nLab SmoothManifolds

Definition

Definition

$SmoothManifolds$ is the category whose

• objects are smooth manifolds

• morphisms are smooth functions between these.

Similarly, for $n \in \mathbb{N}$

$DifferentiableManifolds_n$ is the category whose

• objects are differentiable manifolds

• morphisms are differentiable functions between these

for $n$-fold differentiabiliy.

Each of these categories is also commonly denoted $Man$ or $Mfd$ or Diff etc.

Properties

As a site

Proposition

The category $SmoothManifolds$ 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 $SmoothManifolds$ is given by CartSp${}_{smooth}$.

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 $SmoothManifolds$ is a cohesive topos.

The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over $SmoothManifolds$ 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

$$Sh(SmoothManifolds) \simeq Sh(CartSp_{smooth}) \,.$$

By the discussion at CartSp we have that $CartSp_{smooth}$ 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 $Sh(SmoothManifolds)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves $\hat Sh_{(\infty,1)}(SmoothManifolds)$ (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions

$$\hat Sh_{(\infty,1)}(SmoothManifolds) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth}) \,.$$

Now CartSp${}_{smooth}$ is even an ∞-cohesive site. By the discussion there it follows that $Sh_{(\infty,1)}(CartSp_{smooth})$ (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

$$\cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth}) \,.$$

Remark

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

The cohesive (∞,1)-topos

$$Smooth \infty Grp := Sh_{(\infty,1)}(SmoothManifolds) \simeq Sh_{(\infty,1)}(CartSp_{smooth})$$

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

$$Sh_{(2,1)}(SmoothManifolds) \simeq Sh_{(2,1)}(CartSp_{smooth}) \simeq \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})$$

Related concepts

• CartSp${}_{top}$ , TopMfd

• CartSp${}_{smth}$

• Diff

category: category