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Cyclification of Orbifolds

Hisham Sati, Urs Schreiber

Abstract

Inertia orbifolds homotopy-quotiented by rotation of geometric loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant Tate-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such cyclified stacks has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type. Following our previous formulation of transgression of cohomological charges ("double dimensional reduction"), we explain how cyclification of infinity-stacks is a fundamental and elementary base-change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy type theory). We prove that Ganter/Huan's extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant Tate-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold. As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant Tate-elliptic cohomology; and we close by explaining how this should relate to elliptic M5-brane genera, under our previously formulated Hypothesis H.

Cyclification of Orbifolds

Abstract

Inertia orbifolds homotopy-quotiented by rotation of geometric loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant Tate-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such cyclified stacks has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type. Following our previous formulation of transgression of cohomological charges ("double dimensional reduction"), we explain how cyclification of infinity-stacks is a fundamental and elementary base-change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy type theory). We prove that Ganter/Huan's extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant Tate-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold. As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant Tate-elliptic cohomology; and we close by explaining how this should relate to elliptic M5-brane genera, under our previously formulated Hypothesis H.
Paper Structure (11 sections, 36 theorems, 120 equations)

This paper contains 11 sections, 36 theorems, 120 equations.

Table of Contents

  1. Introduction and Overview
  2. Cyclic inertia orbifolds
  3. GRH's extended intertia orbifold
  4. Cyclic intertia $\infty$-Groupoids
  5. Reproducing GRH's extended inertia orbifold
  6. Transgression as cyclification
  7. Integral 4-classes of equivariant 4-Cohomotopy
  8. Appendix: Technical background
  9. Some simplicial homotopy theory
  10. Group cohomology
  11. Notions of cohesive $\infty$-topos theory

Key Result

Theorem 2.3

For $\mathcal{X} \simeq X \!\sslash\! G \;\in\; \mathrm{Smth} \mathrm{Grpd} _\infty$ any good orbifold GoodOrbifold, GRH's inertia orbifold SkeletonOfHuanInertiaStackOfGoodOrbifold is equivalently the $\mathcal{T} \coloneqq S^1_{\mathrm{coh}}$-cyclic inertia $\infty$-stack in the general sense of SI

Theorems & Definitions (85)

  • Remark 2.1: Components of inertia
  • Remark 2.2: Cohesion knows about essentially constant loops
  • Theorem 2.3: GRH's extended inertia groupoid models the cyclified orbifold
  • proof
  • Remark 2.4: Subtleties
  • Lemma 2.5: Comparison morphism from GRH's inertia orbifold to cyclic orbifold
  • proof
  • Proposition 2.6: Comparison morphism is pullback of shape unit
  • proof
  • Lemma 2.7: Cofibrant resolution of circle group
  • ...and 75 more