### Quinn on Higher-Dimensional Algebra

#### Posted by David Corfield

Frank Quinn kindly wrote to me to point out an essay he is working on – The Nature of Contemporary Core Mathematics (version 0.92). Quinn will be known to many readers here as a mathematician who has worked in low-dimensional topology, and as one of the authors, with Arthur Jaffe, of “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics.

I crop up in the tenth section of the article, which is devoted to a discussion of “a few other accounts of mathematics”, including those of Barry Mazur, Jonathan Borwein, Keith Devlin, Michael Stöltzner, and William Thurston.

One objective is to try to understand why such accounts are so diverse and mostly – it seems to me – irrelevant when they all ostensibly concern the same thing. The mainstream philosophy of mathematics literature seems particularly irrelevant, and the reasons shallow and uninteresting, so only two are considered here. Essays by people with significant mathematical background often have useful insights, and when they seem off-base to me the reasons are revealing. The essay by Mazur is not off-base. (p. 53)

I take it that “irrelevant” is being taken relative to Quinn’s interest in characterising ‘Core Mathematics’.

Section 10.4 is where my work is given a going over. I’ll postpone a discussion of other sections to a later date, but wanted to know what people thought about subsection 10.4.3 (pp. 61-63), which treats the tenth chapter of my book on higher-dimensional algebra. One of the main thrusts is that I have been lured by John into believing that higher-dimensional algebra is more important and powerful than it really is. Evidence is given as to why $n$-categories are unlikely to help resolving issues concerning low-dimensional manifolds. For example,

Topological field theories on 2-manifolds can be characterized in terms of Frobenius algebras. The modular ones (roughly the ones coming form 2-categories) correspond to semisimple Frobenius algebras. Semisimple algebras are ‘measure zero’ in unrestricted algebras and have much simpler structure. this indicates that requiring higher-order decomposition properties corresponding to higher categories enormously constricts the field theories. To get more power we apparently need to reduce the categorical order rather than increase it.

Given we’ve started to get a little more self-reflective at the Café about what (higher) category theory means for us, I’d be interested to hear views on this subsection. No doubt the younger me who wrote that chapter around 8 years ago believed that ‘quantum topology’ would readily extend to knotted spheres in 4-space the account that saw tangles in 3-space as a free braided monoidal category, and invariants cropping up through functors to categories of representations of the same kind. Just devise some candidate braided monoidal 2-categories, and all would be fine.

## Re: Quinn on Higher-Dimensional Algebra

David wrote:

Well, by now that’s well underway. It didn’t happen ‘readily’: it took a lot of work by dozens of mathematicians. After all, you can’t ‘just’ devise interesting braided monoidal 2-categories: doing so requires deep ideas!

But while you may have been overoptimistic concerning the

rateof progress, we now have invariants of 2-dimensional surface in 4d space, obtained by categorifying the representation theory of quantum groups. And this is indeed one of the hottest topics in low-dimensional topology.So, I’d say your younger self was on the right track. For a quick sketch of where we are now, try the section on Khovanov in the prehistory of $n$-categorical physics that Aaron Lauda and I wrote.

And let’s not forget the revolutionary work on TQFT that Jacob Lurie is busy carrying out. Higher category theory is fundamental here — indeed, he’s having to build the foundations of higher categories hand-in-hand with work on this topic.

Quinn wrote:

Well, Quinn’s attempts to crack the Andrews–Curtis conjecture by using computers to find suitable non-semisimple 2d TQFTs seem to have stalled out. But maybe he’s still optimistic? Or maybe he’s hinting at something else? I’m not sure exactly what ‘more power’ he’s hoping to get, and what he hopes to do with it. It would be nice if he were more explicit.