Michael Makkai

On weak higher dimensional categories I: Part 1

Inspired by the concept of opetopic set introduced in a recent paper by John C. Baez and James Dolan, we give a modified notion called multitopic set. The name reflects the fact that, whereas the Baez/Dolan concept is based on operads, the one in this paper is based on multicategories. The concept of multicategory used here is a mild generalization of the same-named notion introduced by Joachim Lambek in 1969. Opetopic sets and multitopic sets are both intended as vehicles for concepts of weak higher dimensional category. Baez and Dolan define weak n-categories as (n+1)-dimensional opetopic sets satisfying certain properties. The version intended here, multitopic n-category, is similarly related to multitopic sets. Multitopic n-categories are not described in the present paper; they are to follow in a sequel. The present paper gives complete details of the definitions and basic properties of the concepts involved with multitopic sets. The category of multitopes, analogs of opetopes of Baez and Dolan, is presented in full, and it is shown that the category of multitopic sets is equivalent to the category of set-valued functors on the category of multitopes.

Admissible Sets and Infinitary Logic

Publisher Summary The relationship of admissible sets to logic can be summarized as the phenomenon that can be called the syntactic completeness of admissible sets. This chapter discusses admissible sets, the model theory of L ω1ω , classical descriptive set theory, effective descriptive set theory, and recursion theory. The main theme is the model theory of admissible fragments of L ω1ω . Further, the chapter features two recent devices: Vaughts use of conjunctive game sentences and Ressayres ∑-saturated structures. The chapter presents examples of admissible sets and also discusses the Kripke–Platek axiom system Hintikka sets, model existence and Z-compactness, Conjunctive game formulas, and Z-saturated structures.

Completeness results for intuitionistic and modal logic in a categorical setting

Abstract Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcans formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere (1992). The completeness theorems are used to conclude results asserting that certain logics are conservatively interpretable in others.

Stone duality for first order logic

Publisher Summary This chapter presents a general theory on the relationship of syntax and semantics of first-order logic. The theory has a close-formal relationship to the Stone-duality theory for Boolean algebras. It subsumes the Godel completeness theorem that occupies a place in it and is analogous to that of the Stone representation inside Stone duality. The theory makes an essential use of ultraproducts. The theory is formulated in the language of category theory. Categories appear in three ways: (1) (first order) theories themselves are made into categories (pretoposes), (2) the collection of models of a fixed theory is made into a category and endowed with an additional structure derived from ultraproducts, resulting in the “ultracategory” of models, and (3) pretoposes on the one hand, and ultracategories on the other, are organized into categories (actually: two categories), and the main result in its final form is stated, in terms of a comparison between these two categories.

Duality and definability in first order logic

Beths theorem in propositional logic Factorizations in

A theorem on barr-exact categories, with an infinitary generalization

2

Applications of Vaught Sentences and the Covering Theorem

-categories Definable functors Basic notions for duality The Stone-type adjunction for Boolean pretoposes and ultragroupoids The syntax of special ultramorphisms The semantics of special ultramorphisms The duality theorem Preparing a functor specification Lifting Zawadowskis argument to ultra

An accessible approach to behavioural pseudometrics

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Lambek's Categorical Proof Theory and Lauchli's Abstract Realizability

morphisms The operations on

A tree argument in infinitary model theory

{\mathcal B} {\mathcal P}^\ast