### Semistrict Infinity-Categories and ω-Semi-Categories

#### Posted by Urs Schreiber

What is a *semistrict* $\infty$-category? The strictest possible version of weak $\infty$-category which is still general enough to satisfy the homotopy hypothesis?

Carlos Simpson in Homotopy types of strict 3-groupoids conjectured (on p. 27) that one remarkable answer is: $\infty$-categories with *strict composition* but possibly *weak identities* – “$n$-snucategories” (*s*trict, *n*on-*u*nital).

Joachim Kock in Weak identity arrows in higher categories remarked that

The conjecture in its strong form has startling consequences, defying all trends in higher category theory: every weak higher category should be equivalent to one with strict composition!

and set out to formulate the conjecture in terms of the notion he calls a *fair $n$-category* inspired by Tamsamani’s $n$-categories. With Joyal he then proved the conjecture up to $n=3$ (with some restrictions) in Weak units and homotopy 3-types.

A little later Simona Paoli in Semistrict Tamsamani $n$-groupoids and connected $n$-types proposed notions of semistrict Tamsamani $n$-categories, showed (as mentioned in TWF 245) that they do satisfy the homotopy hypothesis, and conjectured (p. 69) that one of these versions “corresponds” to Kock’s “fair $n$-categories”.

Of course there may not be *the* semistrict version of $\infty$-categories, and different choices may be useful for different purposes, as remarked in the very last two paragraphs on p. 22,23 of E. Cheng and N. Gurski’s The periodic table of n-categories for low dimensions I.

The two conjectures that Carlos Simpson stated are (p. 27)

Conjecture 1: There are functors $\Pi_n$ and $R$ between the categories of $n$-snugroupoids and $n$-truncated spaces (going in the usual direction) together with adjunction morphisms inducing an equivalence between the localization of $n$-snugroupoids by equivalences, and $n$-truncated spaces by weak equivalences.

Conjecture 2: The localization of the category of $n$-snucategories by equivalences is equivalent to the localization of the categories of weak $n$-categories of Tamsamani and/or Baez-Dolan and/or Batanin by equivalences.

## Re: Semistrict Infinity-Categories and ω-Semi-Categories

Urs wrote:

Answer this question and become famous!Everyone who’s worked hard on $n$-categories has thought about this. A long time ago, Jim Dolan and I thought we’d figured out a nice answer, using iterated enrichment based on a recursively defined generalization of the Gray tensor product. Then Jim found a mistake — the construction broke down when it got to 4-categories, which is precisely when it was just starting to do something new.

Later Brian Day and Ross Street tried a very similar idea, which broke down for the same reason at the same point.

Later Sjoerd Crans invented ‘teisi’, which are somewhat similar. I think he had to define them one dimension at a time, and did it up to dimension 6 or so.

The idea of all these approaches is to keep associativity and unit laws strict but weaken the interchange laws. The Joyal–Kock approach based on weakening only unit laws is also promising. So, as you note, there’s probably not be a unique nice answer to your question. Making some things weak may allow you to keep other things strict.

So, weakness is probably a bit like the lump in the carpet or the twist in the Möbius strip: you can push it around, but you can’t make it go away. When we understand this in detail, we’ll know a lot about $n$-categories.

The first sentence of my reply is not really my advice. My real advice is this:

Avoid this problem, unless you wish to spend a lot of time on it!