Categories with Families: Unityped, Simply Typed, and Dependently Typed

Freyd categories are Enriched Lawvere Theories

This chapter provides an introduction to the interaction between category theory and mathematical logic. Category theory describes properties of mathematical structures via their transformations, or ‘morphisms’. On the other hand, mathematical logic provides languages for formalizing properties of structures directly in terms of their constituent parts—elements of sets, functions between sets, relations on sets, and so on. It might seem that the kind of properties that can be described purely in terms of morphisms and their composition would be quite limited. However, beginning with the attempt of Lawvere [1964; 1966; 1969; 1970] to reformulate the foundations of mathematics using the language of category theory, the development of categorical logic over the last three decades has shown that this is far from true. Indeed it turns out that many logical constructs can be characterized in terms of relatively few categorical ones, principal among which is the concept of adjoint functor. In this chapter we will see such categorical characterizations for, amongst other things, the notions of variable, substitution, prepositional connectives and quantifiers, equality, and various type-theoretic constructs. We assume that the reader is familiar with some of the basic notions of category theory, such as functor, natural transformation, (co)limit, and adjunction: see Poigné’s [1992] chapter on Basic Category Theory in Vol. I of this handbook, or any of the several available introductions to category theory slanted towards computer science, such as [Barr and Wells, 1990] and [Pierce, 1991]. There are three recurrent themes in the material we present. Categorical semantics. Many systems of logic can only be modelled in a sufficiently complete way by going beyond the usual set-based structures of classical model theory. Categorical logic introduces the idea of a structure valued in a category C, with the classical model-theoretic notion of structure [Chang and Keisler, 1973] appearing as the special case when C is the category of sets and functions. For a particular logical concept, one seeks to identify what properties (or extra structure) are needed in an arbitrary category to interpret the concept in a way that respects given logical axioms and rules. A well-known example is the interpretation of simply typed lambda calculus in cartesian closed categories.

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones.

Cartesian double theories: A double-categorical framework for categorical doctrines

We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski-Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a nonlogical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first-order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. In the seventh part, we present an enigma: is variety formalizable in first-order logic without equality? There follow recollections concerning Jan Woleński. This paper is dedicated to his 80th birthday. We end with the bibliography, giving some precise references for those wanting to know more about the topic.

Cartesian closed Dialectica categories

My paper examines some of the many similarities between Mahayana Buddhism and Deweyan philosophy. It builds upon two previously published works. The first is my dialogue with Daisaku Ikeda President of Soka Gakkai International, a UN registered ngo currently active in one hundred ninety-two countries and territories, and the Director Emeritus of the Center for Dewey Studies, Larry Hickman (see Garrison, Hickman, and Ikdea, 2014). My paper will first briefly review some of the many similarities between Buddhism and Deweyan pragmatism. Second, I will also briefly review additional similarities in the published version of my Kneller Lecture to the American Educational Studies Association (see Garrison, 2019). In the present paper, I will introduce some new similarities of interest to educators. Among these are Dewey’s surprisingly Buddhist notions of language and logic as merely useful conventions. Secondly, I examine Dewey’s argument that “causation as ordered sequence is a logical category,” not an ontological category (LW 12: 454). The similarity to the opening chapter of Nagarjuna’s Madhyamaka, or Middle Way, is striking. I will suggest a logical reading has some interesting implications for student-teacher relations.

We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories with families with extra structure corresponding to Martin-Lof type theory with an external tower of universes. We then present a generalized algebraic theory for level-indexed categories with families with extra structure corresponding to Martin-Lof type theory with explicit universe polymorphism: a theory with universe level judgments, internally indexed universes, and level-indexed products. In this way we get abstract characterizations of the two theories as initial models of their respective generalized algebraic theories. We thus abstract from details of the grammar and inference rules of the type theories and highlight their high-level structure. More broadly, the present work can be viewed as a case study of a uniform approach to categorical logic based on generalized algebraic theories and categories with families. We also discuss the relevance to Voevodsky's initiality conjecture project.

Introduction to extensive and distributive categories

We present in this talk an abstract categorical logic based on an abstraction of quantifier. More precisely, the proposed logic is abstract because no structural constraints are imposed on models (semantics free). By contrast, formulas are inductively defined from an abstraction both of atomic formulas and of quantifiers. In this sense, the proposed approach differs from other works interested in formalizing the notion of abstract logic and of which the closest to our approach are the institutions, which in addition to be semantics free do not also impose any syntactic contingencies on the structure of formulas. To define the semantical framework in which formulas will be interpreted, we propose to follow the idea from categorical logic which defines the semantical interpretation of formulas from context and as subobjects of an object of a given category. In the spirit of Lawvere’s hyperdoctrines, we use a more abstract notion which generalizes the notion of subobject, standard in category theory: Pitt’s prop-categories. Always in the spirit of categorical logic, we propose a sequent calculus of which we show correctness and completeness for all semantical frameworks defined over any prop-categories. We then study some conditions which allow us to get this completeness result for particular classes of prop-categories.

Introduction to categorical logic, classifying toposes and the 'bridge' technique The course will begin by presenting the basic notions and results of first-order categorical logic, with the aim of reaching the theory of classifying toposes by Makkai and Reyes and illustrating the general techniques allowing to use them as unifying 'bridges' for transferring information across distinct mathematical theories. The exposition will be accompanied by several examples and applications. The lectures will require a basic familiarity with the fundamental notions of topos theory, as reviewed in Andre Joyal's lectures on Monday. Lecture 1: First-order logic and its interpretation in categories. Geometric theories and syntactic categories. Universal models and representability.

In 1971, Stenstrom published one of the first papers devoted to the problem of when, for a monoid S and a right S -act A S , the functor A⊗ (from the category of left acts over S into the category of sets) has certain limit preservation properties. Attention at first focused on when this functor preserves pullbacks and equalizers but, since that time, a large number of related articles have appeared, most having to do with when this functor preserves monomorphisms of various kinds. All of these properties are often referred to as flatness properties of acts . Surprisingly, little attention has so far been paid to the obvious questions of when A S ⊗ preserves all limits, all finite limits, all products, or all finite products. The present article addresses these matters.

In this paper we give constructions of pullback, finite product, finite limit, coproduct, colimit, pushout, etc. in a special full subcategory X M o d / L role=presentation> 𝔛 𝔐 𝔬 𝔡 /  X M o d / L \mathfrak{XMod}/\mathcal{L} of the category of Lie-Rinehart crossed modules.

Enriched Lawvere theories are a generalization of Lawvere theories that allow\nus to describe the operational semantics of formal systems. For example, a\ngraph enriched Lawvere theory describes structures that have a graph of\noperations of each arity, where the vertices are operations and the edges are\nrewrites between operations. Enriched theories can be used to equip systems\nwith operational semantics, and maps between enriching categories can serve to\ntranslate between different forms of operational and denotational semantics.\nThe Grothendieck construction lets us study all models of all enriched theories\nin all contexts in a single category. We illustrate these ideas with the\nSKI-combinator calculus, a variable-free version of the lambda calculus.\n

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗-autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus.