Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.19437

Signs in objective linear algebra, exemplified with exterior powers and determinants

Joachim Kock·Jesper Michael Møller
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop objective linear algebra in a new setting with a cardinality functor that can take negative values. The signs arise as little homotopies, as ratios between orientations. To illustrate the workings of the theory we give an objective treatment of exterior powers and determinants.

Explore related subjects

math.CTmath.ATmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joachim Kock, Jesper Michael Møller. 2026-03-19. Signs in objective linear algebra, exemplified with exterior powers and determinants. https://arxiv.org/abs/2603.19437

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↗