Homotopy linear algebra

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 $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into $\infty$-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 $\infty$-groupoids; we hope that they can also be of independent interest.