# nLab shape via cohesive path ∞-groupoid

Contents

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

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

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## Idea

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

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

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

This is true for instance for the case

• $\mathbf{H} =$ Smooth∞Grpd

This is due to (BEBP, see Pavlov, theorem 0.2). For $X$ a stable homotopy type in the tangent (∞,1)-topos $T Smooth\infty Grpd$ 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.