Homotopy hypothesis for contractible operad definitions of weak n-categories

[Jamie Vicary <jamie.vicary@cs.ox.ac.uk>]

Hi,

Batanin, Leinster and other have presented related definitions of weak
n-groupoid in terms of contractible globular operads. I personally find
these definitions of "contractible n-groupoids" extremely beautiful. I am
interested to learn what evidence we have that the homotopy hypothesis
might be true for (at least one of) these definitions.

Some good evidence is provided by Peter LeFanu Lumsdaine's [1] proof that a
homotopy type gives rise to an infinity-groupoid in the sense of Leinster.
There is other work along similar lines.  But, as far as I am aware, it
remains possible that contractible n-groupoids might in general be weaker
structures than homotopy n-types.

A fun way to investigate this would be to verify small instances of
phenomena associated to the periodic table in contractible n-groupoids. For
example, Christoph Dorn has shown me a proof that the Eckmann-Hilton
argument holds in a Leinster 2-category; that is, for an object X, and for
2-morphisms f,g:id[X]-->id[X], we have f.g=g.f, thereby establishing one of
the first phenomena predicted by the periodic table.

Have any higher phenomena from the periodic table been verified? Or, is
there other evidence that contractible n-groupoids behave "homotopically"
in general?

Best wishes,
Jamie

[1]
http://peterlefanulumsdaine.com/research/Lumsdaine-Weak-omega-cats-from-ITT-LMCS.pdf


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