Homotopy linear algebra

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 148; no. 2; pp. 293 - 325
• Main Authors: Gálvez-Carrillo, Imma, Kock, Joachim, Tonks, Andrew
• Format: Journal Article Publication
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.04.2018; Cambridge University Press
• Subjects: 15 Linear and multilinear algebra; matrix theory; 18 Category theory; homological algebra; 18G Homological algebra; 37 Dynamical systems and ergodic theory; 37F Complex dynamical systems; 46 Associative rings and algebras; 46A Topological linear spaces and related structures; 55 Algebraic topology; 55P Homotopy theory; Algebra; Algebra, Homological; Algebraic topology; Classificació AMS; duality; Homotopia, Teoria d; homotopy cardinality; homotopy finiteness; Homotopy theory; infinity-groupoids; Linear algebra; Matemàtiques i estadística; Topologia; Topologia algebraica; Vector spaces; Àlgebra homològica; Àrees temàtiques de la UPC; homotopy cardinality; infinity-groupoids; homotopy finiteness; Secondary 18-XX; 55Pxx; 15Axx; 37F20; 46A20; duality; Primary 20L05; linear algebra
• ISSN: 0308-2105; 1473-7124; 1473-7124
• DOI: 10.1017/S0308210517000208

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