Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Log homotopy types are homotopy types with modulus

Shane Kelly

TL;DR

The work proves that log homotopy types form a full subcategory of homotopy types with modulus, via a canonical localization: $ ext{ olinebreak} ext{ extunderline{M}H}_k[oldsymbol{ m extLambda}^{-1}] \\stackrel{\sim}{\to} \, ext{logH}_k$, with a fully faithful embedding $ ext{ olinebreak} ext{ extunderline{M}H}_k\hookrightarrow \text{logH}_k$. This is achieved by connecting modulus-pair orbit categories $ ext{ extunderline{P}Sm}_k$ to log-schemes $ ext{SmlSm}_k$ through localization along $(\overline{X}, nX^{\infty})\to(\overline{X}, X^{\infty})$ and establishing Nisnevich descent, smooth blowup calculus, and $ ext{CI}$-inversions. The paper also describes a scheme-theoretic portrayal of log homotopy types, extends to a colimit construction $ ext{mH}_k\cong\varinjlim_n\text{ olinebreak} ext{ extunderline{M}H}_k[\text{CI}(n)^{-1}]$, and discusses stable versions and transfer phenomena, including a counterexample showing limits of transfers in this setting. These results illuminate how ramification/pole data can be encoded in motivic-homotopy frameworks and bridge modulus- and log-theoretic approaches.

Abstract

We show that the category of log homotopy types is a full subcategory of a category of homotopy types with modulus.

Log homotopy types are homotopy types with modulus

TL;DR

The work proves that log homotopy types form a full subcategory of homotopy types with modulus, via a canonical localization: , with a fully faithful embedding . This is achieved by connecting modulus-pair orbit categories to log-schemes through localization along and establishing Nisnevich descent, smooth blowup calculus, and -inversions. The paper also describes a scheme-theoretic portrayal of log homotopy types, extends to a colimit construction , and discusses stable versions and transfer phenomena, including a counterexample showing limits of transfers in this setting. These results illuminate how ramification/pole data can be encoded in motivic-homotopy frameworks and bridge modulus- and log-theoretic approaches.

Abstract

We show that the category of log homotopy types is a full subcategory of a category of homotopy types with modulus.

Paper Structure

This paper contains 8 sections, 12 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Log schemes are modulus pairs with completed modulus
  4. Nisnevich Topologies
  5. Admissible modifications
  6. Homotopy categories
  7. Rational divisors
  8. Transfers

Key Result

Theorem 1.1

The canonical comparison functor induces an equivalenceHere $(-)[\Lambda^{-1}]$ means localisation in the category $\mathcal{P}{\operatorfont{r^L}}$ of presentable $\infty$-categories and colimit preserving functors, HTT. where $\Lambda$ is the class of (images of) morphisms of the form for $(\overline{X}, X^\infty) \in \mathrm{\underline{P}Sm}_k^{\mathrm{}}$, Def.defi:PSmnc and $n \geq 1$. In pa

Theorems & Definitions (65)

  • Theorem 1.1: Theorem \ref{['theo:main']}
  • Proposition 1.2: Proposition \ref{['prop:PSmLambdaSmlSm']}
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7: Theorem \ref{['prop:mH']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • ...and 55 more