Hammock localization via Segal animae

Kensuke Arakawa, Bastiaan Cnossen

arXiv:2608.19870v1 Primary: math.CT Secondary: math.AT

Abstract

We give a short, conceptual account of Dwyer--Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer--Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.

Disclosure

“ly on combinatorics of quasicategories (Lemmas 3.19 and 3.27). The results in the remaining (sub)sections use only very formal properties of ∞-categories and are not bound to a specific model of ∞-categories. AI disclosure: Coding agents (Claude/Codex) were used in the revision of the manuscript, which also led us to a simpler proof of Lemma 3.22.”
PDF page 3

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-22T18:12:54.961117Z

Structural counts

Pages 41 pdf

Theorems 10 source

Lemmas 15 source

Propositions 11 source

Corollaries 7 source

Definitions 18 source

Displayed equations 170 source

Bibliography entries 40 source

Appendix pages 41 estimated

Count notes

Source counts use the expanded primary TeX file hammock_4.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.