Construction of Categories for Mathematical Conflict Theory II
Construction of Predicate(L)
In the previous hub, http://conflicttheory.hubpages.com/hub/Construction-of-Two-Categories-for-Mathematical-Conflict-Theory, we constructed the category Pred(L) to be used in the construction of the category Predicate(L) below.
The objects of Predicate(L) are to be the same as the objects of Pred(L): see http://conflicttheory.hubpages.com/hub/Construction-of-Two-Categories-for-Mathematical-Conflict-Theory for the definition, and description, of the objects of Pred(L).
Let A and B be non-empty predicates. Then a morphism f:A → B is a function f that sends each word predicate A' of arity 0 in A to one, and only one, nonempty word predicate B' in B, or to one, and only one, blank in B, and each blank in A to either, one, and only one, nonempty word predicate B' of arity zero in B, or one, and only one, blank _____ in B. If a nonempty word predicate A' of arity zero (respectively a blank _____) in A is sent to a nonempty word predicate B' of arity 0 (respectively a blank _____) in B, then the mapping is via a morphism f:A'→B' (respectively, g:____→B', h:A'→____, k:____→____) in Pred(L).
If P is a predicate, the identity morphism is the identity predicate mapping each nonempty word predicate/blank to itself via the respective identity morphism in
Morphisms from the Empty Predicate:
Let Q be a predicate in Predicate(L), if Q is the empty predicate, and C is any other predicate in Predicate(L), then we allow only the inclusion morphism (via morphisms in Pred(L) just as above) including the empty predicate in every predicate. This defines the empty predicate as an initial object of Predicate(L).
Composition of morphism in Predicate(L) are via composition of component morphisms in
Associativity of Composition and Identity axioms of a Category:
Associativity of composition of morphisms in Predicate(L) is inherited from associativity of composition in Pred(L), and the identity axiom in Predicate(L) is satisfied via satisfaction of the identity axiom in Pred(L).
Thus Predicate(L) is a category.