nLab
shape via cohesive path ∞-groupoid

Contents

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

Given a cohesive (∞,1)-topos H\mathbf{H}, then then shape ʃXʃ X of any object XX behaves like the path ∞-groupoid of XX. For some H\mathbf{H} this is true verbatim, in that the shape operation is represented by a geometric singular simplicial complex

ʃXim nMaps(Δ n,X), ʃ X \;\simeq\; \underset{\longrightarrow}{\im}_n Maps(\Delta^n, X) \,,

where on the right we have a homotopy colimit over internal homs from a cohesive incarnation of the n-simplex into XX.

This is true for instance for the case

This is due to (BEBP, see Pavlov, theorem 0.2). For XX a stable homotopy type in the tangent (∞,1)-topos TSmoothGrpdT Smooth\infty Grpd this was observed in Bunke-Nikolaus-Voelkl 13

References

  • Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)

  • Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves Manuscript in preparation.

  • Dmitri Pavlov, Brown representability via concordance (pdf)

Last revised on July 3, 2017 at 15:42:03. See the history of this page for a list of all contributions to it.