The n-Category Café

Gosh, that looks interesting. OK, so my “thirdly” above is nullified :).

I wonder lazily to what extent this Riehl and Verity result is constructive; i.e. giving you an algorithm to actually perform the promotion on a particular structure, in a traditional globular $n$-category setting.

It’s been a while since I looked at it, but my memory is that everything is pretty explicit. Maybe Emily will show up and enlighten us.

I’d also like to know what can be said using these (or other) methods about making equivalences coherent.

I’d also like to know what “homotopy coherent adjunction” means!

“Homotopy coherent adjunction” is defined in the Riehl-Verity paper. I think you should be able to apply the theory to an equivalence by first doing the usual “equivalence-to-adjoint-equivalence” at the bottom level to get an “adjunction up to homotopy” and then do their full coherentification.

Thanks. The term is used freely in the abstract in a way that suggests it is standard. I see now how they define it.

I would expect that ‘coherent’ here means that all equations hold. I’m not quite sure how to express that in the simplicial category terminology, however, and I’m having trouble finding a theorem in an appropriate early part of the paper that I can map to this suggestion.

Regarding equivalences: I understand one can do the usual trick. But I had in mind that an equivalence in fact gives rise to an ambidextrous adjunction, with a correspondingly stronger notion of coherence.

Jamie wrote:

I would expect that ‘coherent’ here means that all equations hold.

In homotopy theory, ‘homotopy coherent’ means ‘obeying the desired equations up to homotopies which obey the desired equations up to homotopies which obey the desired equations up to…’

In any particular example of homotopy coherence, once has to either figure out what equations are ‘desired’ at each step, or, preferably, invent a machine that somehow takes care of them all with the flick of a switch.

I saw Piotr’s Masters thesis in August 2014, in which he works out some of the details. Chris Douglas and Chris Schommer-Pries were aware of it (considerably) earlier, I expect.

Hi John,

It’s great that Mike and separately Emily and Dominic and separately Nick Gurski and collaborators have been doing significant work in this space. Obviously their work goes (in three separate directions!) far beyond the swallowtail issue we thought about, discussed below.

Like you, I’m not interested in priority discussions (much less disputes!) and would normally steer well clear of this, but since you asked directly and explicitly, I’ll say a bit about our related work.

Andre Henriques and I proved a swallowtail coherence, as Jamie refers to above, (that, for instance, you can modify cusps that don’t a priori satisfy swallowtail so that they do) in February 2008.

The reason we never published anything about that is the following. For us, this was a step in our program to construct 3-dimensional local field theories. Roughly speaking, in our approach, there were four steps to this: 1) build a 3-category (for us this was conformal nets — everything about conformal nets is joint work with Arthur Bartels), 2) prove some object there is in a certain sense (incoherently!) dualizable, 3) prove that such an incoherently dualizable object can be promoted to a coherently dualizable object, 4) prove that a coherently dualizable object gives a local field theory.

What we managed to do in Feb 08 is step (3) but only in the case of incoherently invertible objects. (We already understood step (4).) (The swallowtail coherence is part of step (3), and occurs even in the invertible case.) Now, the way we were thinking of it was that step (4) was a form of the 3-dimensional cobordism hypothesis. And in fact, the combination of steps (3) and (4) [as we conceived them] is quite close to the modern form of the cobordism hypothesis for the structure ‘stable framing’, ie for the structure group \Omega(O(infty)/O(3)).

When I gave a talk about these things at MIT in May of 08, I chatted to Mike and it started to become clear the remarkable approach he and Jacob had managed to figure out in this arena, and I think sometime that summer Jacob announced the proof of the cobordism hypothesis in all dimensions and for all structure groups. At that point, even though our approach was rather different (in particular it was constructive, which was both a feature and a liability), we felt it no longer made sense to pursue steps (3) and (4) and henceforth relied on Mike and Jacob’s cobordism hypothesis for that part of the story; in particular, everything about the coherence results in step (3) got excised from the conformal nets series of papers and put in a dusty drawer.

It’s nice to see some of these issues seeing the light of day.

Best, Chris

The wigwam is the simplest interesting coherence relation satisfied by the butterfly. (The cusp is what you get by looking at a plane curve with two morse critical points, the swallowtail with three, the butterfly with four, the wigwam with five, …) Won’t try to draw it here in ascii …

Chris wrote:

Similarly, the very existence of the wigwam tells us that swallowtails can be modified to satisfy butterfly.

Jamie wrote:

Chris, what do you mean by the wigwam?

Chris wrote:

The wigwam is the simplest interesting coherence relation satisfied by the butterfly.

Darn, I had hoped you meant that the very existence of this traditional native American dwelling relied implicitly on aspects of catastrophe theory that we’re only now understanding in an $n$-categorical way.

One answer (but probably not the one Dominic had in mind (-: ) is that adjointification of equivalences can be done in the internal homotopy type theory of simplicial spaces, and therefore it can automatically be done “in any context”. Emily and I mostly worked this out last spring, as part of developing further the simplicial-spaces approach to “directed homotopy type theory” (blog post, talk).

This way of looking at it is really quite pretty, and has some interesting consequences. For instance, if you apply the “has both a left and right inverse” characterization to the equivalences of hom-spaces, it implies that if you have transformations $\eta:1\to G F$ and $\epsilon_1: F G \to 1$ and $\epsilon_2 : F G \to 1$ such that $\eta$ and $\epsilon_1$ satisfy one triangle identity and $\eta$ and $\epsilon_2$ satisfy the other triangle identity, then in fact $\epsilon_1 =\epsilon_2$ and you have an adjunction. I don’t think I knew that before; did anyone else?

I don’t think I knew that, and it’s nice. But it might follow from the usual string diagram argument that if $\eta$ and $\epsilon_1$ obey both triangle identities and $\eta$ and $\epsilon_2$ obey both triangle identities then $\epsilon_1 = \epsilon_2$. In other words: do that string diagram proof and see which of the four assumptions you’re actually using.

if you have transformations $\eta\colon 1 \to GF$ and $\epsilon_1\colon FG \to 1$ and $\epsilon_2\colon FG \to 1$ such that $\eta$ and $\epsilon_1$ satisfy one triangle identity and $\eta$ and $\epsilon_2$ satisfy the other triangle identity, then in fact $\epsilon_1 = \epsilon_2$

That seems very similar to the fact that given an associative binary operation on a set, if an element $\eta$ is left inverse to $\epsilon_1$ and right inverse to $\epsilon_2$ then $\epsilon_1 = \epsilon_2$. I don’t know whether it’s literally an instance of it.

I intended to say (but maybe was not sufficiently clear) that it essentially comes from applying that fact objectwise to the hom-set-bijection characterization of adjunctions.