cohesion of global- over G-equivariant homotopy theory in nLab

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Cohesive ∞\infty-Toposes

Homotopy theory

Representation theory

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Definition

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

(2)GloH≔Sh ∞(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 G∈Grp(FinSet)G \,\in\, Grp(FinSet), write

(3)GH≔Sh ∞(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}.

Lemma

(0-truncated objects reflective in slice over GG-orbi-singularity)
For G∈Grp(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 BH→Bi HBGB H \xrightarrow{\;\; B i_H \;\;} B G between delooping groupoids,

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

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

(4)Sngrlt /≺G⊥↩⟶τ 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

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 BH→Bi 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{\,.} }

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.).

Proof

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