# Goodwillie's Calculus of Functors and Higher Topos Theory

## TL;DR

This paper advances Goodwillie's calculus of functors by integrating higher topos theory, introducing fiberwise orthogonality, and establishing new results on $n$-excisive maps and the Goodwillie tower.

## Contribution

It introduces fiberwise orthogonality within higher topos theory to characterize $n$-excisive maps and derives new properties of the Goodwillie calculus framework.

## Key findings

- Pushout product of $P_n$- and $P_m$-equivalences yields a $P_{m+n+1}$-equivalence.
- Established a Blakers-Massey theorem for the Goodwillie tower.
- Rederived foundational theorems like delooping of homogeneous functors.

## Abstract

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.