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October 7, 2008

Yet Another Model ω-Question

Posted by Urs Schreiber

In the context of \infty-Lie theory one wants to relate smooth \infty-groupoids with \infty-Lie-algebroids. The relation between the two seems to be induced by two adjunctions: the first relates \infty-groupoids to spaces, the second relates spaces to \infty-Lie algebroids.

With this in mind, it is interesting to compare the two standard model category structures on \infty-groupoids and on \infty-Lie algebroids (i.e. on DGCAs). It seems to me as if there is a chance that the model structure on \infty-Lie algebroids is induced from transporting that on \infty-groupoids through these two adjunctions. Which would nicely fit into the picture.

But is it true? Or something similar?

Below I repeat the question with all technical details given.


First, the adjunctions that I mentioned:

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

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

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

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

Identify the objects dual to qDGCAs with \infty-Lie algebroids, aka L L_\infty-algebroids, by definition, and address the (contravariant) functor CE:L AlgebroidsC qDGCAsCE : 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 Ω \Omega^\bullet be the sheaf on CartesianSpaces which sends each cartesian space UU to the set underlying its qDGCA Ω (U)\Omega^\bullet(U) of smooth differential forms: the Chevalley-Eilenberg qDGCA of the tangent Lie algebroid TUT U.

finite paths: Let Π ω\Pi_\omega be the co-presheaf on CartesianSpaces which sends each cartesian space UU to the (globular) set underlying the fundamental ω\omega-groupoid Π ω(U)\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: kk-morphisms in Π ω(U)\Pi_\omega(U) are thin-homotopy classes of smooth images of the standard kk-disk in UU, 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 ω\omega-categories, we obtain a model structure on ωCategories(Spaces)Sh(CartesianSpaces,ωCategories(Sets))\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 ωCategories(Sets)\omega Categories(Sets) comes from the generating cofibrations (definition 1) given by the inclusions of the boundary of the nn-globe into the nn-globe: inclusions of (n1)(n-1)-spheres into nn-disks.

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

The standard model category structure on DGCAsDGCAs 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)D(n)S(n) \hookrightarrow D(n), where S(n)S(n) is the DGCA on a single degree nn generator, while D(n)D(n) is the DGCA free on the cochain complex with a single generator in degree (n1)(n-1) mapped by the differential to the single generator in degree nn. 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)S(n) is Π n(S(S(n)))=B n, \Pi_n(S(S(n))) = \mathbf{B}^n \mathbb{R} \,, the ω\omega-groupoid which is trivial everywhere except in degree nn, where it is the abelian group of real numbers under addition. (Accordingly, I tend to address S(n)S(n) as CE(b n1u(1)))CE(b^{n-1}u(1))) in the \infty-Lie context.)

- the Lie integration of D(n)D(n) is Π n(S(D(n)))=B n1()=:BEB n2, \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 ω\omega-groupoid trivial everywhere except in degrees (n1)(n-1) and nn.

Including also the morphism in the integration procedure, the integrated cofibration does become a fibration, Π n(S()):(S(n)D(n))(BBB n2BEB n2) \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 ω\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 ω\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 B n2\mathbf{B}^{n-2} \mathbb{R}-bundle EB n2BB n2. \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.

Posted at October 7, 2008 5:28 PM UTC

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