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$\mathbb{E}_{\infty}$-coalgebras and $p$-adic homotopy theory

Tom Bachmann, Robert Burklund

Abstract

We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.

$\mathbb{E}_{\infty}$-coalgebras and $p$-adic homotopy theory

Abstract

We show that for any separably closed field of characteristic , the canonical functor from nilpotent -adic spaces to -coalgebras over (given by singular chains with coefficients in ) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.
Paper Structure (8 sections, 31 theorems, 73 equations)

This paper contains 8 sections, 31 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Coalgebras
  3. The Frobenius
  4. Polynomial Functors
  5. Pro-objects
  6. Idempotents in coalgebras
  7. The Artin--Schreier pullback square
  8. Main result

Key Result

Theorem 1.2

Let $k$ be a separably closed field of characteristic $p > 0$. The chains functor restricted to $p$-complete, nilpotent spaces is fully faithful.

Theorems & Definitions (83)

  • Definition 1.1
  • Theorem 1.2: Fully faithfulness theorem, see Theorem \ref{['thm:main']}
  • Theorem 1.3: Characterization theorem, see Theorem \ref{['thm:charact']}
  • Remark 1.4
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.3
  • ...and 73 more