Cyclification of Orbifolds

TL;DR

This paper develops a rigorous mathematical framework for the cyclification of orbifolds using higher topos theory, connecting it to equivariant cohomology theories and providing new insights into transgression and elliptic cohomology.

Contribution

It formalizes the cyclification process of orbifolds within cohesive higher topos theory and proves its equivalence to existing models like Ganter/Huan's inertia groupoid.

Findings

Cyclification is shown to be a fundamental base-change construction in higher topos theory.

Ganter/Huan's inertia groupoid models the intrinsic cyclification of orbifolds.

Transgression in group cohomology is implemented via cyclification, linking twists in different cohomology theories.

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.