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model structure on equivariant dgc-algebras

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Rational homotopy theory

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 GG-equivariant enhancement (Scull 08) models GG-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 GG: GG-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(GOrbits,dgcAlgebras 0) Functors \big( G Orbits \,,\, dgcAlgebras^{\geq 0}_{\mathbb{Q}} \big)

of connective GG-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:

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

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

(Scull 08, Theorem 3.2)

Properties

Proposition

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

Let GG be a finite group.

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

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

(Scull 08, Prop. 5.1)

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.