A short proof of the universality of the relative Rezk nerve

Prior work [11] established a commutativity result for the Hoare power construction and a modified version of the Smyth power construction consisting of strongly compact sets, which is defined for Us-admitting dcpos, where Us-admissability is well-filteredness with compact sets replaced by strongly compact sets. In this paper, we consider the Hoare power construction H and the Smyth power construction Q on the category WF of well-filtered dcpos with Scott-continuous maps. Actually, the functors H and Q can be extended to monads. We prove that H and Q commute, that is, HQ(L) is isomorphic to QH(L) for a well-filtered dcpo L, if and only if L satisfies a property similar to consonance that we call (KC) and the Scott topology coincides with the upper Vietoris topology on Q(L). We also investigate the Eilenberg-Moore category of the monad composed by H and Q under a distributive law on WF and characterize it to be a subcategory of the category Frm, which is composed of all frames and all frame homomorphisms.

math.CT↗

arXiv2026

Algebraic Type Theory, Part 1: Martin-Löf algebras

A new algebraic treatment of dependent type theory is proposed using ideas derived from topos theory and algebraic set theory.

math.CT↗

arXiv2026

Groupoidal polygraphic homology

We show that for a 1-category C, the (ω, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict (ω, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict ω-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict (ω, k)-categories, building on previous work by the first-named author.

math.CT↗