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Marcelo Fiore

Publications and source records attributed to Marcelo Fiore.

At least 19 recordsLinked to original sources
arXiv2026

Symmetric 2-rigs: coexponentiability and cartesian closure

We study coexponentiability in the context of the cocartesian 2-category RIG of symmetric 2-rigs, symmetric strong monoidal cocontinuous functors, and symmetric monoidal natural transformations. Our results characterize the coexponentiable symmetric 2-rigs as those that are deformation retracts of presheaf categories over small categories. As an application, we give an account of the cartesian closure of two full sub-2-categories of the dual of RIG arising from the theory of combinatorial species and the theory of symmetric operads.

math.CT↗
arXiv2025

Substructural Abstract Syntax with Variable Binding and Single-Variable Substitution

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.

cs.LO↗
arXiv2024

Logical Structure on Inverse Functor Categories

Inspired by recent work on the categorical semantics of dependent type theories, we investigate the following question: When is logical structure (crucially, dependent-product and subobject-classifier structure) induced from a category to categories of diagrams in it? Our work offers several answers, providing a variety of conditions on both the category itself and the indexing category of diagrams. Additionally, motivated by homotopical considerations, we investigate the case when the indexing category is equipped with a class of weak equivalences and study conditions under which the localization map induces a structure-preserving functor between presheaf categories.

math.CT↗
arXiv2023

Fixpoint constructions in focused orthogonality models of linear logic

Orthogonality is a notion based on the duality between programs and their environments used to determine when they can be safely combined. For instance, it is a powerful tool to establish termination properties in classical formal systems. It was given a general treatment with the concept of orthogonality category, of which numerous models of linear logic are instances, by Hyland and Schalk. This paper considers the subclass of focused orthogonalities. We develop a theory of fixpoint constructions in focused orthogonality categories. Central results are lifting theorems for initial algebras and final coalgebras. These crucially hinge on the insight that focused orthogonality categories are relational fibrations. The theory provides an axiomatic categorical framework for models of linear logic with least and greatest fixpoints of types. We further investigate domain-theoretic settings, showing how to lift bifree algebras, used to solve mixed-variance recursive type equations, to focused orthogonality categories.

cs.LO↗
arXiv2023

Stabilized profunctors and stable species of structures

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.

cs.LO↗
arXiv2022

Semantic Analysis of Normalisation by Evaluation for Typed Lambda Calculus

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 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.

cs.LO↗
arXiv2022

Formal Metatheory of Second-Order Abstract Syntax

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.

cs.LO↗
arXiv2021

Free Commutative Monoids in Homotopy Type Theory

We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space.

cs.LO↗
arXiv2020

Algebraic models of simple type theories: a polynomial approach

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 $\lambda$-calculi, the computational $\lambda$-calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed $\lambda$-calculus and cartesian-closed categories, to arbitrary simple type theories.

cs.LO↗
arXiv2019

Constructing Infinitary Quotient-Inductive Types

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.

cs.LO↗
arXiv2019

A type theory for cartesian closed bicategories

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 bicategorification 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.

cs.LO↗
arXiv2016

Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures

We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.

math.CT↗
arXiv2014

Functorial Semantics of Second-Order Algebraic Theories

The purpose of this work is to complete the algebraic foundations of second-order languages from the viewpoint of categorical algebra as developed by Lawvere. To this end, this paper introduces the notion of second-order algebraic theory and develops its basic theory. A crucial role in the definition is played by the second-order theory of equality $\M$, representing the most elementary operators and equations present in every second-order language. The category $\M$ can be described abstractly via the universal property of being the free cartesian category on an exponentiable object. Thereby, in the tradition of categorical algebra, a second-order algebraic theory consists of a cartesian category $\Mlaw$ and a strict cartesian identity-on-objects functor $\M \to \Mlaw$ that preserves the universal exponentiable object of $\Mlaw$. Lawvere's functorial semantics for algebraic theories can then be generalised to the second-order setting. To verify the correctness of our theory, two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models.

math.CT↗
arXiv2013

An Equational Metalogic for Monadic Equational Systems

The paper presents algebraic and logical developments. From the algebraic viewpoint, we introduce Monadic Equational Systems as an abstract enriched notion of equational presentation. From the logical viewpoint, we provide Equational Metalogic as a general formal deductive system for the derivability of equational consequences. Relating the two, a canonical model theory for Monadic Equational Systems is given and for it the soundness of Equational Metalogic is established. This development involves a study of clone and double-dualization structures. We also show that in the presence of free algebras the model theory of Monadic Equational Systems satisfies an internal strong-completeness property.

math.CT↗
arXiv2013

Second-Order Algebraic Theories

Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.

cs.LO↗
arXiv2013

Analytic functors between presheaf categories over groupoids

The paper studies analytic functors between presheaf categories. Generalising results of A. Joyal and of R. Hasegawa for analytic endofunctors on the category of sets, we give two characterisations of analytic functors between presheaf categories over groupoids: (i) as functors preserving filtered colimits, quasi-pullbacks, and cofiltered limits; and (ii) as functors preserving filtered colimits and wide quasi-pullbacks. The development establishes that small groupoids, analytic functors between their presheaf categories, and quasi-cartesian natural transformations between them form a 2-category.

math.CT↗