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Linearizing Combinators

Banff International Research Station for Mathematical Innovation and Discovery

Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category, where the differential combinator is defined using linearization. From the Cartesian differential category perspective, the BJORT construction is backwards. In any Cartesian differential category it is always possible to define the notion of a linear map and to linearize a map using the differential combinator. BJORT constructed their differential combinator using an already established notion of linear map and linearization. In this talk, we reverse engineer BJORT's construction by abstracting the notion of linear approximation by introducing linearizing combinators. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed à la BJORT from a system of linearizing combinators in context. Therefore, linearizing combinators provide an equivalent alternative axiomatization of Cartesian differential categories. This is joint work with Robin Cockett.

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: Mount Allison University

2023-10-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86396

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/