Mathematical Conflict Theory Development: A Topos Theory Perspective
Example Suggesting Topos for Mathematical Conflict Theory
In this section, I want to bring up a previous example (see example 3 in http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict)
which will motivate the use of topoi as one way to develop and understand mathematical conflicts of the form we seen in http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict.
In all the examples listed in http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict, I argued those conflicts, and conflicts of the form given in those examples, arose because we attempted to force two distinct things to occupy a container that was only large enough to hold one distinct thing.
Let us now interpret distinct things as elements, and containers for elements as sets. Then the above statement, "attempting to fit two distinct things in a container large enough for only one distinct thing" translates to "attempting to fit a set containing exactly two elements into a set that contains only one element" which in set theory language translates to "attempting to have a two element set S as a subset of a one element set T."
Beginning of an aside:
Now any two element set S constitutes a single classical bit of information . Moreover, any two element set can be taken to be the subobject classifier Ω in the topos Set, whose objects are sets and whose morphisms are functions between sets. Thus, one may define the classical bit as simply being the subobject classifier of any topos isomorphic to the topos Set (that is, any boolean topos). This suggests defining a generalized bit of information, in terms of a given topos, as simply being the subobject classifier of that topos. To continue this aside, in quantum physics, one may attempt to find a suitable topos where the quantum bit is the subobject classifier, and use this topos to understand the nature of the quantum bit. Similarly, one may attempt to understand many other kinds of bits by studying those bits defined as subobject classifiers of the corresponding topoi.
End of aside.
Now in Set, the two object set is the subobject classifier, and the one object set is the terminal object in Set.
So a conflict in Set constitutes attempting to force a classical bit to encode a single classical state, which translates into attempting to force the subobject classifier in Set to be a subset of the one element set in Set. The analogue of a subset, subspace,..etc, in a topos is called a subobject. A subobject of an object S in a topos is simply an equivelance class of monics with codomain S . One may abuse the terminology and think of a subobject of S as simply being a monic into S (or with codomain S) .
Taking a two element set S as a subset of a one element set T in the topos Set translates, in topos theory language, to claiming that the unique arrow δ: Ω→Τ, from the subobject classifier Ω in Set to the terminal object Τ in Set, is a monic. Now the unique arrow from δ: Ω→Τ mentioned above is not a monic in reality, but what we wish to do is to ask what the consequences would be if it were.
To this end,
Let M be a machine whose job is to simply look at toposes and ask various "what if " questions about the topoi being studied by M and figure out what the consequences of those hypothetical speculations would lead to if they were true.
Suppose M looks at Set and asks what the logical consequences would be if δ: Ω→Τ were a monic. The answer is as follows:
Theorem 1. Let X be a topos, Ω the subobject classifier of X, 1 the terminal object of X, and
δ: Ω→1 the unique arrow from the subobject classifier Ω in X to the terminal object 1 in X. Then if δ: Ω→1 is monic, then Ω is isomorphic to 1 in X.
Proof: Let A be any object in X, and let f:A→Ω and g:A→Ω be a set of parallel arrows from A to Ω in X. Then since δ: Ω→1 and f:A→Ω are arrows in X, then δof:A→1 is an arrow from A to the terminal object 1 in X. Similarly, since δ: Ω→1 and g:A→Ω are arrows in X, then δog:A→1 is an arrow from A to the terminal object 1 in X. But by definition of terminal object in a category:
Fact (1) There is a unique arrow from any object B in X to the terminal object 1 in X. Since δof:A→1 and δog:A→1 are both arrows from the same object A in X to the terminal object 1 in X, then by Fact (1), δof = δog. But since δ is monic, then δof = δog implies that f=g. Thus there is at most one morphism from any object A in X to the subobject classifier Ω in X. To see that there is at least one morphism from any object A in X to the subobject classifier Ω in X, note that by the definition of subobject classifier in a category C, there exists a universal monic
true:1→Ω in X. Let A be any object of X, and let δ: A→1 the unique arrow from A in X to the terminal object 1 in X. Then by the axioms of composition operation in a category, the arrow
true o δ: A→Ω exists in X, and so there exists at least one arrow from every object A to the subobject classifier Ω in X. That is, there exists a unique arrow from every object A in X to the subobject classifier Ω in X, and so Ω is a terminal object in X by the definition of a terminal object in a category. But terminal objects in a category are unique up to cannonical isomorphism, and thus, since Ω is a terminal object in X, and 1 is a terminal object in X, Ω and 1 must be isomorphic. This completes the proof.
Now if X were taken to be Set, and Ω a two element set in Set, and 1 the terminal object in Set: then a hypothetical isomorphism (which does not exist in reality since the terminal object of Set contains one element, and the subobject classifier of Set contains exactly two distinct elements) from Ω a two element set in Set, to 1 in Set, is a pair of functions f,g in Set with f:Ω→1 and g:1→Ω with fog=id1 and gof=idΩ.
One may interpret fog=id1, the identity on the terminal object (a one element set), as corresponding to the first cannonical conflict solution of removing the distinction between the two elements in Ω to make Ω a one element set. Similarly, one may interpret gof=idΩ as the second cannonical solution of adding one bit of information to the terminal object to transform the terminal set into a two element set there by making 1 a two element set.
In conclusion, the mathematical conflicts we saw in
can be viewed as arising from our attempting to force a monic from the subobject Ω of a topos to the terminal object 1 in the same topos (for our examples of conflicts, the topos is Set, or any boolean topos). This suggests defining a conflict as a monic from the subobject classifier Ω in a topos to the terminal object in the same topos, and the resulting isomorphism as encoding the two cannonical resolutions of the conflict.
One application is to see if one may uncover additional Russel like paradoxes using the logic of toposes via the mathematical conflicts defined above.
 Understanding Modern Physics Video: classical mechanics video lecture series by Professor Leonard Susskind, http://www.openculture.com/2008/07/susskindlecture.html.
 Sheaves in Geometry and Logic: A First Introduction to Topos Theory, by Saunders MacLane, and Ieke Moerdjk.