Abstract. Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero self-distance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Non-expansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable point-based reasoning for locales. 1.

Abstract not found

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.

The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current article is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as n-fold loop spaces and pointed G-spaces. This language also includes the diagram categories of spectra from [19] as well as spectra in the sense of [17], which will be discussed in some detail. 1.

Let C , D and E be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right q-transfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, does a functor C\Omega D ! E induce a functor D\Omega C ! E? For c; c 0 elements of C whose (k \Gamma 1)-sources and (k \Gamma 1)-targets agree, does a q-transfor C ! D induce a q-transfor C (c;c 0 ) ! D (d;d 0 ), for appropriate d;d 0 2 D ? For c; c 0 2 C and d;d 0 2 D whose (k \Gamma 1)-sources and (k \Gamma 1)-targets agree, does a q-transfor C\Omega D ! E induce a (q+k+1)-transfor C (c;c 0 )\Omega D (d;d 0 ) ! E(e;e 0 ), for appropriate e; e 0 2 E? I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and in some cases for n up to 5 which do not need all data and axioms...

in noncommutative geometry

We give various internal descriptions of the category !-Cpo of !-complete posets and !-continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domain-like sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...

Plotkin's dual characterization of strongly algebraic domains -- by sets of minimal upper bounds and by sequences of finite posets -- is stated and proved in the topical setting. 1.

The simply typed -calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed -theory T, T ` t 1 = t 2 i for all T-models [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. In this thesis, I will describe explicitly how this more powerful completeness result follows from a result in [2]. As models of the form Sets for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for non-category theorists to understand. We hope that the simpler semantics result in new applications of the simply typed -calculus. We also describe how this gives a complete semantics of the simply typed -calculus in a certain category of posets.