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

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

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

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

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

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

Definition

Let SmoothSmooth be the category whose

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

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

Let SmoothSmooth be a site with the standard notion of coverage.

Then let

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

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

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

Remarks

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

Proposition

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

Proof

Cofibrant objects in SPSh(C) proj locSPSh(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)SPSh(C).

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

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

hocolim kU k kΔ kU k. hocolim_k U_k \simeq \int^{k} \Delta^k \cdot U_k \,.

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

Given an objectwise acyclic fibration ABA \to B and a morphism kΔ kU kB\int^k \Delta^k \cdot U_k \to B we construct the lift

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

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

To start with, we pick a lift of aB(U 0)a \in B(U_0) \to a^A(U 0)\hat a \in A(U_0).

This yields two lifts π 0 *a^\pi_0^* \hat a and π 1 *a^\pi_1^* \hat a of π 0 *a\pi_0^* a and π 1 *a\pi_1^* a. By the acyclic Kan fibration property this guarantees a lift g^:π 0 *aπ 1 *a\hat g : \pi_0^* a \to \pi_1^* a of the given g:π 0 *aπ 1 *ag : \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.