Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.23413

Nice exact categories are coexact

James Richard Andrew Gray
View original source ↗Read source document ↗Export metadata ↓

Abstract

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by Tierney's characterization of them as additive exact categories. The next type consists of elementary toposes which are well-known to be exact, but have also been shown to be coexact and coprotomodular by Bourn. In this paper we study a condition weaker than extensivity and equivalent to additivity for pointed categories. We show that for a finitely cocomplete category this condition together with exactness implies coexactness and coprotomodularity. As a special case we obtain that a finitely cocomplete pretopos is coexact.

Explore related subjects

math.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

James Richard Andrew Gray. 2026-03-27. Nice exact categories are coexact. https://arxiv.org/abs/2506.23413

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Commutation of Smyth and Hoare Power Constructions in Well-filtered Dcpos

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

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↗