By Cyrus F. Nourani

This ebook is an advent to a functorial version conception in line with infinitary language different types. the writer introduces the homes and origin of those different types prior to constructing a version conception for functors beginning with a countable fragment of an infinitary language. He additionally provides a brand new strategy for producing regular types with different types by way of inventing countless language different types and functorial version thought. furthermore, the booklet covers string versions, restrict versions, and functorial models.

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Additional info for A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos

Example text

6 Let M be a set and F a finite family of functions on M. We say that (F, M) is a monomorphic pair, provided for every f in F, f is injective, and the sets {Image(f):f in F} and M are pair-wise disjoint. This definition is basic in defining induction for abstract recursiontheoretic hierarchies and inductive definitions. We define generalized standard models with monomorphic pairs. 7 A standard model M, of base M and functionality F, is a structure inductively defined by provided the forms a monomorphic pair.

Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and ZY is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the curried function g : X → ZY defined by g(x)(y) = f(x, y) for all x in X and y in Y. The category of finite sets, with functions as morphisms, is cartesian closed for the same reason.

Conditions 5, 6 and 7 are and conditions. Conditions 8, 9 and 10 are or conditions. Condition 11 is a false condition. Of course, if a different set of axioms were chosen for logic, we could modify ours accordingly. The free Heyting algebra over one generator (aka Rieger–Nishimura lattice) Every Boolean algebra is a Heyting algebra, with p  q given by ^ p ^ q. 38 A Functorial Model Theory Every totally ordered set that is a bounded lattice is also a Heyting algebra, where p  q is equal to q when p > q, and 1 otherwise.