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cohesion of global- over G-equivariant homotopy theory

Contents

Context

Cohesive \infty-Toposes

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

Contents

Idea

In broad generality, the relation between (non-stable) global equivariant homotopy theory and GG-equivariant homotopy theory for any fixed admissible equivariance group GG may be organized and formalized as follows:

The slice of global equivariant homotopy theory (Def. ) over the archetypical GG-orbi-singularity G\prec\!\! G (Def. ) is cohesive over GG-equivariant homotopy theory. In particular:

  1. GG-equivariant homotopy theory faithfully embeds into the G\prec\!\! G-slice of the global theory in two different ways, one of them interpreted as the inclusion of G-spaces XX as global orbispaces XGX \!\sslash\! G,

  2. these inclusions have a compatible pair of reflections, one of which forms the spaces of sections FixLocΓ GFixLoc \,\coloneqq\, \Gamma_{{}_{\prec G}} over the GG-singularity G\prec\!\! G:

This observation is due to Rezk 2014. Below we amplify how this is a formal consequence (as in Rezk 14, Sec. 7.1-7.2) of the evident reflection (Prop. ) of the GG-orbit category inside the G\prec\!\! G-slice of the “global orbit category” (see Rem. below), which immediately implies (Prop. ) the cohesive relation, by \infty -Kan extension.

Preliminaries

Remark

equivariance groups)
Throughout we consider discrete equivariance groups, not necessarily finite (though subgroups of interest will be finite, as in proper equivariant homotopy theory). Much of the following also works for equivariance groups which are compact Lie groups, but some definitions become a tad more laborious to state and the relation to smooth cohesion gets messed up.

Definition

(canonical orbi-singularities) We write

(1)Snglrt Grpd 1,1 fin Grp G BG \array{ Snglrt &\coloneqq& Grpd^{fin}_{1, \geq 1} &\xhookrightarrow{\;}& Grp_\infty \\ \prec\!\!G &\mapsto& B G }

for the full sub- \infty -category of all \infty -groupoids on those that are

  1. 0-connected

  2. 1-truncated

  3. π \pi -finite,

hence, equivalently:

the full sub-(2,1)-category of Grpd on the delooping groupoids BGBG\mathbf{B}G \,\simeq\, B G of finite groups GG, with functors as 1-morphisms and natural transformations (necessarily natural isomorphisms) as 2-morphisms.

Remark

(terminology: singularities vs. “global orbits”)
The (2,1)(2,1)-category in Def. is sometimes called the global orbit category, though other times that name refers to its non-full subcategory on the faithful functors. But neither of these two actually is a “category of orbits” – instead, orbit categories are full subcategories of their slices, by Prop. below. On the other hand, application of global equivariant homotopy theory to orbifolds identifies the category in Def. with the category of archetypical local models for orbi-singularities. Therefore the choice of notation in Def. .

Definition

(global- and GG-equivariant homotopy theory)
For H\mathbf{H} an \infty write

(2)GloHSh (Singlrt,H) Glo \mathbf{H} \;\coloneqq\; Sh_\infty( Singlrt ,\, \mathbf{H} )

for the \infty -topos of \infty -presheaves on SnglrtSnglrt (Def. ), to be called the global equivariant homotopy theory over H\mathbf{H}.

Moreover, for GGrp(FinSet)G \,\in\, Grp(FinSet), write

(3)GHSh (GOrbt,H) G{}\mathbf{H} \;\coloneqq\; Sh_\infty( G{}Orbt ,\, \mathbf{H} )

for the \infty -topos of \infty -presheaves on the GG-orbit category GOrbtG{}Orbt (Def. ), to be called the GG-equivariant homotopy theory over H\mathbf{H}.

Statement

The adjoint pair between sites

Lemma

(0-truncated objects reflective in slice over GG-orbi-singularity)
For GGrp(FinSet)G \,\in\, Grp(FinSet), the full sub- \infty -category of the slice of SnglrtSnglrt (Def. ) over G\prec\!\! G (1) on the 0-truncated objects

  1. consists precisely of the faithful functors BHBi HBGB H \xrightarrow{\;\; B i_H \;\;} B G between delooping groupoids,

    hence those which are deloopings of subgroup-inclusions Hi HGH \xhookrightarrow{\;\; i_H \;\;} G;

  2. is reflective, with reflector being the image-factorization of group homomorphisms:

(4)Sngrlt /Gτ 0(Snglrt /G) τ 0{H K i H i K G} Sngrlt_{/\prec G} \underoverset {\underset{}{\hookleftarrow}} {\overset{\tau_0}{\longrightarrow}} {\;\;\;\;\bot\;\;\;\;} \big( Snglrt_{/\prec G} \big)_{\tau_0} \;\simeq\; \left\{ \array{ \prec\!\!H && \longrightarrow && \prec\!\!K \\ & {}_{\mathllap{\prec i_H}}\searrow && \swarrow_{\mathrlap{\prec i_K}} \\ && \prec\!\!G } \right\}

Proof

That 0-truncated morphisms between 1-groupoids are equivalently the faithful functors is this Prop.. With this in hand, it is immediate to check the hom-equivalence (here just a natural bijection of hom-sets) which characterizes the adjunction.

Lemma

(GG-orbits as 0-truncated objects in slice over GG-orbi-singularity)
For GGrp(FinSet)G \,\in\, Grp(FinSet) there is an equivalence of \infty -categories (here in fact: an equivalence of categories) between

  1. the 0-truncated objects in the slice of SnglrtSnglrt (Def. ) over G\prec\!\! G (1),

  2. the GG-orbit category (the full subcategory of G-sets on the transitive actions, hence the coset sets G/HG/H):

(Sngrtl /G) τ 0 GOrbt (H i H G) G/H. \array{ (Sngrtl_{/\prec G})_{\tau_0} & \xleftrightarrow{\;\; \sim \;\;} & G{}Orbt \\ \left( \array{ \prec\!\!H \\ \downarrow^{\mathrlap{\prec i_H}} \\ \prec\!\!G } \;\; \right) &\mapsto& G/H } \,.

Proof

It is straightforward, to check this directly. But it also follows abstractly by this Prop. about the general relation between slicing over BGB G and \infty -actions of GG:

The functor which assigns to BHBi HBGB H \xrightarrow{\;\; B i_H\;\;} B G its homotopy fiber is a fully faithful functor into the G-sets among all GG- \infty -actions (by the 0-truncation condition). But the homotopy fiber of Bi HB i_H is the coset set (by this Example):

(Snglrt /G) τ 0 ((Grp 1,1) /BG) τ 0 hofib() GSet (H i H G) (BH Bi H BG) G/H. \array{ \big( Snglrt_{/\prec G} \big)_{\tau_0} &\simeq& \big( (Grp_{1,\geq 1})_{/B G} \big)_{\tau_0} & \xhookrightarrow{\;\; hofib(-) \;\;} & G Set \\ \left( \array{ \prec\!\!H \\ \downarrow^{\mathrlap{ \prec i_H }} \\ \prec\!\!G } \;\; \right) &\mapsto& \left( \array{ B H \\ \downarrow^{\mathrlap{ B i_H }} \\ B G } \;\;\; \right) &\mapsto& G/H \mathrlap{\,.} }

Lemma

The free coproduct completions of the (2,1)(2,1)-categories (4) have finite products and the unique coproduct-preserving extension of τ 0\tau_0 to these preserves finite products.

Proof

The category on the right is equivalently the GG-orbit category (by Lem. ) whose free coproduct completion is (using here our assumption that GG is a discrete group) the category of all G G -sets (as in this remark).

Similarly, the free coproduct completion of the category on the left is readily seen to be that of all 1-truncated in Grpd /BG\infty Grpd_{/B G}. Hence the coproduct-preserving extension of τ 0\tau_0 to these is just the 0-truncation functor in this slice \infty -topos and as such preserves finite products (by this Prop., see this Exp.).

In conclusion:

Proposition

(GG-orbits are reflective in slice over GG-orbi-singularity)
For GGrp(FinSet)G \,\in\, Grp(FinSet) the GG-orbit category is canonically a full sub- \infty -category of the slice of SnglrtSnglrt (Def. ) over G\prec\!\! G (1) whose reflector τ 0\tau_0 preserves finite products when extended to the free coproduct completions, where all finite products exist:

Proof

By the immediate combination of Lem. with Lem. and Lem. .

The adjoint quadruple between homotopy theories

Proposition

(slice of Glo(H)Glo(\mathbf{H}) over G\prec\!\!G is cohesive over GHG\mathbf{H})
For H\mathbf{H} any \infty -topos and GGrp(FinSet)G \,\in\, Grp(FinSet), the slice of the global equivariant homotopy theory Glo(H)Glo(\mathbf{H}) (2) over the GG-orbi-singularity G\prec\!\!G (1) is cohesive over the GG-equivariant homotopy theory GHG\mathbf{H} (3) in that there exists an adjoint quadruple of \infty -functors of the form

where those going to the left are fully faithful and the top one preserves finite homotopy products.

Proof

The general fact that \infty -slices of \infty -presheaves are \infty -presheaves on the \infty -slice (by this Prop. or this Prop.) means in the present case that the operation of extracting systems of fixed loci is an equivalence of (infinity,1)-categories as follows:

(GloH) /GPSh (Sngrlt,H) /GPSh (Sngrlt /G,H). (Glo \mathbf{H})_{/\prec G} \;\coloneqq\; PSh_\infty \big( Sngrlt ,\, \mathbf{H} \big)_{/\prec G} \;\; \simeq \;\; PSh_\infty \big( Sngrlt_{/\prec G} ,\, \mathbf{H} \big) \,.

With this, the statement follows – via this Prop. – by \infty -Kan extension of the adjoint pair from Prop. .

References

The observation is due to:

where further properties of this cohesive situation are proven, revolving around further characterization of the full inclusion of GG-orbispaces.

Some of the above notation (e.g. for “G\prec\!\! G”) follows:

which expands on application of the above singular cohesion – in the special case G=1G = 1 but combined with smooth cohesion – to differential-geometric orbifolds and orbifold cohomology.

Diagrams and discussion as presented above are taken from:

Last revised on October 26, 2021 at 04:50:38. See the history of this page for a list of all contributions to it.