nLab cohesion of global- over G-equivariant homotopy theory

Idea

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

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

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

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

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 $G$-orbit category inside the $\prec\!\! G$-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)$$\array{ Snglrt &\coloneqq& Grpd^{fin}_{1, \geq 1} &\xhookrightarrow{\;}& Grp_\infty \\ \prec\!\!G &\mapsto& B G }$$

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

1. 0-connected

2. 1-truncated

3. $\pi$-finite,

hence, equivalently:

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

Remark

(terminology: singularities vs. “global orbits”)
The $(2,1)$-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 $G$-equivariant homotopy theory)
For $\mathbf{H}$ an $\infty$-topos write

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

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

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

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

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

Statement

The adjoint pair between sites

Lemma

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

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

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

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

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

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

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

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

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

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

Proof

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

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

$$\array{ \big( Snglrt_{/\prec G} \big)_{\tau_0} &\simeq& \big( (Grp_{1,\geq 1})_{/B G} \big)_{\tau_0} & \xhookrightarrow{\;\; hofib(-) \;\;} & G Set \\ \left( \array{ \prec\!\!H \\ \downarrow^{\mathrlap{ \prec i_H }} \\ \prec\!\!G } \;\; \right) &\mapsto& \left( \array{ B H \\ \downarrow^{\mathrlap{ B i_H }} \\ B G } \;\;\; \right) &\mapsto& G/H \mathrlap{\,.} }$$

Lemma

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

Proof

The category on the right is equivalently the $G$-orbit category (by Lem. ) whose free coproduct completion is (using here our assumption that $G$ is a discrete group) the category of all $G$-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 $\infty Grpd_{/B G}$. Hence the coproduct-preserving extension of $\tau_0$ 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

($G$-orbits are reflective in slice over $G$-orbi-singularity)
For $G \,\in\, Grp(FinSet)$ the $G$-orbit category is canonically a full sub-$\infty$-category of the slice of $Snglrt$ (Def. ) over $\prec\!\! G$ (1) whose reflector $\tau_0$ 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 $Glo(\mathbf{H})$ over $\prec\!\!G$ is cohesive over $G\mathbf{H}$)
For $\mathbf{H}$ any $\infty$-topos and $G \,\in\, Grp(FinSet)$, the slice of the global equivariant homotopy theory $Glo(\mathbf{H})$ (2) over the $G$-orbi-singularity $\prec\!\!G$ (1) is cohesive over the $G$-equivariant homotopy theory $G\mathbf{H}$ (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:

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

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

Related concepts

• singular cohesion

References

The observation is due to:

• Charles Rezk, Global Homotopy Theory and Cohesion, 2014 (pdf, pdf)

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

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

• Hisham Sati, Urs Schreiber: Proper Orbifold Cohomology [arXiv:2008.01101]

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

Diagrams and discussion as presented above are taken from:

• Hisham Sati, Urs Schreiber: Equivariant principal $\infty$-bundles [arXiv:]