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A Model Categoric Equivalence for Crossed Simplicial Modules

Haydar Can Kaya, Atabey Kaygun

TL;DR

For a field $\Bbbk$ of characteristic $0$ and locally finite crossed groups $\{G_n\}$, the paper establishes a model-categorical equivalence between simplicial $\Bbbk$-vector spaces $\mathrm{Mod}-\Delta$ and representations of a crossed simplicial group $\mathrm{Mod}-\Delta G$. It builds a bridge via a change of basis from $\Delta$ to $\Omega$, the Moore functor $N_*$, and adjunctions (Ind, CoInd, Res), recasting Dold–Kan in a DG–categorical setting and extending it to crossed structures by introducing the bimodule $\Upsilon=\Delta G/\epsilon_G$. By transferring model structures through these adjunctions and leveraging the finite $G_n$ hypothesis (which yields semisimplicity), it proves a Quillen equivalence between $\mathrm{Mod}-\Delta G$ and $\mathrm{Mod}-\Delta$, thereby equipping the homotopy theories of crossed-simplicial representations with a robust model-categorical framework. The work unifies classical results (Dold–Kan, Dwyer–Kan) within a common model-categorical lens and provides concrete tools (coinvariants, $\Upsilon$-induced adjunctions) to analyze homotopy equivalences in crossed simplicial settings.

Abstract

We construct a model categorical equivalence between the category of simplicial vector spaces and the category of representations of a crossed simplicial group $ΔG$ when each $G_n$ is finite and the characteristic of the ground field is 0.

A Model Categoric Equivalence for Crossed Simplicial Modules

TL;DR

For a field of characteristic and locally finite crossed groups , the paper establishes a model-categorical equivalence between simplicial -vector spaces and representations of a crossed simplicial group . It builds a bridge via a change of basis from to , the Moore functor , and adjunctions (Ind, CoInd, Res), recasting Dold–Kan in a DG–categorical setting and extending it to crossed structures by introducing the bimodule . By transferring model structures through these adjunctions and leveraging the finite hypothesis (which yields semisimplicity), it proves a Quillen equivalence between and , thereby equipping the homotopy theories of crossed-simplicial representations with a robust model-categorical framework. The work unifies classical results (Dold–Kan, Dwyer–Kan) within a common model-categorical lens and provides concrete tools (coinvariants, -induced adjunctions) to analyze homotopy equivalences in crossed simplicial settings.

Abstract

We construct a model categorical equivalence between the category of simplicial vector spaces and the category of representations of a crossed simplicial group when each is finite and the characteristic of the ground field is 0.
Paper Structure (18 sections, 16 theorems, 31 equations)

This paper contains 18 sections, 16 theorems, 31 equations.

Table of Contents

  1. Preliminaries
  2. Simplicial and cosimplicial modules
  3. Crossed simplicial modules
  4. Differential graded modules
  5. A change of basis
  6. Induction, coinduction and restriction
  7. Homology and Homotopy of Crossed Simplicial Modules
  8. Point objects
  9. Quasi-isomorphisms and homotopy equivalences
  10. Homotopy equivalence for crossed simplicial modules
  11. The Dold-Kan Equivalence Re-imagined
  12. The Moore functor
  13. The classical Dold-Kan equivalence via coinduction and restriction
  14. An equivalence for crossed simplicial groups
  15. Model Categorical Reformulation
  16. ...and 3 more sections

Key Result

Lemma 1.1

The categorical algebra ${\Delta}$ has a basis that consists of monomials of the form again with $j_r>\cdots > j_n$ and $\ell_r >\cdots > \ell_m$.

Theorems & Definitions (28)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 18 more