### The (∞,1)-Category of (∞,n)-Categories

#### Posted by John Baez

There’s a new paper on the problem of comparing different definitions of $(\infty,n)$-category:

- Clark Barwick and Chris Schommer-Pries, On the unicity of the homotopy theory of higher categories.

My thesis advisor often used the word ‘unicity’ instead of the more common ‘uniqueness’. I’ve never seen anyone else do that until now! But that’s not the most exciting thing about this paper…

According to an email from Chris, they give a few simple axioms that an $(\infty,1)$-category should satisfy to be a reasonable candidate for the $(\infty,1)$-category of $(\infty,n)$-categories. Then they prove that the space of $(\infty,1)$-cateories satisfying these axioms is equivalent to $B(\mathbb{Z}/2)^n$—that is, the product of $n$ copies of the classifying space of $\mathbb{Z}/2$, namely

$B(\mathbb{Z}/2) \simeq \mathbb{RP}^\infty$

For starters, $B(\mathbb{Z}/2)^n$ is connected, so any two theories obeying their axioms are equivalent. But the fun part is the $(\mathbb{Z}/2)^n$. This comes from functors that send an $(\infty,n)$-category to one of its opposites. So, their result can also be seen as sophisticated generalization of the fact that the group of automorphisms of $Cat$ is $\mathbb{Z}/2$.

(A more sensible thing to look at is the *2-group* of autoequivalences of $Cat$ and natural isomorphisms between these, but I’m pretty sure this is still equivalent to $\mathbb{Z}/2$.)

They also show that a lot of proposed theories of $(\infty,n)$-categories satisfy their axioms, and thus are equivalent. These include Barwick’s model of $n$-fold complete Segal spaces (use by Lurie in his proof of the cobordism hypothesis), Rezk’s complete Segal $\Theta_n$-spaces, Simpson’s $n$-fold Segal categories, and a bunch of other models.

## Re: The (∞,1)-Category of (∞,n)-Categories

Until recently, I thought that “unicity” was just a mistake, typically made by native French speakers but common enough that it had infected some native English speakers. But then I witnessed someone (maybe Toby?) put up an argument that it was legit; so maybe I was the mistaken one.