model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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.
There is a model category-structure on the category
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).
(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
the opposite of the projective model structure on equivariant connective dgc-algebras
the model structure on equivariant simplicial sets
(i.e.: the global projective model structure on functors from the opposite of the orbit category to the classical model structure on simplicial sets)
Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)
Marek Golasiński, Equivariant rational homotopy theory as a closed model category, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)
(for Hamiltonian groups)
Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)
Last revised on October 2, 2020 at 19:57:52. See the history of this page for a list of all contributions to it.