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I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, 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 :)

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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 $M \cup_B N$.

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  • $\begingroup$ Thanks a lot Theo! I thought of sth similar at some point...But then dropped it. Usually for a bordism the boundary is the disjoint union of the domain and the codomain (dom&cdom). In the example of End(I), however, the circle being the boundary is somewhat the disjoint union of I with itself except for I's dom and cdom (the two copies of I are glued together to form the circle). Am I right that this detail (of a modified definition of how the boundary corresponds to dom and cdom for bordisms with doms and cdoms with boundary) is sth that is hidden in the more formal layers of the definition? $\endgroup$
    – sani
    Commented Jul 26, 2018 at 17:46
  • $\begingroup$ @A.Nietner Yes, any reasonable definition must provide some version of the idea that a bordism between manifolds with boundary is a "bordism relative to the boundary". There are various ways to do this. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 27, 2018 at 3:43

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