Goodwillie calculus of the category of non-unital algebras, and application to algebraic cobordism

TL;DR

This paper extends Goodwillie calculus to non-presentable $mbda$-categories and applies it to show that tilting functors induce equivalences in algebraic cobordism for perfectoid algebras, with broader applications to $K$-theory.

Contribution

It generalizes Goodwillie calculus to include non-presentable categories and applies this to algebraic cobordism and perfectoid algebras, providing new tools for functor approximation.

Findings

Tilting functor induces weak equivalence of algebraic cobordisms for perfectoid algebras.

Extended Goodwillie calculus to non-presentable $mbda$-categories.

Developed a theory of functor approximation with applications to $K$-theory.

Abstract

This paper reformulates Goodwillie calculus of $\infty$-categories including non-presentable $\infty$-categories. In the case of presentable $\infty$-categories our definition is equivalent to Heuts's~\cite{Heuts2018} work. As an application of this work, we prove that, for any perfectoid algebra, the tilting functor induces a weak equivalence of the left Kan extensions algebraic cobordisms on the category of non-unital algebras. Furthermore, we give a theory of approximation of functors along given natural transformation of them, and applications to the algebraic cobordism to the $K$-theory.