The n-Category Café

The relevant part is:

If $T$ is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category $A$, then the category of algebras $A^T$ is locally presentable. In particular, if $A$ is locally presentable and $i:B\to A$ is a reflective subcategory, then $B$ is locally presentable if $i$ is accessible.

since $sSet$ is locally presentable, and hence accessible (I’ve just learned), then all we need is that the endofunctor $U\circ F:sSet \to sSet$ underlying the ‘algebraic fibrant object’ monad is $\kappa$-continuous for a regular cardinal $\kappa$. I presume this means preserving limits of diagrams of shape ‘bounded by $\kappa$’.

But since limits and colimits in $AlgKan$ are calculated as in $sSet$, then $U\circ F$ preserves them, and we are done: $AlgKan$ is locally presentable. Nice!

But since limits and colimits in $Alg Kan$ are calculated as in $sSet$

The limits and filtered colimits are. But I think that’s what we need for the proof that you sketch:

to deduce that $ALg Kan$ is locally presentable we want that $U \circ F$ preserves filtered colimits.

But $F$ is left adjoint and hence preserves all colimits. And $U$ has the special property that it preserves not only all limits but also all filtered colimits (prop 2.11).

there’s at least an intuitive sense in which algebraic Kan complexes are only “barely algebraic”

That seems to depend on your intuition. Or can you formalize this at least roughly?

Because my intuition is very different. Think about what an algebraic Kan complex is: it is (among other things)

• a choice of composition operation on all 1-morphisms;

• a choice of associator 2-morphisms for all triples of 1-morphisms;

• a choice of pentagonator 3-morphism for all quadruples of 1-morphisms;

and so on.

That’s as algebraic as it gets, it seems to me. It is in a sense more algebraic than all the operadic definitions even! Because this makes specific unique choices for all this data, where operadic definition keep around the spaces of choices without picking unqiue composites etc. The operadic definitions are really more like geometric definitions with a strong algebraic control on how the various spaces of choices interact not with an algebraic specification of the choices themselves.

Perhaps someone should contact Maltsiniotis and suggest that he revise this sentence?

I can’t find his email address. His website seems to have been down all day at least. (?)

Well having had a bit more of a look at the paper I think you could argue either way as to whether the source and target maps are rolled in or whether they are there “beforehand” as it were. A Grothendieck-style omega-groupoid in $\mathbf{Set}$ is defined as a model of a “Gr-coherator”, which is a category $\mathbf{C}$ equipped with a functor $\mathbb{G} \to \mathbf{C}$ satisfying some properties. Given such a coherator, a model of it is a functor $\mathbf{C}^op \to \mathbf{Set}$ which preserves certain kinds of limits (“globular products”). But any such model will certainly give rise to a functor $\mathbb{G}^op \to \mathbf{Set}$ by precomposition with $\mathbb{G} \to \mathbf{C}$. So any model is a globular set. I guess it is fair to argue that it is only a globular set after the fact: but I think that if you want to, you can switch things around as follows.

One of the properties of a “Gr-coherator” $F : \mathbb{G} \to \mathbf{C}$ is that we can take the left Kan extension of $F$ along the inclusion $\mathbb{G} \to \Theta_0$ to get a functor $\bar F \colon \Theta_0 \to \mathbf{C}$. In terms of this functor we can define a model to be the following: it’s a presheaf in $[\mathbf{C}^op, \mathrm{Set}]$ which when restricted along $\bar F$ to a presheaf in $[\Theta_0^op, \mathrm{Set}]$, lies in the image of the singular functor $\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}]$ induced by the inclusion $\Theta_0 \to \hat{\mathbb{G}}$. That is, the category of models $\mathcal{M}$ for the coherator is given by the pullback of $\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}]$ along $(\bar{F})^\ast \colon [\mathbf{C}^op, \mathrm{Set}] \to [\Theta_0^op, \mathrm{Set}]$. But this latter functor $(\bar{F})^\ast$ is monadic so long as we assume that $\bar{F}$ is bijective on objects (something which seems entirely sensible to do to me, I didn’t see it in the paper but perhaps I didn’t look closely enough); in any case, assuming $(\bar{F})^\ast$ is monadic then when we pull it back along $\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}]$, we get a functor $\mathcal{M} \to \hat{\mathbb{G}}$ which is monadic as long as it has a left adjoint, which it clearly does (mumble mumble locally presentable mumble mumble filtered colimits). And once we know that $\mathcal{M}$ is monadic over globular sets, we can equally well view a model of a coherator as a globular set with extra structure, and can recast all the stuff about coherators as stuff about the corresponding monads on globular sets.

(This sort of argumentation is pretty much how Clemens recasts all the globular operad stuff in the things he’s written on this topic).

Ah right well this is part of the nerve theorem which Mark Weber proved. Suppose you have a category $\mathcal{D}$ and a small dense subcategory $\mathcal{C}$ of it. Then there is a technical notion of a monad $T$ on $\mathcal{D}$ having arities from $\mathcal{C}$ which allows you to prove all sorts of wonderful things about it. The condition is very roughly that if when you restrict $T$ to a functor $\mathcal{C} \to \mathcal{D}$ and then left Kan extend back to an endofunctor of $\mathcal{D}$, you get back the $T$ you started with. Actually, this isn’t quite strong enough, it turns out; what you actually need is that when you take the singular functor $\mathcal{D}(T,1) \colon \mathcal{D} \to [\mathcal{D}^op, \mathrm{Set}]$, restrict it along $\mathcal{C} \hookrightarrow \mathcal{D}$ and then left Kan extend, you get back to where you started. The whole thing is in in Mark’s paper called “Familial 2-functors” etc.

Ah, okay, thanks. Does it agree with what I guessed in the case of a p.r.a. functor?

That’s pretty much what it is in the paper anyway: you just need to read pp. 3–4 for the omega-groupoids, and pp. 18–19 for the omega-categories.

To me it seems instead there is the claim that a certain important problem is open and an attempt to learn what precisely the statement of the problem is supposed to be.

Maybe we can just state the problem as “prove the homotopy hypothesis for all definitions of ∞-groupoid,” or more generally “prove the equivalence of all definitions of ∞-category”?

There are two possibilities: the simple solutions are really too simple to be good. Or the complicated approaches are actually too complicated to be good.

Or a third possibility, which I think is likely to be the case: the simple solutions are more tractable and thus easier for many purposes, but there are some purposes for which they are too simple, and for which more complicated solutions are necessary.

This is the case in some other situations where there are many different definitions of a concept, e.g. spectra. The definition of an $\Omega$-spectrum is the simplest and easiest to write down. But they don’t give you a monoidal model category. Probably the simplest monoidal model category of spectra is symmetric spectra. But the homotopy groups of a symmetric spectrum are hard to calculate, and they don’t “equivariantize” very well. Orthogonal spectra are a little more complicated and solve these problems, but not all orthogonal spectra are fibrant and so the 0th space functor can be tricky to deal with. EKMM S-modules solve that problem, but are yet more complicated.

Or a third possibility, which I think is likely to be the case: the simple solutions are more tractable and thus easier for many purposes, but there are some purposes for which they are too simple, and for which more complicated solutions are necessary.

Okay, great, this is what I am trying to find out here, while coming across as somebody who is rehashing old disagreements.

I want to know: what is a good motivation for these complicated definitions of $\infty$-groupoids?

I’ll play advocatus diaboli , to highlight what I am after, so what I say now is intentionally made to sound a bit strong, to trigger reactions:

we have complicated structures here, whose definition alone fills articles. There are no examples for these objects given. There are no applications of these objects given. There is no theory about these objects.

All we have is one single motivation: the desire to find an “algebraic” left hand in the homotopy hypotesis $\infty Grpd \simeq Top$. But then, the definitions given are not shown to solve this!

So what is it we have achieved here? How do we know this is not a dead end, with so little in our hands?

But now I look at it and find: there is already something solving this motivational problem. And it is elegant, useful, has examples and applications and comes with a good theory.

So of course I am wondering: where are we headed: why do we need these complicated definitions?

There is probably a good answer to this. Probably if one formulates sharply what one is really looking for, what it really is that existing algebraic models do not satisfy, why one needs this and what it is good for, then probably one will have good motivation.

Cause, there must be better motivation than just “Grothendieck said so, too”. If in physics you say this with “Einstein” instead, you score on the crackpot index.

All right, so that was the harsh version. I don’t actually strictly feel this way. But maybe putting it this way can serve a purpose.