The n-Category Café

Instead he thinks of the action property in terms of connections. But this is really taking the terminology from just the special case where the $L_\infty$-algebroid in question is the tangent Lie algebroid $T X$ of some space, or something alike. Representations of this tangent-like Lie algebroids happen to be flat connections on vector bundles. This is precisely the differential version of the statement which in Lie integrated version says [SW] that representation of the fundamental groupoid $\Pi_1(X)$ of a space are flat vector bundles with connection. This, in turn, is the oidification of the statement that flat vector bundles on $X$ are given by representations of the fundamental group $\pi_1(X,x) = Aut_{\Pi_1(X)}(x) \,.$

A while ago, after a little back and forth here on the blog, I had proposed what I considered to be a good notion (as opposed to just a working notion) of representations of $L_\infty$-algebroids in On $\infty$-Lie theory. It turns out that this is equivalent to Jonathan Block’s definition.

Some words on the two aspects of actions.

The point is that there are two different somewhat complementary ways to look at the concept of representations (this has emerged here in blog discussion in the form I give now in discussion within John Baez’s Geometric Representation Theory seminar and in particular builds on observations David Corfield made here.) The first point of view is the representation as an action. Representations of an $n$-groupoid $Gr$ are $n$-functors $$\rho : Gr \to (n-1)Cat \,.$$ The other point of view is the representation as the weak or homotopy quotient $V//Gr$ induced by the action. This can nicely be thought of, as David Corfield pointed out, as the pullback of the universal $(n-1)\mathrm{Cat}$-bundle $$\array{ F \\ \downarrow \\ T_{pt} (n-1)Cat \\ \downarrow \\ (n-1)Cat }$$ along the action functor $$\array{ V &\to&F \\ \downarrow &&\downarrow \\ V//Gr &\to&T_{pt} (n-1)Cat \\ \downarrow &&\downarrow \\ Gr &\stackrel{\rho}{\to}&(n-1)Cat }$$ which can be read (along the lines of David Roberts and my discussion here) as the $n$-categorical version of the $n$-bundle which is $\rho$-associated to the universal $Gr$ $n$-bundle. In any case, the whole information of the action is just as well encoded in the extension of the $n$-groupoid $Gr$ by the $(n-1)$category $V$ it acts on in terms of the action $n$-groupoid $V//Gr$.

For the simple toy case where $Gr = \mathbf{B}G$ is a one-object groupoid a long pedagogical discussion of this is for instance here.

Now it just so happens that the differential algebraic formulation of the analogue of the action $n$-functor is awkward, while the differential algebraic formulation of the action Lie $n$-groupoid extension is straightforward:

The definition

For $CE_A(g) \simeq \wedge^\bullet_A N$ a qDGCA with commutative algebra $A$ in degree 0 and $N$ an $\mathbb{N}$-graded $A$-module (I’ll consider all DGAs here as Chevalley-Eilenberg-algebras of $L_\infty$-algebroids $g$ and I restrict attention to the (graded)-commutative case, since the noncommutative case is then straightforward and just adds another layer of sophistication which is not really relevant for the central ideas here) and $V^\bullet$ a complex of $A$-module (the CE-algebra of the action $L_\infty$-algebroid of) an action of $g$ on $V$ is an extension of DGCAs

$$\wedge_A^\bullet V^\bullet \leftarrow CE_A(g,V) \leftarrow CE_A(g) \,.$$

(Here and throughout I am writing, $\wedge^\bullet_A(\cdot)$ for graded-symmetric tensor powers over $A$.)

Unwrapping this it says that as a GCA the middle piece is of the form $$\wedge^\bullet_A ( V \oplus g^*) \,,$$ that the differential on $\wedge^\bullet_A g^*$ is that of $CE_A(g)$ and the co-nullary part with respect to $g^*$ of the differential on $V$ is that of $V^\bullet$, while, finally, the part $$d : V \to V \otimes g^*$$ encodes the (dual map of) the action of $g$ on $V$ (or possibly on $V^*$ if that is preferred).

A quick review of Jonathan Block’s definition is given from p. 4 of the second article on. The original definition is def. 2.6 in the first article. He doesn’t mention these DGCA extensions there, but it is directly seen to be the same concept (for the case of flat connections, in his sense).

The point of DGCAs: $\infty$-Lie integration

I am emphasizing the point of DGCA extensions because in the world of DGCAs we have $\infty$-Lie theory. In particular, every such extension can be integrated to a sequence of $\infty$-Lie groupoids internal to a general notion of smooth spaces (which in nice special cases may or may not factor through (infinite dimensional) manifolds or the like).

Recalling that if the $L_\infty$-algebroid $g$ is of the kind of tangent Lie algebroids, the Lie version of fundamental path ($\infty$-)groupids, a representation is a flat connection on something, we are dealing in this case with the differential and linearized version of the groupoid extension denoted $tra^* V//G \to \Pi(X)$ here.

Parallel transport, from this perspective, is lifts of morphisms in $\Pi(X)$ through this extension. So given any extension of qDGCAs $$\wedge^\bullet V^\bullet \leftarrow CE_A(g,V) \leftarrow CE_A(g)$$ we Lie integrate it by first sending it to the world of smooth spaces $$S(\wedge^\bullet V^\bullet) \to S(CE_A(g,V)) \to S(CE_A(g)) \,,$$ where $S$ is the smooth classifying space functor, and then further to the world of smooth $\infty$-groupoids by taking fundamental $\infty$-groupoids $$\Pi_\infty(S(\wedge^\bullet V^\bullet)) \to \Pi_\infty(S(CE_A(g,V))) \to \Pi_\infty(S(CE_A(g))) \,,$$ (Usually we’ll want to truncate at some finite sufficiently large $n \lt \infty$.) Again, parallel transport, in this picture, is lift of $n$-morphisms through this extension.

It is this relation between $L_\infty$, DGCAs, smooth spaces and smooth $\infty$-groupoids which is the rationale behind looking at extensions of qDGCAs.

Jonathan Block’s results.

Block takes the obious DG-version of the category of modules over differential algebras thus defined and then passes to its homotopy category, i.e. to the derived category in which the weak equivalences are turned into invertible morphisms. The main point of his studies is that the module categories thus obtained provide a unifying picture for all kinds of triangulated categories that appear in various famous duality results which involve equivalences of derived categories in one way or another, of the kind Fourier-Mukai, T-duality and other linear 2-maps.

This is very nice. I am certainly glad to learn about this.

From there on it comes not as a surprise, but is of course even nicer, that these dualities are implemented by the bi-modules generalizing the above modules in the obvious way. I think that in the qDGCA case these bimodules for $CE(g)$ and $CE(g')$ are nothing but $CE(g)^op\otimes CE(g')$ -modules, corresponding to the fact that on the integrated level $Gr$-$Gr'$ bi-representations are ($\infty$)-functors from $Gr^{op} \times Gr$.

A concrete example is worked out in the second article.