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Our seminar here at UCSD this quarter is about the Morrison-Walker “blob complex” and the corresponding notion of “disc-like” n-category. I have been trying to understand why (or whether) the latter definition should be regarded as “fully coherent”. The brief discussion at the end of the paper constructs the constraint isomorphisms, but doesn’t verify the coherence axioms. So far, haven’t been able to see how to deduce the unit coherence axiom (the “triangle equation”) for bicategories from the disc-like 2-category axioms, but it could just be that I don’t understand the latter very well! Has anyone around here thought about this?
I haven’t thought about this. I had tried to dig a bit deeper into this last semester, when we made some notes at blob n-category, but then I was being distracted by too many other things.
So you are looking for the coherence law
$\array{ (g I) f &&\stackrel{a}{\to}&& g (I f) \\ & {}_{\mathllap{r \cdot id}}\searrow && \swarrow_{\mathrlap{id \cdot l}} \\ && g f }$for composable 1-morphisms $g$, $f$, I suppose? Isn’t there an evident homeomophism of 2-balls that exhibits this? (I don’t know, I haven’t really sat down and worked through this.)
The homeomorphism of 2-balls isn’t evident to me. What trips me up is that the unit isomorphisms $r$ and $l$ are obtained by “product maps” corresponding to “pinched products”, and then one of them has to be glued with the associator. But there is only one axiom relating product morphisms to gluing, and it doesn’t seem to apply: it requires that two pinched product maps be glued together to produce a third one, and I haven’t been able (yet) to express this gluing in that form.
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