A Functorial Model Theory: Newer Applications to Algebraic by Cyrus F. Nourani

By Cyrus F. Nourani

This booklet is an creation to a functorial version conception according to infinitary language different types. the writer introduces the homes and origin of those different types prior to constructing a version idea for functors beginning with a countable fragment of an infinitary language. He additionally provides a brand new method for producing wide-spread versions with different types by way of inventing countless language different types and functorial version conception. furthermore, the booklet covers string versions, restrict versions, and functorial models.

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

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If G is a group, then the category of all G-sets is cartesian closed. y)) for all g in G, F:Y → Z and y in Y. The category of finite G-sets is also cartesian closed. The category Cat of all small categories (with functors as morphisms) is cartesian closed; the exponential CD is given by the functor category consisting of all functors from D to C, with natural transformations as morphisms. Categorical Preliminaries 45 If C is a small category, then the functor category Set C consisting of all covariant functors from C into the category of sets, with natural transformations as morphisms, is cartesian closed.

The purpose of this article is to give a survey of the classical results on ultraproducts of first order structures in order to provide some background for the papers in this volume. Over the years, many generalizations of the ultraproduct construction, as well as applications of ultraproducts to nonfirst order structures, have appeared in the literature. To keep this paper of reasonable length, we will not include such generalizations in this survey. For earlier surveys of ultraproducts see Kiseler (1967).

Given a natural transformation Φ: Hom(A, –) → F, the corresponding element u  F(A) is given by u = F A (id A). Conversely, given any element u  F(A) we may define a natural transformation Φ: Hom(A, –) → F via ΦX (f) = (Ff) (u). 28 A Functorial Model Theory where f is an element of Hom(A, X). In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism. This leads to the following definition: The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.

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