arXiv subjects

Luis Gambarte

Publications and source records attributed to Luis Gambarte.

3 recordsLinked to original sources

arXiv2026

(2-dep,$Σ$)-categories are not generalised categories with families

The notion of a generalised category with families, introduced by Coraglia and Emmenegger is one of the most general notions introduced to capture categorically the notion of dependent typing. It is shown that these generalised categories with families are biequivalent to comprehension categories. We will show that the notion of a (2-dep,$Σ$)-category, an extension of the notion of a (dep,$Σ$)-category, introduced by Petrakis, is not biequivalent to generalised categories with families, but instead is equivalent to a direct generalisation of that notion.

math.CT↗

arXiv2026

Categories with a Base of Computability

The notion of a base of computability $\mathscr{C}$ in a category $\mathscr{C}$ was introduced as a tool to generate computability models, in the sense of Longley and Normann, from categories. In this paper we introduce the category $\mathsf{CatBaseComp}$ of categories with a base of computability, and we show that $\mathsf{CatBaseComp}$ has all pie limits. We prove that a Grothendieck fibration lifts a base of computability in the base category to a base of computability in the total category of the fibration, and conversely, a pullback-preserving Grothendieck fibration maps a base of computability in the total category to a base of computability in the base category of the fibration. Connecting $\mathsf{CatBaseComp}$ with the semantics of dependent type theory, we show that $\mathsf{CatBaseComp}$ is a type-category, or a (fam, $Σ$)-category with a terminal object. Moreover, we prove that CatBaseComp is a (2-fam, $Σ$)-category, a 2-categorical generalisation of a (fam, $Σ$)-category. Finally, we describe the canonical (2-dep, $Σ$)-structure of CatBaseComp, i.e., the canonical dependent arrows of CatBaseComp that are compatible with its (2-fam, $Σ$)-structure.

math.CT↗

arXiv2024

The Grothendieck computability model

Translating notions and results from category theory to the theory of computability models of Longley and Normann, we introduce the Grothendieck computability model and the first-projection-simulation. We prove some basic properties of the Grothendieck computability model, and we show that the category of computability models is a type-category, in the sense of Pitts. We introduce the notion of a fibration and opfibration-simulation, and we show that the first-projection-simulation is a split opfibration-simulation.

math.CT↗