Inspired by Plotkin and Power’s algebraic treatment of computational effects and the principle of notions of computations as monoids, we propose a categorical framework for equational theories and models of monoids equipped with operations. This framework generalises Plotkin and Power’s algebraic treatment of effectful operations taking or returning values as input or output to operations that may take or return computations as input or output. Additionally, to give semantic models of computational effects in a modular way, we introduce a formal theory of modular constructions of algebraic structures based on the framework of lifting functors along fibrations.

Inspired by algebraic effects and the principle of notions of computations as monoids, we study a categorical framework for equational theories and models of monoids equipped with operations. The framework covers not only algebraic operations but also scoped and variable-binding operations. Appealingly, in this framework both theories and models can be modularly composed. Technically, a general monoid-theory correspondence is shown, saying that the category of theories of algebraic operations is equivalent to the category of monoids. Moreover, more complex forms of operations can be coreflected into algebraic operations, in a way that preserves initial algebras. On models, we introduce modular models of a theory, which can interpret abstract syntax in the presence of other operations. We show constructions of modular models (i) from monoid transformers, (ii) from free algebras, (iii) by composition, and (iv) in symmetric monoidal categories.

This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is refined further by considering intensional Kripke relations (in the form of Artin–Wraith glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently typed functional programming language is synthesised.

Algebraic effects offer a versatile framework that covers a wide variety of effects. However, the family of operations that delimit scopes are not algebraic and are usually modelled as handlers, thus preventing them from being used freely in conjunction with algebraic operations. Although proposals for scoped operations exist, they are either ad-hoc and unprincipled, or too inconvenient for practical programming. This paper provides the best of both worlds: a theoretically-founded model of scoped effects that is convenient for implementation and reasoning. Our new model is based on an adjunction between a locally finitely presentable category and a category of functorial algebras . Using comparison functors between adjunctions, we show that our new model, an existing indexed model, and a third approach that simulates scoped operations in terms of algebraic ones have equal expressivity for handling scoped operations. We consider our new model to be the sweet spot between ease of implementation and structuredness. Additionally, our approach automatically induces fusion laws of handlers of scoped effects, which are useful for reasoning and optimisation.

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

The implementation and semantics of dependent type theories can be studied in a syntax-independent way: the objective metatheory of dependent type theories exploits the universal properties of their syntactic categories to endow them with computational content, mathematical meaning, and practical implementation (normalization, type checking, elaboration). The semantic methods of the objective metatheory inform the design and implementation of correct-by-construction elaboration algorithms, promising a principled interface between real proof assistants and ideal mathematics. In this dissertation, I add synthetic Tait computability to the arsenal of the objective metatheorist. Synthetic Tait computability is a mathematical machine to reduce difficult problems of type theory and programming languages to trivial theorems of topos theory. First employed by Sterling and Harper to reconstruct the theory of program modules and their phase separated parametricity, synthetic Tait computability is deployed here to resolve the last major open question in the syntactic metatheory of cubical type theory: normalization of open terms.

We present two proofs of coherence for cartesian closed bicategories. Precisely, we show that in the free cartesian closed bicategory on a set of objects there is at most one structural 2-cell between any parallel pair of 1-cells. We thereby reduce the difficulty of constructing structure in arbitrary cartesian closed bicategories to the level of 1-dimensional category theory. Our first proof follows a traditional approach using the Yoneda lemma. For the second proof, we adapt Fiore’s categorical analysis of normalisation-by-evaluation for the simply-typed lambda calculus. Modulo the construction of suitable bicategorical structures, the argument is not significantly more complex than its 1-categorical counterpart. It also opens the way for further proofs of coherence using (adaptations of) tools from categorical semantics.

We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the “gradual guarantee” theorem of gradual typing. Combined with the ordinary extensionality ($𝜂$) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of Findler and Felleisen’s definitions of contracts, and use it to build some concrete domain-theoretic models of gradual typing.

We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism. These dynamism judgements build on prior work in blame calculi and on the “gradual guarantee” theorem of gradual typing. Combined with the ordinary extensionality (eta) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing.

The standard reading of type theory through the lens of category theory is based on the idea of viewing a type system as a category of well-typed terms. We propose a basic revision of this reading: rather than interpreting type systems as categories, we describe them as functors from a category of typing derivations to a category of underlying terms. Then, turning this around, we explain how in fact any functor gives rise to a generalized type system, with an abstract notion of typing judgment, typing derivations and typing rules. This leads to a purely categorical reformulation of various natural classes of type systems as natural classes of functors.

The main purpose of this paper is to describe the general framework (which can also be seen as providing a categorical analysis of refinement types), and to present a few applications. As a larger case study, we revisit Reynolds’ paper on “The Meaning of Types” (2000), showing how the paper’s main results may be reconstructed along these lines.

I chart a line of development from category-theoretic models of programs and logics to automatic program verification/analysis techniques that are in deployment at Facebook. Our journey takes in a number of concepts from the computer science logician’s toolkit – including categorical logic and model theory, denotational semantics, the Curry-Howard isomorphism, sub structural logic, Hoare Logic and Separation Logic, abstract interpretation, compositional program analysis, the frame problem, and abductive inference.

The concept of refinement in type theory is a way of reconciling the “intrinsic” and the “extrinsic” meanings of types. We begin with a rigorous analysis of this concept, settling on the simple conclusion that the type-theoretic notion of “type refinement system” may be identified with the category-theoretic notion of “functor”. We then use this correspondence to give an equivalent type-theoretic formulation of Grothendieck’s definition of (bi)fibration, and extend this to a definition of monoidal closed bifibrations, which we see as a natural space in which to study the properties of proofs and programs. Our main result is a representation theorem for strong monads on a monoidal closed fibration, describing sufficient conditions for a monad to be isomorphic to a continuations monad “up to pullback”.

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI’s proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers [inline image] and [inline image] which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

We show how to solve the word problem for simply typed λβη-calculus by using a few well-known facts about categories of presheaves and the Yoneda embedding. The formal setting for these results is 𝒫-category theory, a version of ordinary category theory where each hom-set is equipped with a partial equivalence relation. The part of 𝒫-category theory we develop here is constructive and thus permits extraction of programs from proofs. It is important to stress that in our method we make no use of traditional proof-theoretic or rewriting techniques. To show the robustness of our method, we give an extended treatment for more general λ-theories in the Appendix.

Having been under the impression that categorial grammars in general and the so-called syntactic calculus in particular had been swept away by the tide of transformational grammar, I was very surprised to learn of the recent revival of interest in these matters, as, for example, by Buszkowski in Poland and by van Benthem in the Netherlands. Stimulated by the renewed activity in this area, I decided to take another look at it, and in particular, to explore the categorical connection, which had been at the back of my mind all along.

Solves the decision problem for the simply typed lambda calculus with a strong binary sum, or, equivalently, the word problem for free Cartesian closed categories with binary co-products. Our method is based on the semantic technique known as “normalization by evaluation”, and involves inverting the interpretation of the syntax in a suitable sheaf model and, from this, extracting an appropriate unique normal form. There is no rewriting theory involved and the proof is completely constructive, allowing program extraction from the proof.