nLab
global equivariant indexing category

Context

Homotopy theory

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Special notions

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Contents

Idea

The global equivariant indexing category is the full subcategory of topological ∞-groupoids on those which are deloopings of compact Lie groups, with hom-spaces being the geometric realization of the internal homs there.

The (∞,1)-presheaves over the global equivariant indexing category is the global equivariant homotopy theory. This is a cohesive (∞,1)-topos over ∞Grpd (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

Glo /BGOrb G. Glo_{/\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 oribispace is cohesive over orbispaces

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

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 December 9, 2015 at 09:33:54. See the history of this page for a list of all contributions to it.