Idea

Analogous to how ordinary cohomology is the cohomology in the (∞,1)-topos of infinity-groupoids, smooth cohomology is the cohomology in the (∞,1)-topos of Lie ∞-groupoids.

Recall that each (∞,1)-topos $\mathbf{H}$ comes with its notion of cohomology. Moreover, every $(\infty,1)$-topos is an (∞,1)-category of (infinity,1)-sheaves, i.e. of ∞-stacks on some site $C$:

$\mathbf{H} = Sh_\infty(C) \,.$

If $C = Op(X)$ is a category of open subsets of some topological space $X$, then $Sh_\infty(C)$ is a petit topos whose objects may be thought of as generalized spaces over $X$.

More generally, when $C$ is a category of general test spaces such as the category Diff of all manifolds, then $Sh_\infy(C)$ is a gros topos whose objects play the role of generalized spaces modeled on the objects in $C$, following the general idea of space and quantity.

Accordingly, smooth cohomology we take to be the cohomology encoded by an $(\infty,1)$-topos $Sh_\infty(C)$ for $C$ a category of smooth test spaces, such as Diff.

More concretely, it turns out that for the smooth structure of the objects of $Sh_\infty(C)$ only the stalks of the ordinary Grothendieck topos $Sh(C)$ matters and that it is sufficient and useful to restrict Diff to a subcategory that contains only ball-like manifolds.

Definition

Let $Smooth$ be the category whose

• objects are pointed smooth manifolds with boundary that are homeomorphic to a ball $D^n$, for $n \in \mathbb{N}$;

• morphisms are point-preserving morphisms of smooth manifolds between these.

Let $Smooth$ be a site with the standard notion of coverage.

Then let

$\mathbf{H}_{smooth} := Sh_\infty(Smooth)$

be the hypercomplete (∞,1)-category of (∞,1)-sheaves on $Smooth$.

The notion of cohomology inside $\mathbf{H}_{smooth}$ we call smooth cohomology.

Properties

Since for computing smooth cohomology we are interested in using the projective local model structure on simplicial presheaves, we collect some useful facts about this.

Definition (good cover)

For $C$ some site, $X \in SPSh(C)$, a Cech cover $\pi : U \to X$ of $X$ – i.e. a morphism such that the homotopy colimit $hocolim_k U_\bullet$ over the Cech never of $U$ is weakly equivalent to $X$ – is called a good cover if $U_\bullet$ is degreewise a coproduct over representables.

Proposition

In the local projective model structure $SPSh(C)_{proj}^{loc}$ the objects $\int^{[k] \in \Delta} \Delta^k \cdot U_k$ obtained from a good cover are cofibrant.

Proof

Cofibrant objects in $SPSh(C)_{proj}^{loc}$ are those objects such that maps out of them lift through all objectwise acyclic Kan fibrations.

This means that in particular all representables are cofibrant in $SPSh(C)$.

By assumption $U_k$ is a coproduct over representables hence cofibrant. Since $U_\bullet$ is assumed to be a Cech cover, the inclusion of degenerate $k$-simplices into all $k$-simplices is the inclusion of $k$-fold intersections that contain self-intersections into all $k$-fold intersections. So it is an inclusion of the form $\coprod_{i\in I} S_i \to \coprod_{j \in J} S_j$ of coproducts over representables for an inclusion $I \hookrightarrow J$ of index sets. Therefore the inlcusion is itself a cofibration. This means that $U_\bullet$ is Reedy cofibrant in $[\Delta^{op}, SPSh(C)^{loc}_{proj}]_{Reedy}$.

Therefore by the Bousfield-Kan map the hocolim over $U^\bullet$ may be expressed as an ordinary coend

$hocolim_k U_k \simeq \int^{k} \Delta^k \cdot U_k \,.$

A morphism $\int^k \Delta^k \cdot U_k \to B$ is the same as an element in $\int_{k} SSet(\Delta^k, B(U_k))$, where we use notation as if $U_k$ were a single representative.

Given an objectwise acyclic fibration $A \to B$ and a morphism $\int^k \Delta^k \cdot U_k \to B$ we construct the lift

$\array{ && A \\ & \nearrow & \downarrow \\ \int^k \Delta^k \cdot U_k &\to& B }$

by inductively lifting the corresponding element in $\int_k SSet(\Delta^k, B(U_k))$ to an element in $\int_k SSet(\Delta^k, A(U_k))$.

To start with, we pick a lift of $a \in B(U_0)$ \to $\hat a \in A(U_0)$.

This yields two lifts $\pi_0^* \hat a$ and $\pi_1^* \hat a$ of $\pi_0^* a$ and $\pi_1^* a$. By the acyclic Kan fibration property this guarantees a lift $\hat g : \pi_0^* a \to \pi_1^* a$ of the given $g : \pi_0^* a \to \pi_1^* a$.

Created on August 21, 2009 at 17:55:11. See the history of this page for a list of all contributions to it.