Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product

TL;DR

This paper introduces the acyclic product, a commutative monoid structure on the poset of left-exact localizations of a higher topos, linking to ideals of rings and homotopical towers.

Contribution

It defines the acyclic product on localizations, establishing a structural analogy with ring ideals and connecting to Goodwillie calculus in homotopy theory.

Findings

Defines the acyclic product as a monoid structure.

Shows how powers of localizations form towers analogous to homotopical towers.

Describes topoi of n-excisive functors as classifying n-nilpotent objects.

Abstract

We define a commutative monoid structure on the poset of left-exact localizations of a higher topos, that we call the acyclic product. Our approach is anchored in a structural analogy between the poset of left-exact localizations of a topos and the poset of ideals of a commutative ring. The acyclic product is analogous to the product of ideals. The sequence of powers of a given left-exact localization defines a tower of localizations. We show how this recovers the towers of Goodwillie calculus in the unstable homotopical setting. We use this to describe the topoi of $n$-excisive functors as classifying $n$-nilpotent objects.