Homotopy linear algebra

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 à la Baez-Hoffnung-Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over $\infty$-groupoids; we hope that they can also be of independent interest.

32 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v2: slight notation changes and expository improvements. Final version, to appear in Proc Royal Soc. Edinburgh A

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• Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
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• Whole-grain Petri nets and processes
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• Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctions
• The incidence comodule bialgebra of the Baez-Dolan construction
• Frobenius and commutative pseudomonoids in the bicategory of spans
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