arXiv subjects

Umberto Tarantino

Publications and source records attributed to Umberto Tarantino.

5 recordsLinked to original sources

arXiv2026

Conceptual completeness for subgeometric logics

We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author. Unlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai's original reconstruction theorem via ultracategories.

math.LO↗

arXiv2026

Profunctorial algebras

We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first describe how to extend pseudomonads on a bicategory to skew monads on its bicategory of two-sided discrete fibrations, and we characterize in terms of exact squares when these extensions are themselves pseudomonads. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories admitting a skew extension to profunctors, and in a few relevant cases we introduce suitable quotients also extending to profunctors. Among the latter, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.

math.CT↗

arXiv2026

Toposes with enough points as categories of étale spaces

We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sense of Makkai, for geometric theories with enough Set-models. The same result has recently been obtained independently by Saadia (arXiv:2506.23935) and by Hamad (arXiv:2507.07922). Both of their proofs rely on groupoid representations of toposes, which our proof here does not assume.

math.CT↗

arXiv2025

Ultracategories via Kan extensions of relative monads

Many structured categories of interest are most naturally described as algebras for a relative monad, but turn out nonetheless to be algebras for an ordinary monad. We show that, under suitable hypotheses, the left oplax Kan extension of a relative 2-monad on categories yields a pseudomonad having the same category of colax algebras. In particular, we apply this to the study of ultracategories to recover the 'ultracompletion' pseudomonad.

math.CT↗

arXiv2025

A category of arrow algebras for modified realizability

In this paper we further the study of arrow algebras, simple algebraic structures inducing toposes through the tripos-to-topos construction, by defining appropriate notions of morphisms between them which correspond to morphisms of the associated triposes. Specializing to geometric inclusions, we characterize subtriposes of an arrow tripos in terms of nuclei on the underlying arrow algebra, recovering a classical locale-theoretic result. As an example of application, we lift modified realizability to the setting of arrow algebras, and we establish its functoriality.

math.CT↗