smooth homotopy type

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 an cohesive (∞,1)-topos $\mathbf{H}$ which models cohesive homotopy type theory for a kind of *smooth* cohesion – such as $\mathbf{H} =$ Smooth∞Grpd – as a *smooth type*. Or *smooth homotopy type*. 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 $Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq \mathbf{H}$. (Also sometumes called a *smooth space*.)

Revised on May 4, 2013 20:37:42
by Urs Schreiber
(150.212.92.50)