set HomΔ(A,Ω) has more than one element. Here HomΔ(A,Ω) is defined to be the set of all morphisms in Δ from the object A to the subobject classifier Ω. Note that for any object A in Δ, there exists at least one morphism in Δ from A to Ω (given by truth o δ).

To explore the properties of Ω further, suppose there exists exactly one object A in Δ such that HomΔ(A,Ω) has exactly two elements and for every other object A' in Δ, HomΔ(A',Ω) contains exactly one element. Then A is not isomorphic to any object (other than A itself) in Δ. The proof of this statement is as follows:

Let f, g:A →Ω be the two distinct morphisms in HomΔ(A,Ω). Consider the unique morphism δ:Ω→1 from the subobject classifier Ω to the terminal object 1 in Δ. Then since 1 is the terminal object, and f and g are morphisms from A to Ω in Δ, we have δ o f = δ o g, where

δ o f:A→1 and δ o g:A→1 must be the unique morphism from A to the terminal object 1 in Δ by the definition of terminal object in a category. So, we have δ o f = δ o g but f is not equal to g, and so δ fails to be a monic. Suppose there existed an epi e:A'→A for some object A' not equal to the object A in Δ. Then f o e and g o e are both elements of HomΔ(A',Ω). Since A' is not equal to A, and we assumed that every set HomΔ(A',Ω). with A' not equal to A is a singleton set, then HomΔ(A',Ω) has only one element, and f o e=g o e. But since e is an epi, f o e=g o e implies that f=g and so HomΔ(A,Ω) is a singleton set, which contradicts our assumption that HomΔ(A,Ω) had exactly two elements in it. This there exists no epimorphism with codomain A. Since every isomorphism in a topos is an epi, this shows that A can not be isomorphic to any other object in Δ other than A itself. So in this special case, Ω has allowed us to identify A from among all other objects in Δ: this is what information does. A bit of information isolates a smaller collection of things from among a larger collection of things [3].

The ideas above are not complete. I will continue to work on the concept, and will further elaborate via a hub if I find any insights.





Sources:

[1] Understanding Modern Physics Video: classical mechanics video lecture series by Professor Leonard Susskind, http://www.openculture.com/2008/07/susskindlecture.html.

[2] Sheaves in Geometry and Logic: A First Introduction to Topos Theory, by Saunders MacLane, and Ieke Moerdjk.

[3] http://en.wikipedia.org/wiki/Bit




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