arXiv subjects

Jacopo Emmenegger

Publications and source records attributed to Jacopo Emmenegger.

13 recordsLinked to original sources

arXiv2026

Free constructions for comprehension categories

Jacobs comprehension categories subsume a large class of categorical models of type dependency, supporting also the description of morphisms between types. We study the relationship between comprehension categories and a particular subclass, which we call Lawvere-Ehrhard comprehension categories. First, we characterize this subclass by comparing a fibration of terms and a fibration of type morphisms associated to a given comprehension category. Next, we provide the construction of the free comprehension category over a fibration. Finally, we construct the free Lawvere-Ehrhard comprehension category over a Jacobs comprehension category.

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arXiv2025

Toward the effective 2-topos

A candidate for the effective 2-topos is proposed and shown to include the effective 1-topos as its subcategory of 0-types.

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arXiv2024

A 2-categorical analysis of context comprehension

We consider the equivalence between the two main categorical models for the type-theoretical operation of context comprehension, namely P. Dybjer's categories with families and B. Jacobs' comprehension categories, and generalise it to the non-discrete case. The classical equivalence can be summarised in the slogan: "terms as sections". By recognising "terms as coalgebras", we show how to use the structure-semantics adjunction to prove that a 2-category of comprehension categories is biequivalent to a 2-category of (non-discrete) categories with families. The biequivalence restricts to the classical one proved by Hofmann in the discrete case. It also provides a framework where to compare different morphisms of these structures that have appeared in the literature, varying on the degree of preservation of the relevant structure. We consider in particular morphisms defined by Claraimbault-Dybjer, Jacobs, Larrea, and Uemura.

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arXiv2024

A comonad for Grothendieck fibrations

We prove that cloven Grothendieck fibrations over a fixed base $\ct{B}$ are the pseudo-coalgebras for a lax idempotent 2-comonad on $\ct{Cat}/\ct{B}$. We show this via an original observation that the known colax idempotent 2-monad for fibrations over a fixed base has a right 2-adjoint. As an important consequence, we obtain an original cofree construction of a fibration on a functor. We also give a new, conceptual proof of the fact that the forgetful 2-functor from split fibrations to cloven fibrations over a fixed base has both a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the left splitting and the 2-comonad yields the right splitting. Moreover, we show that the constructions induced by these coherence theorems recover Giraud's explicit constructions of the left and the right splittings.

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arXiv2023

Categorical models of subtyping

Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for categories with families, categories with attributes, and natural models; in particular, all of them can be traced back to certain discrete Grothendieck fibrations. We extend this intuition to the case of general, non necessarily discrete, fibrations, so that over a given context one has not only a set but a category of types. We argue that the added structure can be attributed to a notion of subtyping that shares many features with that of coercive subtyping, in the sense that it is the product of thinking about subtyping as an abbreviation mechanism: we say that a given type $A'$ is a subtype of $A$ if there is a unique coercion from $A'$ to $A$. Whenever we need a term of type $A$, then, it suffices to have a term of type $A'$, which we can `plug-in' into $A$. For this version of subtyping we provide rules, coherences, and explicit models, and we compare and contrast it to coercive subtyping as introduced by Z. Luo and others. We conclude by suggesting how the tools we present can be employed in finding appropriate rules relating subtyping and certain type constructors.

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arXiv2022

The solutions to single-variable polynomials, implemented and verified in Lean

In this work, we describe our experience in learning the use of a computer proof assistant - specifically, Lean - from scratch, through proving formulae for the solutions of polynomial equations. Specifically, in this work we characterize the solutions of quadratic, cubic, and quartic polynomials over certain fields, specifically, fields with operations returning square and cubic roots of characteristic other than two or three. The purpose of this work is thus twofold. Firstly, it describes the learning experience of a starting Lean user, including a detailed comparison between our work in Lean and very closely related work in Coq. Secondly, our results represent a modest improvement over the aforementioned related work in Coq, which we hope will be of some scientific interest.

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arXiv2021

Algebraic Presentations of Type Dependency

C-systems were defined by Cartmell as the algebraic structures that correspond exactly to generalised algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky's construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky. We construct this equivalence as the restriction of an equivalence between more general structures, called CE-systems and E-systems, respectively. To this end, we identify C-systems and B-systems as "stratified" CE-systems and E-systems, respectively; that is, systems whose contexts are built iteratively via context extension, starting from the empty context.

math.CT↗

arXiv2021

W-types in setoids

We present a construction of W-types in the setoid model of extensional Martin-Löf type theory using dependent W-types in the underlying intensional theory. More precisely, we prove that the internal category of setoids has initial algebras for polynomial endofunctors. In particular, we characterise the setoid of algebra morphisms from the initial algebra to a given algebra as a setoid on a dependent W-type. We conclude by discussing the case of free setoids. We work in a fully intensional theory and, in fact, we assume identity types only when discussing free setoids. By using dependent W-types we can also avoid elimination into a type universe. The results have been verified in Coq and a formalisation is available on the author's GitHub page.

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arXiv2021

Exact completion and constructive theories of sets

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e. objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

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arXiv2020

A characterisation of elementary fibrations

Grothendieck fibrations provide a unifying algebraic framework that underlies the treatment of various form of logics, such as first order logic, higher order logics and dependent type theories. In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along a diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality. The present paper provides a characterisation of elementary fibrations based on particular structures in the fibres, called transporters. The characterisation is a substantial generalisation of the one already available for faithful fibrations. There is a close resemblance between transporters and the structures used in the semantics of the identity type of Martin-Löf type theory. We close the paper by comparing the two.

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arXiv2020

The Fullness Axiom and exact completion of homotopy categories

We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy category of any model category satisfying mild requirements, thus obtaining in particular the local cartesian closure of the exact completion of topological spaces and homotopy classes of maps. Under a type-theoretic reading, these results provide a general motivation for the local cartesian closure of the category of setoids. However, results and proofs are formulated solely in the language of categories, and no knowledge of type theory or constructive set theory is required on the reader's part.

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arXiv2020

On the local cartesian closure of exact completions

This paper presents a necessary and sufficient condition on a category with weak finite limits for its exact completion to be (locally) cartesian closed. A paper by Carboni and Rosolini already claimed such a characterisation using a different property on the base category, but we shall show that weak finite limits are not enough for their proof to go through. We shall also indicate how to strengthen the hypothesis for that proof to work. It will become clear that, in the case of ex/lex completions, their characterisation is still valid and it coincides with the one presented here.

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arXiv2018

A category-theoretic version of the identity type weak factorization system

Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an enrichment of Joyal's notion of tribe, which we call a tribe with weakly stable path objects, and prove a purely category-theoretic version of the identity type weak factorization system, thus generalizing Gambino and Garner's result. We conclude showing that this structure subsumes also the weak factorization systems coming from the topological and simplicial models of identity types obtained by van den Berg and Garner.

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