# nLab smooth homotopy type

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### 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

In view of the congruence of the notions of homotopy type and type in homotopy type theory it makes sense to refer to an object in a cohesive (∞,1)-topos $\mathbf{H}$ such as as Smooth∞Grpd as a smooth homotopy type or smooth infinity-groupoid. Accordingly then an n-truncated object in $\mathbf{H}$ is a smooth $n$-type.

For instance a smooth 0-type is then an object in the sheaf topos $Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq \mathbf{H}$ of smooth sets.

## References

Revised on November 6, 2014 07:58:13 by Urs Schreiber (89.92.25.218)