Homotopy linear algebra
Imma G\'alvez-Carrillo, Joachim Kock, Andrew Tonks

TL;DR
This paper develops a framework called homotopy linear algebra, studying linear functors between slices of the infinity-category of infinity-groupoids, establishing dualities and a notion of homotopy cardinality relevant for incidence algebras and Möbius inversion.
Contribution
It introduces a new homotopy linear algebra framework that models dualities and defines homotopy cardinality, extending classical linear algebra concepts to infinity-groupoids.
Findings
Established duality between vector spaces and profinite-dimensional vector spaces in the homotopy setting.
Defined a global notion of homotopy cardinality compatible with the duality.
Supported the development of incidence algebras and Möbius inversion over infinity-groupoids.
Abstract
By homotopy linear algebra we mean the study of linear functors between slices of the -category of -groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into -categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality \`a la Baez-Hoffnung-Walker compatible with this duality. We needed these results to support our work on incidence algebras and M\"obius inversion over -groupoids; we hope that they can also be of independent interest.
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