Abstract Traditional category theory is typically based on set-theoretic principles and ideas, which are often nonconstructive. An alternative approach to formalizing category theory is to use e -category theory, where hom sets become setoids. Our work reconsiders a third approach – p -category theory – from Čubrić et al. ( Mathematical Structures in Computer Science 8(2) 153–192, 1998) emphasizing a computational standpoint. We formalize in Rocq a modest library of p -category theory – where homs become subsetoids – and apply it to formalizing algorithms for normalization by evaluation, which are purely categorical but, surprisingly, do not use neutral and normal terms. Čubrić et al. ( Mathematical Structures in Computer Science 8(2) 153–192, 1998) establish only a soundness correctness property by categorical means; here, we extend their work by providing a categorical proof also for a strong completeness property. For this, we formalize the full universal property of the free Cartesian-closed category, which is not known to have been performed before. We further formalize a novel universal property of unquotiented simply typed $\lambda$ -calculus syntax and apply this to a proof of correctness of a categorical normalization by evaluation algorithm. We pair the overall mathematical development with a formalization in the Rocq proof assistant, following the principle that the formalization exists for practical computation. Indeed, it permits extraction of synthesized normalization programs that compute (long) $\beta$ $\eta$ -normal forms of simply typed $\lambda$ -terms together with a derivation of $\beta$ $\eta$ -conversion.

We develop a unified categorical theory of substructural abstract syntax with variable binding and single-variable (capture-avoiding) substitution. This is done for the gamut of context structural rules given by exchange (linear theory) with weakening (affine theory) or with contraction (relevant theory) and with both (cartesian theory). Specifically, in all four scenarios, we uniformly: define abstract syntax with variable binding as free algebras for binding-signature endofunctors over variables; provide finitary algebraic axiomatisations of the laws of substitution; construct single-variable substitution operations by generalised structural recursion; and prove their correctness, establishing their universal abstract character as initial substitution algebras.

Abstract A categorical axiomatic theory of creation/annihilation operators on symmetric Fock space is introduced, and the combinatorial model that motivated it is presented. Commutation relations and coherent states are considered in both frameworks.

We introduce a bicategorical model of linear logic which is a novel variation of the bicategory of groupoids, profunctors, and natural transformations. Our model is obtained by endowing groupoids with additional structure, called a kit, to stabilize the profunctors by controlling the freeness of the groupoid action on profunctor elements. The theory of generalized species of structures, based on profunctors, is refined to a new theory of \emph{stable species} of structures between groupoids with Boolean kits. Generalized species are in correspondence with analytic functors between presheaf categories; in our refined model, stable species are shown to be in correspondence with restrictions of analytic functors, which we characterize as being stable, to full subcategories of stabilized presheaves. Our motivating example is the class of finitary polynomial functors between categories of indexed sets, also known as normal functors, that arises from kits enforcing free actions. We show that the bicategory of groupoids with Boolean kits, stable species, and natural transformations is cartesian closed. This makes essential use of the logical structure of Boolean kits and explains the well-known failure of cartesian closure for the bicategory of finitary polynomial functors between categories of set-indexed families and cartesian natural transformations. The paper additionally develops the model of classical linear logic underlying the cartesian closed structure and clarifies the connection to stable domain theory.

This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

Abstract 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.

Despite extensive research both on the theoretical and practical fronts, formalising, reasoning about, and implementing languages with variable binding is still a daunting endeavour – repetitive boilerplate and the overly complicated metatheory of capture-avoiding substitution often get in the way of progressing on to the actually interesting properties of a language. Existing developments offer some relief, however at the expense of inconvenient and error-prone term encodings and lack of formal foundations. We present a mathematically-inspired language-formalisation framework implemented in Agda. The system translates the description of a syntax signature with variable-binding operators into an intrinsically-encoded, inductive data type equipped with syntactic operations such as weakening and substitution, along with their correctness properties. The generated metatheory further incorporates metavariables and their associated operation of metasubstitution, which enables second-order equational/rewriting reasoning. The underlying mathematical foundation of the framework – initial algebra semantics – derives compositional interpretations of languages into their models satisfying the semantic substitution lemma by construction.

Abstract 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.

Abstract This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel’s size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem prover.

Abstract The glueing construction, defined as a certain comma category, is an important tool for reasoning about type theories, logics, and programming languages. Here we extend the construction to accommodate ‘2-dimensional theories’ of types, terms between types, and rewrites between terms. Taking bicategories as the semantic framework for such systems, we define the glueing bicategory and establish a bicategorical version of the well-known construction of cartesian closed structure on a glueing category. As an application, we show that free finite-product bicategories are fully complete relative to free cartesian closed bicategories, thereby establishing that the higher-order equational theory of rewriting in the simply-typed lambda calculus is a conservative extension of the algebraic equational theory of rewriting in the fragment with finite products only.

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed λ-calculi, the computational λ-calculus, and predicate logic.

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.

Abstract We investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, second-order and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument.

We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal property, thereby lifting the Curry-Howard-Lambek correspondence to the bicategorical setting. Our approach is principled and practical. Weak substitution structure is constructed using a bicategori-fication of the notion of abstract clone from universal algebra, and the rules for products and exponentials are synthesised from semantic considerations. The result is a type theory that employs a novel combination of 2-dimensional type theory and explicit substitution, and directly generalises the Simply-Typed Lambda Calculus. This work is the first step in a programme aimed at proving coherence for cartesian closed bicategories.

We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial we recover Levy’s notion of CBPV adjunction model, while when the effect structure is trivial we have Benton’s linear/non-linear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the [E]EC ([Enriched] Effect Calculus) models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones. To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. We explain possible advantages of this style of calculi for the operational semantics of effects.

We formalise and study the notion of polymorphic algebraic theory, as understood in the mathematical vernacular as a theory presented by equations between polymorphically-typed terms with both type and term variable binding. The prototypical example of a polymorphic algebraic theory is System F, but our framework applies more widely. The extra generality stems from a mathematical analysis that has led to a unified theory of polymorphic algebraic theories with the following ingredients: ; polymorphic signatures that specify arbitrary polymorphic operators (e.g. as in extended λ-calculi and algebraic effects); ; metavariables, both for types and terms, that enable the generic description of meta-theories; ; multiple type universes that allow a notion of translation between theories that is parametric over different type universes; ; polymorphic structures that provide a general notion of algebraic model (including the PL-category semantics of System F); ; a Polymorphic Equational Logic that constitutes a sound and complete logical framework for equational reasoning. Our work is semantically driven, being based on a hierarchical two-levelled algebraic modelling of abstract syntax with variable binding.

The paper develops a mathematical theory in the spirit of categorical algebra that provides a model theory for second-order and dependently-sorted syntax. The theory embodies notions such as alpha-equivalence, variable binding, capture-avoiding simultaneous substitution, term metavariable, meta-substitution, mono and multi sorting, and sort dependency. As a matter of illustration, a model is used to extract a second-order syntactic theory, which is thus guaranteed to be correct by construction.

We introduce a mathematical structural operational semantics that yields a congruence result for bisimilarity and is suitable for investigating rule formats for name-passing systems. Indeed, we instantiate this general abstract model theory in a framework of nominal sets and extract from it a GSOS-like rule format for name-passing process calculi for which the associated notion of behavioural equivalence - given by a form of open bisimilarity - is a congruence

We present a notion of η-long β-normal term for the typed lambda calculus with sums and prove, using Grothendieck logical relations, that every term is equivalent to one in normal form. Based on this development we give the first type-directed partial evaluator that constructs %able to construct normal forms of terms in this calculus.

This paper gives the first decidability results on type isomorphism for recursive types, establishing the explicit decidability of type isomorphism for the type theory of sums and products over an inhabited generic recursive polynomial type. The technical development provides connections between themes in programming-language theory (type isomorphism) and computational algebra (Gröbner bases).

Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculi with empty and sum types from this viewpoint. We close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit type) is not finitely axiomatisable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in some particular cases including that of type isomorphism with the empty type and equality with zero.

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 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 meta-theory.

We consider the problem of automatically verifing infinite-state cryptographic protocols. Specifically, we present an algorithm that given a finite process describing a protocol in a hostile environment (trying to force the system into a "bad" state) computes a model of traces on which security properties can be checked. Because of unbounded inputs from the environment, even finite processes have an infinite set of traces: the main focus of our approach is the reduction of this infinite set to a finite set by a symbolic analysis of the knowledge of the environment. Our algorithm is sound (and we conjecture complete) for protocols with shared-key encryption-decryption that use arbitrary messages as keys; further it is complete in the common and important case in which the cryptographic keys are messages of bounded size.

Provides a semantic framework for (first-order) message-passing process calculi by combining categorical theories of abstract syntax with binding and operational semantics. In particular, we obtain abstract rule formats for name and value passing with both late and early interpretations. These formats induce an initial-algebra/final-coalgebra semantics that is compositional, respects substitution and is fully abstract for late and early congruence. We exemplify the theory with the /spl pi/-calculus and value-passing CCS (calculus of communicating systems).

A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles, we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general notion of weak open-map bisimulation is introduced, and results relating various notions of strong and weak bisimulation are provided. The abstract theory is accompanied by fine concrete study of two key models for concurrency, the interleaving model of synchronisation trees and the independence model of labelled event structures.

We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with compatible algebra and substitution structures. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.

We develop a 2-categorical theory for recursively defined domains. In particular we generalise the traditional approach based on order-theoretic structures to category-theoretic ones. A motivation for this development is the need of a domain theory for concurrency, with an account of bisimulation. Indeed, the leading examples throughout the paper are provided by recursively defined presheaf models for concurrent process calculi. Further we use the framework to study (open-map) bisimulation.

This paper presents two basic results on game-based semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G, offering a fundamental structural analysis of the category as well as a transparent way to prove computational adequacy. As a consequence we obtain an intensional full-abstraction result through a standard definability argument. Next we extend the category G by introducing a category of games G/sub i/ with optimised strategies; we show that the denotational semantics in G/sub i/ gives a compilation of FPC terms into core Pict codes (the asynchronous polyadic /spl pi/-calculus without summation). The process representation follows a pioneering idea of Hyland and Ong (1995). However we advance their representation by introducing semantically well-founded optimisation techniques; we also extend the setting to encompass the rich type structure of FPC. The resulting code gives basic insight on the relationship between the abstract, categorical, types and their possible implementations.

We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, via, that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domain-theoretic structures.

Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. In this paper we prove an enrichment theorem showing that every axiomatic domain-theoretic category can be endowed with an intensional notion of approximation, the path relation, with respect to which the category Cpo-enriches. Our analysis suggests more liberal notions of domains. In particular, we present a category where the path order is not ω-complete, but in which the constructions of domain theory (such as, for example, the existence of uniform fixed-point operators and the solution of domain equations) are available.

We study recursive types from a syntactic perspective. In particular, we compare the formulations of recursive types that are used in programming languages and formal systems. Our main tool is a new syntactic explanation of type expressions as functors. We also introduce a simple logic for programs with recursive types in which we carry out our proofs.

This paper provides both a fully abstract (domain-theoretic) model for the /spl pi/-calculus and a universal (set-theoretic) model for the finite /spl pi/-calculus with respect to strong late bisimulation and congruence. This is done by: considering categorical models, defining a metalanguage for these models, and translating the /spl pi/-calculus into the metalanguage. A technical novelty of our approach is an abstract proof of full abstraction: The result on full abstraction for the finite /spl pi/-calculus in the set-theoretic model is axiomatically extended to the whole /spl pi/-calculus with respect to the domain-theoretic interpretation. In this proof, a central role is played by the description of non-determinism as a free construction and by the equational theory of the metalanguage.

Motivated by a desire to treat non-termination directly in the semantics of computation, the notion of approximation between programs is studied in the context of categories of partial maps. In particular, contextual approximation and specialisation are considered and shown to coincide. Moreover, after exhibiting the approximation between total maps as a primitive notion, from an arbitrary (or axiomatic) approximation order on total maps a computationally natural approximation order on partial maps is derived. The main technical contribution is a characterisation of when this approximation order between partial maps is domain-theoretic (in the sense that the category of partial mapsCpo-enriches) provided that the approximation order between total maps is also.

Categorical models of the metalanguage FPC (a type theory with sums, products, exponentials and recursive types) are defined. Then, domain-theoretic models of FPC are axiomatised and a wide subclass of them-the absolute ones-are proved to be both computationally sound and adequate. Examples include: the category of cpos and partial continuous functions and functor categories over it.>

The concept of bisimulation from concurrency theory is used to reason about recursively defined data types. From two strong-extensionality theorems stating that the equality (resp. inequality) relation is maximal among all bisimulations, a proof principle for the final coalgebra of an endofunctor on a category of data types (resp. domains) is obtained. As an application of the theory developed, an internal full abstraction result for the canonical model of the untyped call-by-value lambda -calculus is proved. The operations notion of bisimulation and the denotational notion of final semantics are related by means of conditions under which both coincide.>