Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in Higher Operads, Higher Categories of Tom Leinster, but I haven't found anything.
This would probably be very useful for interpreting Martin-Löf type theory in the category of Batanin/Leinster weak $\omega$-groupoids.
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$\begingroup$ Excellent question. There is a definition of a fibration of 2-groupoids (Hardie-Kamps-Kieboom) which is too strict, and also Hermida's definition, but I think there is another definition out there, which should be exemplified by the map of fundamental bigroupoids induced by a Dold fibration of spaces. My gut feeling is that this low-dimensional example will give good insight into the $\infty$-case. $\endgroup$– David Roberts ♦Jan 4, 2012 at 23:39
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2$\begingroup$ I don't know of one. $\endgroup$– Tom LeinsterJan 9, 2012 at 12:50
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$\begingroup$ I haven't seen one written down, but in the case of $\omega$-groupoids (as opposed to $\omega$-categories) it should be easy to define: just stipulate that every k-morphism lifts with every possible domain (or codomain), as for an isofibration. The tricky part would probably be showing that these are the right class of a WFS. $\endgroup$– Mike ShulmanJan 13, 2012 at 17:21
2 Answers
There isn't one existing. There is something on the completely strict omega-categorical case in Michael Warren's article "The strict ω-groupoid interpretation of type theory" (available from his web page at IAS).
However we have a PhD student here at Macquarie working on higher fibrations and all that so hopefully there will be something on the weak case in due course!
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$\begingroup$ Welcome to MO, Richard! That's good news; I look forward to seeing the thesis. $\endgroup$ Jan 24, 2012 at 18:17
There is a discussion of the strict cubical case in
R. Brown and R. Street, `Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product', Cah. Top. G\'eom. Diff. Cat., 52 (2011) 188-208.
(arXiv:1009.5609) in which the notion of fibration is just a Kan fibration of cubical sets. However I understand a cubical version of weak $\omega$-groupoids has not been developed (attempted?).