Only had time to very quickly look at this paper. But here is a quick question.

One of the puzzles mentioned in the introduction is that thinking of a $k$-fold monoidal $r$-catgory as a $(k+r)$-category with all $(i \leq k)$-morphisms trivial runs us into the problem that the former gadget seems to want to live in a $(r+1)$-category, while the latter lives in an $(k+r+1)$-category.

But maybe this is rather telling us that our expectation about the home of $k$-fold monoidal $r$-categories is wrong?

The reason I am saying this is that only recently I had a long discussion with somebody which crucially involved the 2-category whose

- objects are groups

- morphisms are group homomorphisms

- 2-morphisms are “intertwiners” of these, namely precisely those 2-morphisms which we obtain by thinking of the groups as 1-object categories and of their morphisms as fucntors.

The application we were talking about crucially demanded to take this 2-category serious. I noticed that it took me a while to make the structure of this 2-category transparent to my discussion partner. And I thought by myself that we should all better get used to thinking of groups as 1-object groupoids generally.

Precisely the same issue, in its analogous incranation, plays a crucial role in the entire field of von Neumann algebras. There it is very important to consider 2-categories (even though these are not always identified as such) whose objects are algebras, whose morphisms are algebra homomorphisms and whose 2-morphisms are intertwiners. In other words, to regard algebras as 1-object Vect-enriched categories.

In a couple of introductory talks to von Neumann algebra theory that I heard recently on this subject lots of time was spent with explaining what these intertwiners are and how their horizontal and vertical compositon works.

To the uninitiated eyes, the entire construction here is bound to look intricate and ad hoc. But it all comes down to a triviality once we seriously think of algebras as 1-object categories: these intertwiners are precisely natural transformations, henece 2-morphisms in Cat. That explains everything that is ever done with them.

So, my question is this: maybe it is “evil” (in the sense John Baez uses this word) to regard $k$-fold monoidal $r$-categories as anything else than $k$-tuply stabilized $(k+r)$-categories. Maybe we should not try to do that in the first place. A couple of applications do suggest so.

(By the way: Bruce once had a remark/question on precisely this issue here: Algebras as 2-Categories and its Effect on Algebraic Geometry).

## Re: Degeneracy

Only had time to very quickly look at this paper. But here is a quick question.

One of the puzzles mentioned in the introduction is that thinking of a $k$-fold monoidal $r$-catgory as a $(k+r)$-category with all $(i \leq k)$-morphisms trivial runs us into the problem that the former gadget seems to want to live in a $(r+1)$-category, while the latter lives in an $(k+r+1)$-category.

But maybe this is rather telling us that our expectation about the home of $k$-fold monoidal $r$-categories is wrong?

The reason I am saying this is that only recently I had a long discussion with somebody which crucially involved the 2-category whose

- objects are groups

- morphisms are group homomorphisms

- 2-morphisms are “intertwiners” of these, namely precisely those 2-morphisms which we obtain by thinking of the groups as 1-object categories and of their morphisms as fucntors.

The application we were talking about crucially demanded to take this 2-category serious. I noticed that it took me a while to make the structure of this 2-category transparent to my discussion partner. And I thought by myself that we should all better get used to thinking of groups as 1-object groupoids generally.

Precisely the same issue, in its analogous incranation, plays a crucial role in the entire field of von Neumann algebras. There it is very important to consider 2-categories (even though these are not always identified as such) whose objects are algebras, whose morphisms are algebra homomorphisms and whose 2-morphisms are intertwiners. In other words, to regard algebras as 1-object Vect-enriched categories.

In a couple of introductory talks to von Neumann algebra theory that I heard recently on this subject lots of time was spent with explaining what these intertwiners are and how their horizontal and vertical compositon works.

To the uninitiated eyes, the entire construction here is bound to look intricate and ad hoc. But it all comes down to a triviality once we seriously think of algebras as 1-object categories: these intertwiners are precisely natural transformations, henece 2-morphisms in Cat. That explains everything that is ever done with them.

So, my question is this: maybe it is “evil” (in the sense John Baez uses this word) to regard $k$-fold monoidal $r$-categories as anything else than $k$-tuply stabilized $(k+r)$-categories. Maybe we should not try to do that in the first place. A couple of applications do suggest so.

(By the way: Bruce once had a remark/question on precisely this issue here: Algebras as 2-Categories and its Effect on Algebraic Geometry).