1. Defining $Cob_n$
Let $n$ be a positive integer. Define the cobordism category $C=Cob_n$ as follows:
- Objects are $(n-1)$-dimensional closed, smooth oriented manifolds.
- A morphism from $M \in Obj(C)$ to $N\in Obj(C) $ is a certain equivalence class of $n$-bordisms from $M \in Obj(C)$ to $N \in Obj(C).$
- A $n$-bordism from $M \in Obj(C)$ to $N \in Obj(C)$ is a tuple $(B, M, N, \phi_B)$ with $B$ a $n$-dimensional smooth oriented manifold with boundary (empty ir non-empty) together with an orientation preserving diffeomorphism $\phi_B: \overline M \coprod N \rightarrow \partial B$. Here, $\overline M$ denotes the manifold $M$ with opposite orientation.
- We call two bordisms $(B, M, N, \phi_B)$ and $(B', M, N, \phi_{B'})$ equivalent if there exists an orientation preserving diffeomorphism $\phi$ such that $\phi \circ \phi_B= \phi_{B'}$. One checks that this defines indeed an equivalence relation on all $n$-bordisms from $M$ to $N$ for given $M,N \in Obj(C)$.
- Composition of bordisms is defined by "gluing along the common boundary." (A formal exposition of "gluing along a boundary" can apparently be found in Lee's Introduction to Topological Manifolds.) For a given $M \in Obj(C)$ the identity morphism is given by the cylinder $(M \times [0,1], M, M)$.
- One checks that this defines a category. In particular, one shows that the composition is well-defined, i.e. independent of the choice of representative.
- The disjoint union of manifolds endows $C$ with the structure of a monoidal category. (The monoidal unit is given by the empty set considered as a $(n-1)$-dimensional manifold. Associator, as well as left and right unit constraint are defined appropriately.)
2. Rigidity in $Cob_n$
I want to show that the above monoidal caregory is left rigid. Namely, given a manifold $M \in Obj(C)$ I am trying to prove that $\overline M$ is its left dual object. To that end consider the manifold $B:= M \times [0,1]$. Define $M's$ birth and death morphism by the equivalence class of the representatives $b_M:=(B, \emptyset, \overline M \coprod M)$ and $d_M:=(B, M \coprod \overline M, \emptyset)$ respectively. All that is left to verify are the zigzag identities. That is, I need to prove the existence of two diffeomorphisms $\psi_1: (B \coprod B) \coprod_{(M \coprod \overline M)\coprod M} (B \coprod B) \xrightarrow \sim B$ and $\psi_2: (B \coprod B) \coprod_{\overline M \coprod (M \coprod \overline M)} (B \coprod B) \xrightarrow \sim B$. (Does that notation even make any sense?)
3. Question
- When I am trying to picture what the manifolds $(B \coprod B) \coprod_{(M \coprod \overline M)\coprod M} (B \coprod B)$ and $(B \coprod B) \coprod_{\overline M \coprod (M \coprod \overline M)} (B \coprod B)$ "look like" (i.e. embedding them them in $\mathbb R^3$) in $Cob_2$ (or $Cob_1$ for that matter) with oriented $M=S^1$ (or oriented $M=S^0$), I think of them as something like a bended cylinder/tube. That a diffeomorphism to the cylinder exists "seems clear". But this not a proof.
How do I make these pictures formal? How do I give those diffeomorphisms precisely?