Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Introduction to Framed Correspondences

Marc Hoyois, Nikolai Opdan

TL;DR

This work develops framed correspondences as a robust, computation-friendly framework for motivic homotopy theory. It constructs the ∞-category of framed correspondences and establishes a coherent theory of framed transfers via lci morphisms and the virtual tangent bundle, culminating in a reconstruction principle that SH(S) is equivalent to the framed version SH^{fr}(S). The framework yields powerful results for algebraic cobordism, providing framed models for Thom spectra and showing how MGL and related spectra arise from framed presheaves and derived moduli stacks. Overall, the approach unifies transfers, recognition, and cobordism within framed correspondences, enabling explicit computations of infinite loop spaces and Thom spectra in the motivic setting.

Abstract

We give an overview of the theory of framed correspondences in motivic homotopy theory. Motivic spaces with framed transfers are the analogue in motivic homotopy theory of $E_{\infty}$-spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite $\mathbb{P}^1$-loop spaces. We will discuss the foundations of the theory (following Voevodsky, Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the computations of the infinite loop spaces of the motivic sphere and of algebraic cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open problems.

Introduction to Framed Correspondences

TL;DR

This work develops framed correspondences as a robust, computation-friendly framework for motivic homotopy theory. It constructs the ∞-category of framed correspondences and establishes a coherent theory of framed transfers via lci morphisms and the virtual tangent bundle, culminating in a reconstruction principle that SH(S) is equivalent to the framed version SH^{fr}(S). The framework yields powerful results for algebraic cobordism, providing framed models for Thom spectra and showing how MGL and related spectra arise from framed presheaves and derived moduli stacks. Overall, the approach unifies transfers, recognition, and cobordism within framed correspondences, enabling explicit computations of infinite loop spaces and Thom spectra in the motivic setting.

Abstract

We give an overview of the theory of framed correspondences in motivic homotopy theory. Motivic spaces with framed transfers are the analogue in motivic homotopy theory of -spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite -loop spaces. We will discuss the foundations of the theory (following Voevodsky, Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the computations of the infinite loop spaces of the motivic sphere and of algebraic cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open problems.
Paper Structure (6 sections, 13 theorems, 69 equations, 1 figure)

This paper contains 6 sections, 13 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transfers for motivic spectra
  3. The $\infty$-category of framed correspondences
  4. The recognition principle
  5. Algebraic cobordism
  6. Framed models for effective Thom spectra

Key Result

theorem 1.4

The evaluation functor is an equivalence.

Figures (1)

  • Figure 1: A cobordism from two circles to one.

Theorems & Definitions (41)

  • definition 1.3
  • theorem 1.4
  • proof
  • example 1.5: Atiyah duality
  • definition 1.6
  • theorem 1.7
  • definition 2.1
  • proof : Sketch
  • definition 2.3
  • example 2.4
  • ...and 31 more