The n-Category Café

First, the adjunctions that I mentioned:

Let $CartesianSpaces$ be the full subcategory of $SmoothManifolds$ on objects of the form $\mathbb{R}^n$ for all $n$, equipped with their standard smooth structure. This is naturally a site.

Let sheaves on this be our model for “smooth spaces”, $Spaces := Sheaves(CartesianSpaces)$ and let strict $\infty$-groupoids internal to $Spaces$, $\omega Groupoids(Spaces)$, be our model for “smooth $\infty$-groupoids”.

Finally, let co-presheaves on $CartesianSpaces$ with values in qDGCAs be our model for “$C^\infty$ quasi-free differential graded commutative algebras”, $C^\infty qDGCAs := qDGCAs^{CartesianSpaces}$.

Here a qDGCA is a tuple consisting of i) a commutative algebra $A$, ii) a non-negatively graded cochain complex $g^*$of $A$-modules, iii) a differential $d$ on the free-over-$A$ graded-antisymmetric algebra $\wedge^\bullet_A g^*$ extending the free differential induced from $g^*$.

Identify the objects dual to qDGCAs with $\infty$-Lie algebroids, aka $L_\infty$-algebroids, by definition, and address the (contravariant) functor $CE : L_\infty Algebroids \stackrel{\simeq}{\to} C^\infty qDGCAs$ as “forming the Chevalley-Eilenberg qDGCA”.

There are then two important “ambimorphic” objects: finite paths and infinitesimal paths.

infinitesimal paths: Let $\Omega^\bullet$ be the sheaf on CartesianSpaces which sends each cartesian space $U$ to the set underlying its qDGCA $\Omega^\bullet(U)$ of smooth differential forms: the Chevalley-Eilenberg qDGCA of the tangent Lie algebroid $T U$.

finite paths: Let $\Pi_\omega$ be the co-presheaf on CartesianSpaces which sends each cartesian space $U$ to the (globular) set underlying the fundamental $\omega$-groupoid $\Pi_\omega(U)$. I can give more details on that by private LaTeX mail if desired, but only the general idea should be relevant here: $k$-morphisms in $\Pi_\omega(U)$ are thin-homotopy classes of smooth images of the standard $k$-disk in $U$, suitably well behaved along the boundary so that composition by gluing makes sense.

By homming into or out of these ambimorphic objects, one obtains the functors indicated in the above diagram. As far as I understand what Todd Trimble taught me, this yields two examples of Stone-like dualities. In particular two adjunctions. But I am hoping Todd finds the time to make a sanity check of my statements here.

Now concerning the model structure:

Using the result from T. Beke: Sheafifiable homotopy model categories, about model structures on categories of sheaves, applied to the folk model structure on $\omega$-categories, we obtain a model structure on $\omega Categories(Spaces) \simeq Sh(CartesianSpaces, \omega Categories(Sets))$ by directly internalizing everything from sets to sheaves. I suppose.

Now, the folk model structure on $\omega Categories(Sets)$ comes from the generating cofibrations (definition 1) given by the inclusions of the boundary of the $n$-globe into the $n$-globe: inclusions of $(n-1)$-spheres into $n$-disks.

Compare this to the standard model category structure on $L_\infty Algebroids \simeq C^\infty qDGCAs$, which I suppose we obtain again using Beke’s result from the standard model strure on $DGCAs$ (But careful here: do I need to consider co-sheaves instead of co-presheaves? And even then?).

The standard model category structure on $DGCAs$ is for instance recalled in K. Hess: Rational homotopy theory: a brief introduction on page 6:

Here the generating cofibrations are the inclusions $S(n) \hookrightarrow D(n)$, where $S(n)$ is the DGCA on a single degree $n$ generator, while $D(n)$ is the DGCA free on the cochain complex with a single generator in degree $(n-1)$ mapped by the differential to the single generator in degree $n$. Geometrically, this again looks like spheres and disks, only that everything is dualized now!

Is there a relation?

To check, we can chase these generating cofibrations through the above diagram and see what results. We get:

- the $\infty$-Lie integration of $S(n)$ is $$\Pi_n(S(S(n))) = \mathbf{B}^n \mathbb{R} \,,$$ the $\omega$-groupoid which is trivial everywhere except in degree $n$, where it is the abelian group of real numbers under addition. (Accordingly, I tend to address $S(n)$ as $CE(b^{n-1}u(1)))$ in the $\infty$-Lie context.)

- the Lie integration of $D(n)$ is $$\Pi_n(S(D(n))) = \mathbf{B}^{n-1} (\mathbb{R} \to \mathbb{R}) =: \mathbf{B} \mathbf{E} \mathbf{B}^{n-2} \mathbb{R} \,,$$ the $\omega$-groupoid trivial everywhere except in degrees $(n-1)$ and $n$.

Including also the morphism in the integration procedure, the integrated cofibration does become a fibration, $$\Pi_n(S(-)) : (S(n) \hookrightarrow D(n) ) \mapsto ( \mathbf{B}\, \mathbf{B}\mathbf{B}^{n-2} \mathbb{R} \leftarrow \mathbf{B}\, \mathbf{E} \mathbf{B}^{n-2} \mathbb{R} )$$ as one checks easily using the characterization of fibrations of $\omega$-groupoids as in section 2 of Brown, Golasinski: A model structure for the homotopy theory of crossed complexes (and assuming that their model structure coincides on $\omega$-groupoids with the folk-model structure, something everybody seems to expect but nobody has shown, as far as I am aware).

In fact, if we loop this once this becomes the universal $\mathbf{B}^{n-2} \mathbb{R}$-bundle $$\mathbf{E} \mathbf{B}^{n-2} \mathbb{R} \to \mathbf{B} \mathbf{B}^{n-2} \mathbb{R} \,.$$

So it seems something nice is going on.