I am currently reading in [Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors](https://arxiv.org/abs/1409.5723), and have a (I guess) pretty basic question for my understanding of the (∞, n) category of Cobordisms...


Their (informal) definition of Bord(n) is 



> 
> A genuine example of an (∞, n)-category with n > 0 is given by Bord(n), the ∞-category of cobordisms, which can be informally described as consisting of having points as objects, 1-dimensional
>  bordisms as 1-morphisms, 2-dimensional bordisms between bordisms as 2-morphisms, and so on until we arrive at n-dimensional bordisms as n-morphisms, from where higher morphisms are given by diffeomorphisms
>  and isotopies: more precisely, the (n + 1)-morphisms are diffeomorphisms which fix the boundaries, (n + 2)-morphisms are isotopies of diffeomorphisms, (n + 3)-morphisms are isotopies of isotopies, and so on.
> 
> 
> 


I am wondering about the 2-morphism (and therefore any higher morphism): Lets take the interval I=[0,1] as the bordism between two points {0} and {1}, and similarly I'=[3,4]. What would be the 2-morphism between I and I' (or equivalently, what would be the endomorphisms of I)? (informally). The reason why I am asking simply is that I cannot think of any 2 dimensional compact manifold M with boundary where the boundary is the disjoint union of two intervals, I and I'...


Thanks in advance!


Alex :)



2-morphisms for Bord(n)
The endomorphisms of the interval are, at this informal level of discussion, surfaces with $S^1$ boundary. More generally, if you want to talk about $\hom(M,N)$, where $M$ and $N$ are $k$-dimensional bordisms, then certainly $M$ and $N$ had better have the same domain and codomain $B = \partial M = \partial N$ (since they had better be $k$-morphisms between the same $(k-1)$-morphisms). A *bordism* from $M$ to $N$ is then a manifold with boundary $M \cup\_B N$.


