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I extracted the definition of “n-category with all duals” from Scott Morrison and Kevin Walker’s “Blob homology” at blob n-category.
This is to some extent a take on defining hyperstructures.
added examples: fundamental blob $n$-category of a topological space and bordism $n$-category
In the definition of $\underset{\to}{C}_{k-1}(\partial X)$, is the colimit over the groupoid of balls in the (k-1)-sphere? That doesn’t seem right, but it’s the only thing I see how to define, since $C_{k-1}$ is only defined on that groupoid. I couldn’t find where in their paper they define $\underset{\to}{C}$.
This is around def. 6.3.2 in their article. I have added a liitle bit more detail to the entry blob n-category, but still not everything.
This definition seems very circular! The “boundary restriction” transformation is defined in terms of the extension to general shapes, which is defined in terms of the composition operation, which uses the boundary restriction maps to define when things are composable. Is there some implicit induction on dimensions going on?
This definition seems very circular!
Yes, indeed. And the fact that the writeup is very non-linear itself makes it hard to plough through.
I think that, as you suggest, we are supposed to define this inductively: first we define composition of 1-morphisms/1-blobs, that allows us to extend the blob $n$-graph to 1-spheres, then using this we define compostion of 2-morphisms/2-blobs, and so forth.
As Urs said, the category definition is spirally, not circular. The colimit construction which extends from balls to any manifold (in particular, to spheres) is described in section 6.3. To state the “boundary restriction” axiom for k-morphsims requires all of the axioms for (k-1)-morphisms. So if we were presenting things in linear order, we would give all the axioms for 0-morphsims, then all the axioms for 1-morphisms, and so on.
The colimit which defines $\underset{\to}{C}_{k-1}(\partial X)$ is over the poset of all decompositions of $\partial X$ into (k-1)-balls.
I have edited the entry blob n-category a little to reflect this fact better.
But Idon’t quite have the time right now to add more details.
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