omega-something should be as weak as necessary for the greatest possible generality

Okay, I see. I am aware that some people use the “omega” this way, but I find it a bit odd, since my impression had been (possibly incorrectly) that the use of “omega” over “infinity” has originally (and still so in the majority of publications) precisely been to emphasize that one refers to the strict version, because the “omega” alludes the the fact that we reach omega-categories by taking the limit of the iterated enrichment

(n+1)Cat = nCat-Cat

starting with 0Cat = Set, which produces strict n-categories at each level.

In any case, why should one otherwise say “omega-category” instead of “infinity-category”?

Well, sorry for the terminological nitpicking. It’s not really important. Feel free to ignore this comment of mine.

I can’t find a paper by Harnik to go with that talk, but I imagine there is one somewhere.

As for omega-groupoids, I don’t know that I’m thinking of one particular definition, except that my default assumption is that an n-something or omega-something should be as weak as necessary for the greatest possible generality (that way, you can add adjectives to denote strict ones). So I guess I’m thinking of something like weak (infinity,1)-categories?

Certainly any time omega-groupoids come up, it also comes up that they classify homotopy types of spaces.

Wait, what precisely do you mean by “omega-groupoid” here? Usually the statement is that these do not classify homotopy types.

We recently talked about that in Semistrict Infinity-Categories.

The omega-groupoids mentioned there are the entirely strict ones: all composition is strictly associative and all k-morphisms have strict inverses for all k. These fail to capture all homotopy types from degree 3 on:

While strict 3-groupoids are insufficient to model all 3-types, Gray-groupoids are sufficients. This is supposed to be just another incarnation of the fact you mention above, that every tricategory is at best equivalent to a Gray category, but not in general to a strict 3-category.

But I know that sometimes people say “omega-groupoid” for something weaker than the “true” omega-groupoids. Which definition precisely do you have in mind?

Harnik described how to generate an omega-category recursively: generate faces of dimension n by freely adjoining some indeterminate cells, which need all these operations telling how they can be stuck together.

Interesting coincidence. I am in the middle of thinking about something related to this. I am mostly following the nice work by Francois Metayer on this topic.

In Resolutions by polygraphs he discusses a procedure to obtain omega-categories which are “degreewise freely generated” from a globular set.

This is pretty much the familiar generators-and-relations description of 1-categories, only that now each relation of generators of k-morphisms is itself a generator of (k+1)-morphisms and subject to further relations in degree (k+2). And so on.

In Cofibrant complexes are free Metayer shows that the omega-categories obtained this way are precisely the cofibrant ones in the folk model structure on omega-cat. This is a nice consistency check on the construction.

Did Harnik mention Metayer’s work? Does he have an article about his work that I can look at?