n-category = (n,n)-category
n-groupoid = (n,0)-category
it was conjectured (page 27) that
For weak unit laws in -categories see
Simpson’s conjecture is proven up to the case of 3-categories with a single object.
One expects several alternative such semi-strictification statements. Eugenia Cheng and Nick Gurski write the following at the end of their paper:
Finally we consider the question of eliminating the distinguished invertible elements by using a stricter form of -category. Generalising from the previous sections, we see that we do not need to restrict all the way to strict -categories – a semistrict version will suffice. One form of semistrictness has everything strict except interchange (cf. Gray-categories and see Crans 2000b, Crans 2000a); another has everything strict except units (Koch 2005, Simpson 1998). These have both been proposed as solutions to the coherence problem for -categories.
However, there are other possible “shades” of semistrictness and the above notions do not appear to be right for the present purposes. Instead, we need a form of semistrict -category in which the units and interchange for -cells are strict, but everything else can be weak. This is to eliminate the constraint -cells that become distinguished invertible elements in our -degenerate situation; we expect that as in the case the associator is automatically forced to be the identity.
Hypothesis 5.3. Semistrictness Let . Then an -degenerate semistrict -category in the above sense is precisely a commutative monoid.
E. Cheng and N. Gurski’s, The periodic table of -categories for low dimensions I (arXiv)