Let $mathrm{Cob}_2$ be the symmetric monoidal $(infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are diffeomorphisms, isotopies, etc.)

Another question asked whether symmetric monoidal $(infty,1)$-functors $mathrm{Cob}_2 to mathcal{C}$ have been classified for general $mathcal{C}$.

The answer seems to be that we have some idea of what’s going on, but we don’t have a full classification.

I’m now wondering whether we know enough to say anything at all, when the target category is $mathrm{Cob}_2$ itself:

What is the $(infty,1)$-category of (strongly) symmetric monoidal endo-functors of this $(infty,1)$-category:

$$

mathrm{Fun}_{(infty,1)}^otimes(mathrm{Cob}_2, mathrm{Cob}_2) = ?

$$

I’m asking this because $mathrm{Cob}_2$ is the simplest bordism category that is not covered by the cobordism hypothesis.

(For the extended surface category $mathrm{Cob}_{langle 0,1,2rangle}$ (an $(infty,2)$-category) this question could be answered by invoking the cobordism hypothesis.)

So this seems like a good way of testing how well we understand $infty$-categories of bordisms beyond the omnipresent cobordism hypothesis.

There are some things one can show about this category of functors by elementary means.

For example, because $mathrm{Cob}_2$ has duals, every natural transformations between symmetric monoidal functors is invertible. Hence the functor category is actually an $infty$-groupoid and one could equivalently ask: what is its homotopy type?

One can also construct a few examples. Let $A$ be compact oriented $0$-manifold (= a finite set $A$ with a map $A to {pm 1}$). Then we can construct a functor:

$$

mathcal{Z}_A: mathrm{Cob}_2 longrightarrow mathrm{Cob}_2, quad

M mapsto M times A, quad (W:M to N) mapsto (W times A:M times A to N times A).

$$

This defines a functor from the groupoid of compact oriented $0$-manifolds to the functor category I’m interested in.

One can also almost define a splitting for this, by sending an arbitrary $mathcal{Z}:mathrm{Cob}_2 to mathrm{Cob}_2$ to $pi_0(mathcal{Z}(S^1))$, though this does not recover the orientation.

(Fun fact: when looking at the homotopy category the functor $hmathrm{Cob}_2 to hmathrm{Cob}_2$

that reverses the orientation is isomorphic to the identity functor via a unique natural isomorphism.)

I would suspect that every endofunctor is of the form $mathcal{Z}_A$ and that we have an equivalence:

$$

mathrm{Fin}_{/{pm1}}^{cong} xrightarrow{ simeq }

mathrm{Fun}_{(infty,1)}^otimes(mathrm{Cob}_2, mathrm{Cob}_2),

qquad A mapsto mathcal{Z}_A.

$$

However, there seem to be no tools available to actually prove this.