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Notes on motivic infinite loop space theory

Tom Bachmann, Elden Elmanto

Abstract

In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory, together with their proofs, in the case of infinite perfect fields. These are our extended notes on these talks.

Notes on motivic infinite loop space theory

Abstract

In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory, together with their proofs, in the case of infinite perfect fields. These are our extended notes on these talks.

Paper Structure

This paper contains 43 sections, 53 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. The reconstruction theorem
  3. Setup
  4. Preliminary reductions
  5. Recollections on prespectra
  6. Spectrification
  7. Prolongation
  8. Equationally framed correspondences
  9. Framed correspondences
  10. Equationally framed correspondences
  11. Relative equationally framed correspondences
  12. Comparison results
  13. Equationally framed versus tangentially framed
  14. The cone theorem
  15. Voevodsky's lemma
  16. ...and 28 more sections

Key Result

Theorem 1.1

For a perfect field $k$, there exists a canonical, symmetric monoidal equivalence of $\infty$-categories

Theorems & Definitions (138)

  • Theorem 1.1
  • Theorem 2.1: reconstruction
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • ...and 128 more