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.
Let $\mathcal{C}$ be the full subcategory of $\emptyset/Cob$ on non-empty connected cobordisms from $\emptyset$. Isomorphisms in $\emptyset/Cob$ are precisely the morphisms that permute the indices of circles.
The main reason to consider the category $C$ is that (the opposite of) the projection functor $\mathcal{C}\to Cob$ is analogous to the functor $\hat{C}:\Delta^{op}\to\Gamma(as)$ from [1]. Hochschild homology of any functor $F:\Gamma(as)\to R-mod$ is defined in [1] as the homology of the functor $F\circ\hat C$.
The category $\mathcal{C}$ can be endowed with a structure of generalized Reedy category:
- 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.
This structure is not dualizable. There is another structure, which is not generalized Reedy structure, since the codomain of the degree map is not an ordinal. Yet it satisfies all the other properties of dualizable generalized Reedy structure:
- The degree is $(-2g+n)$,
- Morphisms in $R_-$ attach surfaces so that each connected component of cobordism (that is projection of the morphism in $R_-$ to $Cob$) 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).
Question 1. Is there similar relation between (monoidal) functors from $Cob$ and functors from $\mathcal{C}$, or from a similar category, with values in $R-mod$ or in $Ch(R-mod)$?
Question 2. There is homotopy category of simplicial presheaves on $\mathcal{C}$. 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.
