Contents

model category

for ∞-groupoids

and

# Contents

## Idea

Where the projective model structure on connective dgc-algebras models the rational homotopy theory of rationally finite type nilpotent topological spaces (by the fundamental theorem of dg-algebraic rational homotopy theory), its $G$-equivariant enhancement (Scull 08) models $G$-equivariant rational homotopy theory of topological G-spaces:

Using Elmendorf's theorem, the underlying category is that of diagrams of connective dgc-algebras parametrized over the orbit category of $G$: $G$-equivariant dgc-algebras. A key technical aspect of this generalization is that not all objects are injective anymore, but otherwise the definitions and properties of the model structure proceed in analogy to Bousfield-Gugenheim‘s projective model structure on connective dgc-algebras. Notably, the minimal cofibrations coincide with the equivariant minimal Sullivan models earlier considered by Triantafillou 82.

## Definition

###### Proposition

There is a model category-structure on the category

$Functors \big( G Orbits \,,\, dgcAlgebras^{\geq 0}_{\mathbb{Q}} \big)$

of connective $G$-equivariant dgc-algebras (i.e. with differential of degree +1) over the rational numbers, whose weak equivalences and fibrations are those of the underlying model structure on equivariant connective cochain complexes, hence:

$\mathrm{W}$weak equivalences are the quasi-isomorphisms over each $G/H \in G Orbits$;

$Fib$fibrations are the morphisms which over each $G/H \in G Orbits$ are degree-wise surjections whose degreewise kernels are injective objects (in the category of vector G-spaces).

## Properties

###### Proposition

(Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras)

Let $G$ be a finite group.

The $G$-equivariant PL de Rham complex-construction is the left adjoint in a Quillen adjunction between

$\big( G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} G SimplicialSets_{Qu}$

## References

Last revised on October 2, 2020 at 15:57:52. See the history of this page for a list of all contributions to it.