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I fixed some of the references at Batanin omega-category.
Here is a question:
around corollary 9.4 of
it is shown for the special case of Gray-categories that the weak $\omega$-functors of
are actually the resolved morphisms with respect of a canonical model category structure, out of cofibrant resolutions into fibrant objects.
Is an analog of this known for general Batanin $\omega$-categories? Is there a canonical model structure on Batanin $\omega$-categories such that the weak $\omega$-functors represent the correct derived hom-space?
AFAIK that is not known.
I added a reference to some papers by Cheng and Leinster on Batanin (and Trimble) infinity-categories.
Is there any good reason why this entry is called “Batanin omega-category” while the one on another approach is called “Grothendieck-Maltsiniotis infinity-category”? Maybe people used to use “omega-category” to mean “weak infinity-category” - I forget. That may old-fashioned. But either way, it seems we should use the same term both for the Batanin infinity-categories and the Grothendieck-Maltsiniotis infinity-categories.
5: This is the self-consistent technical terminology in Batanin’s paper, rather than newspaper’s description or an informal discussion. If one is to change it, one has to do this systematically within the formal framework. Batanin says
A weak $\omega$-category is a globular set together with the structure of algebra over a universal contractible $\omega$-operad.
Hence, only if one disagrees with calling this kind of higher operads $\omega$-operads then it makes sense to disgree with $\omega$-category. So how would you call the kind of higher operads from Batanin’s paper and in Batanin’s context as opposed to other kinds from literature?
Of course, you are right in the sense that Malstiniotis-Grothendieck notion is in essence a variant/sister notion, but is it formally defined in the same setup ?
And maybe it does make sense to keep referring to “omega”-structure unless and until it is proven that they are indeed equivalent to the corresponding infinity-structures.
(Not that “omega” is particularly suggestive of that distinction (or of anything of relevance, for that matter?!) but at least it is a different term than “infinity” that prevents confusion while the hoped-for equivalences have not been established… yet.)
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