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Binomial rings and homotopy theory

Geoffroy Horel

TL;DR

The paper constructs a fully faithful functor from finite-type nilpotent spaces to cosimplicial binomial rings, providing an algebraic model for integral homotopy types via the functor $X\mapsto \mathbf{Z}^X$. It proves a main theorem that the derived unit $X\to\langle\mathbf{Z}^X\rangle$ is a weak equivalence for nilpotent finite-type spaces, and extends this to broader localization results by leveraging pro-space techniques and Isaksen’s framework. It introduces a binomial affine homotopy type, showing $X^{bin}$ is a binomial affine stack and that $X^{bin}$ encodes localizations $X^{bin}(R)$ with $R\subset\mathbf{Q}$ or $\mathbf{F}_p$. As an application, it defines a binomial Grothendieck–Teichmüller group $\mathrm{GT}$ whose $p$- and rational-points recover the classical pro-$p$ and rational GT groups, unifying integral homotopy theory with binomial-algebraic geometry.

Abstract

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the Grothendieck-Teichmüller group.

Binomial rings and homotopy theory

TL;DR

The paper constructs a fully faithful functor from finite-type nilpotent spaces to cosimplicial binomial rings, providing an algebraic model for integral homotopy types via the functor . It proves a main theorem that the derived unit is a weak equivalence for nilpotent finite-type spaces, and extends this to broader localization results by leveraging pro-space techniques and Isaksen’s framework. It introduces a binomial affine homotopy type, showing is a binomial affine stack and that encodes localizations with or . As an application, it defines a binomial Grothendieck–Teichmüller group whose - and rational-points recover the classical pro- and rational GT groups, unifying integral homotopy theory with binomial-algebraic geometry.

Abstract

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the Grothendieck-Teichmüller group.
Paper Structure (10 sections, 33 theorems, 126 equations)

This paper contains 10 sections, 33 theorems, 126 equations.

Table of Contents

  1. Binomial rings
  2. The cobar construction of $\operatorname{Num}[x]$
  3. Cosimplicial binomial rings
  4. Main theorem for Eilenberg-MacLane spaces
  5. Main theorem
  6. Beyond nilpotent spaces
  7. Other localizations
  8. Binomial affine homotopy type
  9. A binomial Grothendieck-Teichmüller group
  10. Conservativity of the bar construction

Key Result

Theorem 1

There is a functor $\Omega^*_{PL}$ from the homotopy category of simplicial sets to the opposite of the homotopy category of commutative differential graded algebras which is a left adjoint and whose right adjoint is denoted $A\mapsto \langle A\rangle$. When restricted to simplicial sets $X$ that ar is a model for the localization with respect to rational homology isomorphisms.

Theorems & Definitions (75)

  • Theorem : sullivaninfinitesimal
  • Theorem : Mandell mandellcochains, Toën toenprobleme
  • Theorem
  • Definition 1.1
  • Remark 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 65 more