Left-exact localizations of ∞-topoi III: The acyclic product
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 n -excisive functors as classifying n n -nilpotent objects.
Fields of Study & Subjects
Homotopy and Cohomology in Algebraic Topology
Rings
Modules
and Algebras
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Transactions of the American Mathematical Society
- Volume
- 379
- Issue
- 9
- Pages
- 6447–6534
- Publication date
- 21 January 2026
- Language
- EN