The n-Category Café

What I am saying here is not supposed to be very deep. But I believe it provides a useful way to look at this situation. We need not restrict ourselves to a particular choice of 2-morphisms in $P_2(X)$. All we need to require is that the 2-morphisms are such that a path $\gamma$ is adjoint to its reverse $\bar \gamma$ $$\begin{aligned} \gamma &: \sigma \mapsto \gamma(\sigma) \\ \bar \gamma &: \sigma \mapsto \gamma(1-\sigma) \,, \end{aligned}$$ such that the adjunction is ambidextrous and special.

(We may for instance choose 2-morphisms to be thin homotopy classes of 2-paths.)

If we arrange for that, then we may “cancel” a path with it’s reverse by inserting the counit 2-morphism $\epsilon_\gamma$ $$\array{ x \stackrel{\gamma}{\to} y \stackrel{\bar \gamma}{\to} x \\ \;\;\;\Downarrow \epsilon_\gamma \\ x \stackrel{\Id}{\to} x } \,.$$ Similarly, we may create a path and its reverse “from nothing” by inserting the unit $i_\gamma$ $$\array{ x \stackrel{\Id}{\to} x \\ \;\;\;\Downarrow i_\gamma \\ x \stackrel{\gamma}{\to} y \stackrel{\bar \gamma}{\to} x } \,.$$ Composition and co-composition of paths using these cancellations and co-cancellations follows the rules of products and coproducts in special Frobenius algebras. From this point of view, the fact that the associator satisfies a pentagon is more or less automatic.

The co-composition operation is particularly interesting. It should induce the product operation on string fields.

That needs to be worked out in details. But here is a sketch.

For simplicity, consider the case where we decide to only cancel and create portions of paths that are precisely half of a former path. Then the above setup yields a category $$P_1^{LW}(X)$$ which is at the same time a co-category.

There are many different choices of co-category structure we can use, each determined by a choice of path for each point of $X$. Later we will want to “integrate over all these choices”, but for the moment just fix one.

To define a string field, let $\mathrm{tra} : P_1(X) \to \mathrm{Bim}(\mathrm{Vect}_\mathbb{C})$ be a gerbe with connection on $X$ #. Let $[0,1]$ be the category obtained from regarding the interval as a poset. Then the functor category $$[[0,1],P_1(X)]$$ is the configuration space of our string #, and a string field is a section of the gerbe with connection pulled back to this configuration space.

It sends each path $\stackrel{\gamma}{\to}$ to a morphism $e_\gamma$ $$\array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \mathbb{C} \\ \downarrow &\Downarrow e_\gamma& \downarrow \\ A_x &\stackrel{\mathrm{tra}(\gamma)}{\to}& A_y } \,.$$ which is essentially a section of some kind of bundle over path space.

Now, given two such sections, $e_1$ and $e_2$, say, we may use the co-composition on paths to produce a new section - their product: $$P_1^{LM}(X) \stackrel{\text{co-compos.}}{\to} P_1^{LM}(X) {}_t \times_s P_1^{LM}(X) \stackrel{e_1 \times e_2}{\to} \tilde T {}_t \times_s \tilde T \stackrel{\text{compos.}}{\to} \tilde T \,.$$

If we had a measure on our paths (like is assumed in string field theory), we could try to integrate this operation over all choices of co-category structures. The result should be the star-product of string field theory #.