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Categorical models of unstable G-global homotopy theory

Tobias Lenz

Abstract

We prove that the category $\textbf{G-Cat}$ of small categories with $G$-action forms a model of unstable $G$-global homotopy theory for every discrete group $G$, generalizing Schwede's global model structure on $\textbf{Cat}$. As a consequence, we prove that $\textbf{G-Cat}$ models proper $G$-equivariant homotopy theory not only when we test weak equivalences on fixed points, but also when we test them on categorical homotopy fixed points.

Categorical models of unstable G-global homotopy theory

Abstract

We prove that the category of small categories with -action forms a model of unstable -global homotopy theory for every discrete group , generalizing Schwede's global model structure on . As a consequence, we prove that models proper -equivariant homotopy theory not only when we test weak equivalences on fixed points, but also when we test them on categorical homotopy fixed points.

Paper Structure

This paper contains 11 sections, 34 theorems, 56 equations.

Table of Contents

  1. Preliminaries
  2. G-global homotopy theory
  3. The Thomason model structure
  4. Transferring model structures
  5. Categories with monoid actions
  6. Equivariant Dwyer maps for groups
  7. Equivariant Dwyer maps for monoids
  8. The equivariant Ex-functor
  9. Categories vs. simplicial sets
  10. The G-global model structure
  11. G-categories as models of G-global homotopy types

Key Result

Proposition 1.7

Let $M$ be a simplicial monoid and let $\mathcal{F}$ be a collection of finite subgroups of $M_0$. Then there exists a unique model structure on $\textbf{$\bm M$-SSet}$ in which a map $f$ is a weak equivalence or fibration if and only if $f^H$ is a weak equivalence or fibration, respectively, in the and generating acyclic cofibrations Moreover it is simplicial (for the obvious enrichment), proper

Theorems & Definitions (86)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.6
  • Proposition 1.7
  • proof
  • Corollary 1.8
  • Theorem 1.9
  • proof
  • Remark 1.10
  • ...and 76 more