Schreiber smooth cohomology

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.

Remarks

• In practice one uses models for ∞-stack (∞,1)-toposes to work with $\mathbf{H}_{smooth}$. Notice that one of the possible model structures on simplicial (pre)sheaves is the projective local model structure on simplicial sheaves.

  In this model, an object is an infinity-groupoid internal to smooth spaces.

• To refine smooth cohomology to differential cohomology? we refine it to a structured (infinity,1)-topos using the path infinity-groupoid?.

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 $g : \pi_0^* a \to \pi_1^* a$.