Marcelo Fiore
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View article: Modular abstract syntax trees (MAST): substitution tensors with second-class sorts
Modular abstract syntax trees (MAST): substitution tensors with second-class sorts Open
We adapt Fiore, Plotkin, and Turi's treatment of abstract syntax with binding, substitution, and holes to account for languages with second-class sorts. These situations include programming calculi such as the Call-by-Value lambda-calculus…
View article: Substructural Abstract Syntax with Variable Binding and Single-Variable Substitution
Substructural Abstract Syntax with Variable Binding and Single-Variable Substitution Open
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) wi…
View article: Formal P-Category Theory and Normalization by Evaluation in Rocq
Formal P-Category Theory and Normalization by Evaluation in Rocq Open
Traditional category theory is typically based on set-theoretic principles and ideas, which are often non-constructive. An alternative approach to formalizing category theory is to use E-category theory, where hom sets become setoids. Our …
View article: ∞-Categorical Models of Linear Logic
∞-Categorical Models of Linear Logic Open
The notion of categorical model of linear logic is now well studied and established around the notion of linear-non-linear adjunction, which encompasses the earlier notions of Seely categories, Lafont categories and linear categories. Thes…
View article: An axiomatics and a combinatorial model of creation/annihilation operators
An axiomatics and a combinatorial model of creation/annihilation operators Open
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 framework…
View article: Logical Structure on Inverse Functor Categories
Logical Structure on Inverse Functor Categories Open
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 …
View article: A finite algebraic presentation of Lawvere theories in the object-classifier topos
A finite algebraic presentation of Lawvere theories in the object-classifier topos Open
Over the topos of sets, the notion of Lawvere theory is infinite countably-sorted algebraic but not one-sorted algebraic. Shifting viewpoint over the object-classifier topos, a finite algebraic presentation of Lawvere theories is considere…
View article: Stabilized profunctors and stable species of structures
Stabilized profunctors and stable species of structures Open
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…
View article: Stabilized profunctors and stable species of structures
Stabilized profunctors and stable species of structures Open
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…
View article: Fixpoint constructions in focused orthogonality models of linear logic
Fixpoint constructions in focused orthogonality models of linear logic Open
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.…
View article: Fixpoint constructions in focused orthogonality models of linear logic
Fixpoint constructions in focused orthogonality models of linear logic Open
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.…
View article: Free Commutative Monoids in Homotopy Type Theory
Free Commutative Monoids in Homotopy Type Theory Open
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 th…
View article: Semantic analysis of normalisation by evaluation for typed lambda calculus
Semantic analysis of normalisation by evaluation for typed lambda calculus Open
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 sh…
View article: Semantic Analysis of Normalisation by Evaluation for Typed Lambda Calculus
Semantic Analysis of Normalisation by Evaluation for Typed Lambda Calculus Open
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 sh…
View article: Quotients, inductive types, and quotient inductive types
Quotients, inductive types, and quotient inductive types Open
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 infinita…
View article: Formal metatheory of second-order abstract syntax
Formal metatheory of second-order abstract syntax Open
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 metat…
View article: Agda code supporting "Quotients, Inductive Types and Quotient Inductive Types"
Agda code supporting "Quotients, Inductive Types and Quotient Inductive Types" Open
Formal proofs for the results in the paper "Quotients, Inductive Types and Quotient Inductive Types" constructed with the Agda theorem proving system, version 2.6.2.\n\nImportant note: There is an updated version of this dataset. Version 2…
View article: Free Commutative Monoids in Homotopy Type Theory
Free Commutative Monoids in Homotopy Type Theory Open
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 th…
View article: Coherence for bicategorical cartesian closed structure
Coherence for bicategorical cartesian closed structure Open
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 biequival…
View article: Quotients, inductive types, and quotient inductive types
Quotients, inductive types, and quotient inductive types Open
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 infinita…
View article: Algebraic models of simple type theories: a polynomial approach
Algebraic models of simple type theories: a polynomial approach Open
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 $…
View article: Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure
Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure Open
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 there…
View article: Algebraic models of simple type theories
Algebraic models of simple type theories Open
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…
View article: Lawvere theories and C-systems
Lawvere theories and C-systems Open
In this paper we consider the class of `-bijective C-systems, i.e., C-systems for which the length function is a bijection. The main result of the paper is a construction of an isomorphism between two categories - the category of `-bijecti…
View article: Code supporting "Constructing Infinitary Quotient-Inductive Types"
Code supporting "Constructing Infinitary Quotient-Inductive Types" Open
Agda code (source and html) supporting the paper "Constructing Infinitary Quotient-Inductive Types" in 23rd International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2020), Dublin, Ireland 2020. See RE…
View article: Classical logic with Mendler induction
Classical logic with Mendler induction Open
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-styl…
View article: A type theory for cartesian closed bicategories
A type theory for cartesian closed bicategories Open
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 proper…