4 papers · 1 filter
Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'
Gabriel Saadia, Umberto Tarantino
Virtual ultracategories are a recent categorification of topological spaces seen as sets of points together with a convergence relation between points and ultrafilters. However, a…
Toposes with enough points as categories of étale spaces
Sam van Gool, Jérémie Marquès, Umberto Tarantino
We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplif…
Ultracategories via Kan extensions of relative monads
Umberto Tarantino, Joshua Wrigley
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, u…
A category of arrow algebras for modified realizability
Umberto Tarantino
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 morp…