Michael Makkai

• McGill University

This document is a re-writing of the book of Makkai-Reyes with the same title. It is due to Francisco Marmolejo, who not only re-wrote it but made several corrections.

In this addendum, we correct some typos and fill a gap in the proof of Theorem 21 of [F. van Breugel, C. Hermida, M. Makkai, J. Worrell. Recursively defined metric spaces without contraction. Theoretical Computer Science 380 (1/2) (2007) 143–163]. We reprove Theorem 21 and fill the gap by Lemmas 2–4 of this paper.

We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition rectifies the situation. We give the modified definition, using non-reflexive rather than reflexive globular sets,...

We compare computads with multitopic sets. Both these kinds of structures have n-dimensional objects (called n-cells and n-pasting diagrams, respectively). The computads form a subclass of the more familiar class of omega-categories, while multitopic sets have been devised by Hermida, Makkai and Power as a vehicle for a definition of the concepts o...

In this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1-bounded complete metric spaces and nonexpansive functions. In contrast to previous approaches, we do not assume that the endofunctors are locally contractive, and our results do not depend on Banach’s fixed-point theorem.Our approach i...

Behavioural pseudometrics are a quantitative analogue of behavioural equivalences. They provide robust models for those concurrent systems in which quantitative data plays a crucial role. In this paper, we show how behavioural pseudometrics can be defined coalgebraically. Our results rely on the theory of accessible categories. We apply our results...

Behavioural pseudometrics are a quantitative analogue of behavioural equivalences. They provide robust models for those concurrent systems in which quantitative data plays a crucial role. In this paper, we show how behavioural pseudometrics can be defined coalgebraically. Our results rely on the theory of accessible categories. We apply our results...

The present third part of a three-part paper gives the definition of a multitopic set, and that of the category of multitopes. A detailed proof is given of the fact that the category of multitopic sets is equivalent to the category of set-valued functors on multitopes.

The present second part of a three-part paper gives the detailed treatment of the new notion of multicategory, and that of the construction of the particular multicategory of function replacement. For the overall purpose of the whole paper, see the abstract in 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...

We introduce the notion of higher dimensional multigraph. This notion extends that of multigraph, which underlies multicategories and is essentially equivalent to the notion of context-free grammar. We develop the definition and explain how it gives a semantically coherent category theoretic approach to the notion of higher order context-free gramm...

We consider the notion of replete object in the category of directed complete partial orders and Scott-continuous functions, and show that, contrary to previous expec- tations, there are non-replete objects. The same happens in the case of !-complete

The concept of sketch is generalized. Morphisms of finite (generalized) sketches are used as sketch-entailments. A semantics and a deductive calculus of sketch-entailments are developed. A General Completeness Theorem (GCT) shows that the deductive calculus is adequate for the semantics. In each of a number of categories of sketches, a particular s...

The notion of anafunctor is introduced. An anafunctor is, roughly, a “functor defined up to isomorphism”. Anafunctors have a general theory paralleling that of ordinary functors; they have natural transformations, they form categories, they can be composed, etc. Anafunctors can be saturated, to ensure that any object isomorphic to a possible value...

Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's 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...

Let C be a small Barr-exact category, Reg(C, Set) the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e:C→[Reg(C, Set), Set] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and f...

On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(≥κ 1 )→I(T,K) avec chaque possibilite se conformant a la condition de Morley

In this paper it is proved that ifT is a countable completeω-stable theory in ordinary logic, thenT has either continuum many, or at most countably many, non-isomorphic countable models.

A selfcontained exposition is given of a part of stability theory in model theory, the part that deals with the concepts of orthogonality, weight and regularity. The necessary background from earlier parts of stability theory is explained but proofs in this part are given in outline only or not at all.

We give a complete axiomatization for admissible fragments ofL {ie257-1}(Q). This axiomatization implies syntactically Gregory’s characterization ofL {ie257-2} sentences with no uncountable models ([5]). This is then extended to stationary logic. To obtain these results, we employ Ressayre’s methods ([16], [17]) augmented with an application of gam...

We investigate the logic L(αα) which allows the second-order quantifier “ααs’ meaning “for almost all countable sets s.” We prove Completeness, Compactness, and Omitting Types Theorems and develop a Gentzen-style proof theory for this logic, as well as for the infinitary version LA(αα). Relations with various sublogics like L(Q) are discussed.

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 Lω1ω, classical descriptive set theory, effective descriptive set theory, and recursion theory. The main theme is the model theory of admissibl...

Without Abstract

A tree argument is used to show that any counterexample to Vaught's conjecture must have an uncountable model. A similar argument replaces the use of forcing by Burgess in a theorem on Σ11 equivalence relations.

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem $0.1^\ast$ below) to derive several known and a few new results related to the logic $L_{\omega_1\omega}$. Among others, we prove that if every countable model in a $PC_{\omega_1\omega}$ class has only co...