nLab
global orbit category

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The global orbit category is the unification of all GG-orbit categories as the groups GG are allowed to vary.

The (∞,1)-presheaves over the global orbit category provide the base (∞,1)-topos over which the global equivariant homotopy theory (see there) is a cohesive (∞,1)-topos (Rezk 14).

Definition

The following defines the global equivariant indexing category GloGlo.

Definition

Write GloGlo for the (∞,1)-category whose

(Rezk 14, 2.1)

Remark

Equivalent models for the global indexing category, def. include the category “O glO_{gl}” of (May 90). Another variant is O gl\mathbf{O}_{gl} of (Schwede 13).

(Rezk 14, 2.4, 2.5)

The following is the global orbit category.

Definition

Write

OrbGlo Orb \longrightarrow Glo

for the non-full sub-(∞,1)-category of the global indexing category, def. , on the injective group homomorphisms.

(Rezk 14, 4.5)

Properties

Relation to the local orbit category

The slice (∞,1)-category of the global orbit category over BG\mathbf{B}G is the local orbit category of GG

Orb /BGOrb G. Orb_{/\mathbf{B}G} \simeq Orb_G \,.

Relation to orbispaces and GG-spaces

The (∞,1)-category of (∞,1)-presheaves over the global orbit category is that of orbispaces.

Accordingly, by the discussion here, the slice (∞,1)-topos of orbispaces over BG\mathbf{B}G is that of G-spaces

PSh (Orb) /BGPSh (Orb /BG)PSh (Orb G)GSpace PSh_\infty(Orb)_{/\mathbf{B}G} \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb_G) \simeq G Space

(where the last step is Elmendorf's theorem).

Relation to equivariant homotopy theory

The (∞,1)-category of (∞,1)-presheaves on the global equivariant indexing category is the global equivariant homotopy theory and under the canonical projection is a cohesive (∞,1)-topos over ∞Grpd. Its slice (∞,1)-topos over the terminal orbispace is cohesive over orbispaces

PSh (Glo) /𝒩PSh (Orb). PSh_\infty(Glo)_{/\mathcal{N}} \to PSh_\infty(Orb) \,.

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G

References

See also

  • Stefan Schwede, Global homotopy theory, 2013 (pdf)

  • Peter May, Some remarks on equivariant bundles and classifying spaces, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

Last revised on August 31, 2018 at 16:11:29. See the history of this page for a list of all contributions to it.