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 MartinLöf type theory in the category of Batanin/Leinster weak $\omega$groupoids.



There isn't one existing. There is something on the completely strict omegacategorical case in Michael Warren's article "The strict ωgroupoid 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! 


There is a discussion of the strict cubical case in R. Brown and R. Street, `Covering morphisms of crossed complexes and of cubical omegagroupoids are closed under tensor product', Cah. Top. G\'eom. Diff. Cat., 52 (2011) 188208. (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?). 

