nLab global orbit category



Homotopy theory

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

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

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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).


The following defines the global equivariant indexing category GloGlo.


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

(Rezk 14, 2.1)


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.



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)


Relation to the local orbit category

The slice (∞,1)-category of the global orbit category OrbOrb (the version with faithful functors as morphisms) 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


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 October 8, 2021 at 06:33:45. See the history of this page for a list of all contributions to it.