Let $Cob$ be the category such that

• $Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$,
• morphisms are (homeomorphism classes of orientable) cobordisms.

Denote by $\emptyset/Cob$ either the slice category under $\emptyset$, or the same slice category minus the object $id_\emptyset$. Isomorphisms in $\emptyset/Cob$ are precisely the morphisms that permute the indices of circles.

The category $\emptyset/Cob$ can be endowed with a structure of generalized Reedy category in at least two ways.

• The degree of the object that corresponds to the surface of genus $g$ with $n$ boundary circles is $(2g+n)$,
• Up to an isomorphism, morphisms in $R_-$ attach disks to some of the boundary circles,
• Morphisms in $R_+$ attach any surfaces other than disks.

or

• The degree is $(-2g+n)$,
• Morphisms in $R_-$ attach surfaces so that each connected component of the cobordism has at most one boundary circle not glued to the boundary of the source,
• Morphisms in $R_+$ attach surfaces of genus $0$ (with at least one boundary circle not glued to the source).

The second structure is dualizable, the first is not.

In a certain sense (there is a precise statement, though not necessarily the right one) the projection functor $\emptyset/Cob\to Cob$ is analogous to the functor $\hat{C}:\Delta\to\Gamma(as)$ from [1]. Hochschild homology of any functor $F:\Gamma(as)\to R-mod$ is defined as the homology of the functor $F\circ\hat C$.

Question 1. Is there similar relation between the (monoidal) functors from $Cob$ and the functors from $\emptyset/Cob$ with values in $R-mod$ or in $Ch(R-mod)$?

Question 2. There is a homotopy category of simplicial presheaves on $\emptyset/Cob$. Has it been studied before?

[1] Pirashvili, T. and Richter, B., 2002. Hochschild and Cyclic Homology via Functor Homology. K-Theory, 25(1), pp.39-49.