nLab
Diff

Definition

Definition

$\mathrm{Diff}$ (also called $\mathrm{Man}$ or $\mathrm{Mfd}$) is the category whose

• objects are (paracompact) smooth manifolds;

• morphisms are smooth functions between these.

Properties

As a site

Proposition

The category $\mathrm{Diff}$ becomes a large site by equipping it with the coverage consisting of open covers.

This is an essentially small site: a dense sub-site for $\mathrm{Diff}$ is given by CartSp${}_{\mathrm{smooth}}$.

Proof

The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).

For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.

Corollary

The sheaf topos over $\mathrm{Diff}$ is a cohesive topos.

The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over $\mathrm{Diff}$ is a cohesive (∞,1)-topos.

Proof

For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories

$$\mathrm{Sh}(\mathrm{Diff})\simeq \mathrm{Sh}({\mathrm{CatSp}}_{\mathrm{smooth}}).$$Sh(Diff) \simeq Sh(CatSp_{smooth}) \,.

By the discussion at CartSp we have that ${\mathrm{CartSp}}_{\mathrm{smooth}}$ is a cohesive site. By the discussion there the claim follows.

For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves $\mathrm{Sh}(\mathrm{Diff}{)}_{\mathrm{loc}}^{{\Delta }^{\mathrm{op}}}$ is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves ${\widehat{\mathrm{Sh}}}_{(\mathrm{\infty },1)}(\mathrm{Diff})$ (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions

$${\widehat{\mathrm{Sh}}}_{(\mathrm{\infty },1)}(\mathrm{Diff})\simeq {\widehat{\mathrm{Sh}}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}}).$$\hat Sh_{(\infty,1)}(Diff) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth}) \,.

Now CartSp${}_{\mathrm{smooth}}$ is even an ∞-cohesive site. By the discussion there it follows that ${\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}})$ (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally

$$\cdots \simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}}).$$\cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth}) \,.

Remark

The cohesive topos $\mathrm{Sh}(\mathrm{Diff})\simeq \mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{smooth}})$ is in particular the home of diffeological spaces. See there for more details.

The cohesive (∞,1)-topos

$$\mathrm{Smooth}\mathrm{\infty }\mathrm{Grp}:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Diff})\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}})$$Smooth \infty Grp := Sh_{(\infty,1)}(Diff) \simeq Sh_{(\infty,1)}(CartSp_{smooth})

is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.

The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos

$${\mathrm{Sh}}_{(2,1)}(\mathrm{Diff})\simeq {\mathrm{Sh}}_{(2,1)}({\mathrm{CartSp}}_{\mathrm{smooth}})\simeq {\tau }_{\le 1}{\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}})$$Sh_{(2,1)}(Diff) \simeq Sh_{(2,1)}(CartSp_{smooth}) \simeq \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})

Related concepts

• CartSp${}_{\mathrm{top}}$ , TopMfd

• CartSp${}_{\mathrm{smth}}$, SmoothMfd

category: category