📝 SummarXiv

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.