The String Coffee Table

Later I had a great talk with Danny Stevenson where we made a lot of progress on understanding the tangent bundle of a 2-space: it’s a category in the category of vector bundles!!!

That’s interesting.

Let’s see. Such a 2-(tangent bundle) of a 2-space would have a vector bundle of objects. This should be the ordinary tangent bundle to the space of objects of the 2-space.

Then it should have a vector bundle of morphism. This is presumeably just the tangent bundle to the space of morphisms of our 2-space.

Hm, then probably source, target and composition maps are just the vector bundle morphisms between these vector bundles that come from the differentials of the source, target and composition map of our original 2-space.

Understanding the tangent bundle of a 2-space and how its sections might form a Lie 2-algebra is probably a prerequisite for understanding Lie 2-algebroids.)

One way to understand semistrict Lie 2-algebroids is in terms of the dg-algebra duals of $L_\infty$-algebras. This way one can see in particular that what is called the Courant algebroid really is a 2-algebroid.

But of course it would be nice to have direct construction.

It seems easy to categorify the definition of an ordinary algebroid, once we know what a 2-(tangent bundle) and a 2-(vector bundle) is supposed to be.

So there should be a 2-space $M$ and a 2-(vector bundle) ‘$E \to M$’ over it, which is apparently a category internalized in the category of vector bundles such that the base of the vector bundle of objects is the object space of $M$ and the base of the vector bundle of morphisms is the morphism space of $M$.

Then there should be the 2-(tangent bundle) $TM$ of $M$ that you mentioned and an anchor functor $E \stackto{\rho} TM$, and so on.

On the other hand, what still seems to be missing is a nicer category-theoretic description of a mere algebroid. If that were available it might lead to a nicer 2-algebroid definition, too.

(BTW, it’s the A-model which gives rise to the Fukaya theory and the B-model which gives rise to the derived category of coherent sheaves.)

One missing link in my thesis is this: I want to differentiate local $p$-holonomy $p$-functors

that go from the $p$-path $p$-groupoid $\mathcal{P}_p(U_i)$ to the structure $p$-group $G_p$. The result should be a $p$-connection $p$-functor

from the $p$-path $p$-algebroid $\mathfrak{g}_p(U_i)$ to the Lie $p$-algebra $\mathfrak{g}_p$ of $G_p$.

Since I didn’t know (and possibly nobody knew) what the direct description of the $p$-path $p$-algebroid should look like, I switched to the dual description of algebroids in terms of dg-algebras, guessed the dg-algebra which should correspond to $\mathfrak{p}_p(U_i)$ and then showed that this (obvious) guess makes sense by demonstrating that it leads to the correct linearized cocycle relations. (That’s the content of section 13 of my thesis.)

Now that you opened my eyes, I should try to substantiate this guess by seeing if I can construct a 2-path 2-algebroid as a category internalized in the category of vector bundles.

Actually, the 1-path 1-algebroid should be nothing but the tangent bundle of base space. If true, the 2-path 2-algebroid should be nothing bu the 2-(tangent bundle) of base space, regarded as a 2-space.

Hm, maybe not. I don’t see how any 2-forms would get into the game this way. Maybe I would have to use the 2-(vector bundle) of the path space of base space.

Hm…