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Conflict Theory Development
A Category Theory of Conflict
In what follows, we will assume that each word being treated as an object will be an object in some category C, and each arrow below, to be a morphism in the same category C. For those who lack a knowledge of category theory, the following link to a series of short video lectures on category theory by theCatsters is an excellent resource to learn some basic category theory quickly: https://www.scss.tcd.ie/Edsko.de.Vries/ct/catsters/linear.php.
To motivate the use of category theory as the mathematical language in which to develop our mathematical theory of conflict, we shall use the following simple example of a conflict:
Let P be the predicate "The cat is _____"
Let h: alive-> P be the arrow that takes the word red and maps it into the predicate P to produce the statement "The cat is alive." Like wise, let the arrow h':dead->P be the arrow that takes the word dead and maps it into the predicate P to produce the statement "The cat is dead."
Notice, that like in our previous examples in the previous hub (see http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict), the claims "The cat is alive" and "The cat is dead" conflict if we assert they cannot both be true at the same time: i.e. if we assert that the same cat cannot be both dead and alive at the same time.
Let q: alive->dead be the arrow that takes to word "alive" to the word "dead": i.e.
Since h'oq(alive): alive->P and h:alive->P are morphisms in our underlying catogory, we might naturally ask whether h=h'oq. The following computation shows h is not equal to h'oq.
h'oq(alive): alive->P is an arrow from alive to P with h'oq(alive)=h'(q(alive))=h'(dead)="The cat is dead."
On the other hand, h(alive): alive->P is an arrow from alive to P with h(alive)="The cat is alive." Since "The cat is alive." is not equal to "The cat is dead.", we see that h'oq(alive) is not equal to h(alive), that is, h'oq ≠ h.
So our conflict has reduced to a diagram in a category C that does not commute. This suggests defining a generalized conflict in a category C to simply be a non-commutative diagram in C.
Definition 1: Let C be a category. A generalized conflict in C is a non-commutative diagram in C.
For those with some knowledge in category theory, a necessary and sufficient condition for the following diagram below to commute would have been for q to be a morphism in the slice category C/P. In our examples from the previous hub, and the example above, the notion of having two distinct items being forced into a space too small to hold both of those items was essential for the conflicts of the form given in examples 1-3 of the following hub: http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict.
The diagram above encodes the notion of having two distinct items mapped into a single space, and the non-commutativity of the same diagram encodes the notion of the space being too small to fit those two distinct items.
This motivates the following definition of a strict S-Conflict in a category C.
Definition 2: Let C be a category, P and object of C, and consider the slice category C/P. A strict S-Conflict in C is a morphism f: x->x' in C such that f is not a morphism in C/P. That is, f is a morphism in C such that the following diagram fails to commute:
There are two cannonical solutions to this conflict corresponding to the two cannonical solutions listed in http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict: the two cannonical solutions listed in the link above are to remove the distinction between two distinguished items x and x', or create two distinct spaces S and S' and put each distinct item, x and x', into a distinct space large enough to hold them: i.e. put x in S, and x' into S', say.
The first solution (or rather a generalization of the first solution) is simply to find a morphism q':alive -> dead in the slice category of C/P. Such a morphism q', by the definition of a morphism in C/P, would make the diagram above commute thus resolving the conflict.
The second cannonical solution is to find an endofunctor G:C->C and a morphism q':alive -> dead in C such that q' is a morphism in the category Coalg(G) of coalgebras of G: the reasoning for this is that if C is a category, then a predicate on C may be thought of as a functor G:C->C and for a morphism q:x->G(x), and q':x'->G(x') (or in our case,
f:alive ->G(alive), g:dead->G(dead)) a morphism from from g:x->x' making q'og=Ggoq would resolve our conflict. Intuitively, the space is defined to be the functor G, and the existance of a G-coalgebra morphism from q:x->G(x) to q':x'->G(x') corresponds to saying the space G is large enough to hold both x and x', thereby, preventing a conflict.
http://conflicttheory.hubpages.com/hub/Beginnings-of-a-Mathematical-Theory-of-Conflict. A dual notion of a strict S-conflict, a strict S-Co-conflict, an interpretation of the meaning of a strict S-Co-conflict, and the interpretation of the two cannonical co-solutions to the conflict will all be elaborated on in a future hub.
For those who lack a background in category theory, the following series of short video lectures by theCatsters is an excellent resource to learning basic category theory quickly: https://www.scss.tcd.ie/Edsko.de.Vries/ct/catsters/linear.php.