## June 30, 2008

### Block on L-∞ Module Categories

#### Posted by Urs Schreiber

Jim Stasheff and Aaron Bergman kindly pointed out to me (here) work by Jonathan Block,

Jonathan Block
Duality and equivalence of module categories in noncommutative geometry I
arXiv:math/0509284

Part II: Mukai duality for holomorphic noncommutative Tori
arXiv:math/0604296

The way I would say it is that what Jonathan Block studies here are modules and bimodules for actions of $L_\infty$-algebroids on $\infty$-vector bundles, even though this is not quite the way he puts it.

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.

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.

Posted at June 30, 2008 10:42 PM UTC

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### Re: Block on L-oo Module Categories

I would find it easier to follow if the classical notion of representation of a group G on SOMETHING were more transparent here. I gather the SOMETHING would be an object V of the designated 0-cat and hence an element of the group maps to a morphism of that object: G to Mor(V).

It helps to distinguish this from the action’ G x V –> V.

Then considering the weak or homtopy quotient V_G surely loses information since it makes sense from a homotopy pov.
How do you recover G from *//G aka BG in the old notation?

You consider the extension of G by V - meaning? and as a ____?

Maybe some of this is covered in the long pedagogical’ ! discusssion referenced.

You write tha the differential algebra analog is awkward? How so? Even for Ainfty algebras, it works rahter well.

co-nullary = ??

smoothness may be relevant BUT not essential

Posted by: jim stasheff on July 1, 2008 1:29 PM | Permalink | Reply to this

### Re: Block on L-oo Module Categories

I would find it easier to follow if the classical notion of representation of a group $G$ on SOMETHING were more transparent here.

The best way is: regard $G$ as a one-object groupoid $\mathbf{B}G$. Then a representation on any $V$ is a functor $\rho : \mathbf{B}G \to \mathbf{B}Aut(V) \,.$

If $V$ sits in some category $C$, for instance $C = Vect$, then this comes from a functor $\mathbf{B}G \to C \,.$

I gather the SOMETHING would be an object $V$ of the designated 0-cat

In general, an $n$-group would act on a $V$ that is an object in an $n$-category.

Then considering the weak or homtopy quotient $V_G$ surely loses information since it makes sense from a homotopy pov.

Yes, but we remember not just the homotopy quotient, but the sequence it sits in.

Take the fundamental representation of a group $G$ on itself by right action. The homotopy quotient $G//G$ is homotopy equivalent to the point $\bullet$. But still, all the information about the representation is encoded in the action groupoid sequence $G \to G//G \to \mathbf{B}G \,.$ To see what is going on, take and realize nerves everywhere. Then this becomes the universal $G$-bundle $G \to E G \to B G \,.$ Now the above statement becomes that the total space $E G$ of the universal $G$-bundle is equivalent to the point. The reason that universal bundles are still interesting, even though their total spaces are contractible, is that it’s not just the total space which is involved, but also the information about the typical fiber and about the group acting on it.

How do you recover $G$ from $*//G$ aka $B G$ in the old notation?

I should emphasize again that I am writing $\mathbf{B}G = *//G$ for the one-object groupoid coming from the group $G$. The realization of its nerve is the space $B G$.

Obtaining the group $G$ from the groupoid $\mathbf{B}G$ is trivial: it is the Hom-space.

More generally, we are talking about encoding the action of a groupoid $Gr$ on a set $V$ by a sequence of groupoids $V \to V//Gr \to Gr \,.$ Here $Gr$ (which might be $\mathbf{B}G$) is part of the data. It need not be reextracted.

What needs to be re-extracted is the map $V \times G \to V$. But that is easy: pick the element $V$ in the left term, inject it into the middle term and then take the morphism lifting $\bullet \stackrel{g}{\to}$ in $\mathbf{B}G$ to the morphism in $V//G$ with that source. The action then consists of passing to the target of that morphism.

You consider the extension of $G$ by $V$ - meaning?

Meaning that $V \to V//G \to \mathbf{B}G$ is an “exact sequence of groupoids”. That means that the composition of the two functors only hits identity morphisms and that the image of the functor on the left hits precisely all morphisms in the middle that are sent to identity morphisms on the right.

Maybe some of this is covered in the ‘long pedagogical’ ! discusssion referenced. #.

Some of it, yes. I was walking with Eric through some of the basics. But the main things I tried to say briefly in On $\infty$-Lie theory #. But I gather I should provide more details and examples.

John Baez has has a lot in his online notes on just action groupoids (without the emphasis of groupoid extensions and universal bundles) . I am having trouble finding the canonical reference to the material he has, but I think probably week 249 is a place to look at.

You write tha the differential algebra analog is awkward? How so? Even for $A_\infty$ algebras, it works rahter well.

I might be just ignorant. But I seem to remember that a while ago we thought about this together without coming to a satisfactory solution. But possibly I just was and still am confused.

The problem I see is this:

one would want to say that a representation of an $L_\infty$-algebra(oid) $g$ on a complex $E^\bullet$ is an $L_\infty$-morphism $g \to end(E^\bullet) \,,$

where $end(E^\bullet)$ denotes something like the $L_\infty$-algebra of endomorphisms of $E^\bullet$.

While in principle this is clear, I never saw a nice way to talk about $end(E^\bullet)$. But maybe I should just think about it again.

co-nullary = ??

The differential in the DGCA is the of the dual to the tower of $n$-ary brackets in the $L_\infty$-algebra. I am calling co-$n$-ary the component of $d$ which is dual to the $n$-ary bracket.

So in the decomposition $d = d_1 + d_2 + d_3 + \cdots$ $d_i : g^* \to \wedge^i g^*$ I call $g_i$ “co $i$-ary”.

Same for the differential encoding an action $d : E^\bullet \to E^\bullet \otimes g^* \,.$

This in general has a co-nullary component which lands just in $E^\bullet \hookrightarrow E^\bullet \otimes g^*$. This gives the differential on the complex $E^\bullet$, which in my point of looking at it is witnessed by the left part of the extension which says that $\wedge E^\bullet \leftarrow CE_A(g,E)$ is a DGCA homomorphism.

Posted by: Urs Schreiber on July 1, 2008 2:50 PM | Permalink | Reply to this
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