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Construction of Categories for Mathematical Conflict Theory I

Updated on June 15, 2012

Category Pred(L) Construction

In the previous hub on conflict theory development (see, we defined the the notion of a conflict and strict S-conflict for an arbitrary category. We mentioned how the two cannonical conflict resolutions we have been discussing can be formalized as finding a morphism in a slice category C/P for some category C and an object P in C, or via finding a morphism in the category Coalg(G) of an endo-functor from a category C' to itself.

In the next two hubs, we shall construct two categories Pred(L) and Predicate(L) (Pred(L) will be used to define Predicate(L) which we shall be working) and explore the notion of conflicts within the category Predicate(L). I must admit, this is a bit of a rabbit trail, and I am not sure what this will lead to, if anything.

In this hub, we shall construct a category, which we shall name Pred(L), whose objects are predicates over the language L (in this case, the language L the set of words over an alphabet ∑={S,___}, where S is a nonempty set of symbols), and whose morphisms are morphisms of predicates over L.

To begin our construction of Predicate(L), we shall need the following definition:

Definition 1.

Let L be a language. A predicate over L is a sequence of words, and zero or more blank spaces, strung together into a string. We do not require the predicate to form a fragment of a valid statement in the given language.

Example, "the cat is ____." is a predicate over the English language, and "cat the dog blue sand" is a also a predicate over the English language. The reason for not requiring predicates to necessarily form syntactically or symmantically valid expressions is that we hope to construct our categories in such a way that they will form toposes: in particular, we will wish to explore (in a later hub) whether taking the set of all sub-predicates of a given predicate S forms a power object in the category Predicate(L): such an operation forms a predicate that is not necessarily a syntactically valid statement in the language L, and such an operation is not necessarily a symmantically valid statement in the language L.

The word "cat" is also a predicate over the English language.

Definition 2. Given a predicate P over the language L, define the arity of P to be the number of blank spaces contained in the predicate P.

Example, the predicate "the cat is ____." has arity 1 because it has one blank. The predicate "cat" has arity zero because it has no blanks. The predicate "___ ____ ____ ____." has arity four because it has four blanks.

Notation and Axioms for Predicates.

If D is a predicate of arity ≥ 1, and C is a predicate, denote the fact that C has been inserted into D via the notation D(C) by some operation. If D is a predicate and the predicate (or subpredicate) sits inside D, then denote this fact by the notation D[C]. Note that we do not require D to have arity ≥ 1 in order to have another predicate sit inside D. For example, the predicate D="The mouse likes cheese." has arity 0, since it has no blanks, but the predicate "mouse" sits inside the predicate "The mouse likes cheese."=D.

Definition of the Empty Predicate:

The empty predicate is defined to be the predicate with no symbols, and no blanks: "".

Before constructing Predicate(L), we shall construct an intermediate category
Pred(L) which will serve as a basis in which to build the category in which we desire to work: Predicate(L).

Construction of Pred(L)

Objects: predicates over L (including the empty predicate "" of arity zero).

The predicate consisting of one blank is not considered to be empty. So the only empty predicate is, in fact, the empty predicate, which by definition, must have arity zero.

Morphisms: Let P and S be predicates, then define a morphism from P to S as follows:

A morphism f from P to S is specified by sending P to S, as in f(P)=S.

The Identity Morphism: given a predicate S in Ob(Pred(L)), define the identity morphism to be i(S)=S, the morphism that maps S to itself.

Composition: Let g:R→S, and f:S→T be morphisms in Pred(L). Then composition of f and g is defined as follows:


Associativity of Composition of morphisms in Pred(L)

Let h:Q-->R, g:R-->S, and f:S-->T be morphisms in Pred(L).



This establishes associativity of composition.

Identity Operation:

Let f:S-->T be a morphism in Pred(L), IdT : T→T be the identity morphism on T, and IdS : S→S be the identity morphism on S. Then

IdT o f=IdT (f(S))=IdT (T)=T=f(S)=f(IdS (S))=f o IdS.

Thus the identity axiom hold, and we have shown that Pred(L) satisfies the axioms of a category. Thus Pred(L) is a category.

In the next hub,, we shall define the category Predicate(L).


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