The n-Category Café

As a (hopefully interesting) remark, this “free braided monoidal category on a commutative monoid object”, called BrSet here, is really useful in the problem of quantization of Poisson Hopf algebras. Namely (see page 5 of the link), a braided lax (again!) monoidal functor $F:\text{BrSet}\to C$ satisfying some conditions (basically those that recognize nerves of groups among all simplicial sets) is equivalent to a Hopf algebra in $C$.

The problem (on which we’re working with my student Jan Pulmann) is to use BrSet for quantization of other things - for the moment it works for groupoids (page 10), but the plan is to make it work for (nice) higher groupoids. Roughly speaking a symmetric lax monoidal functor $\text{FinSet}\to C$ is the same as a (plain) functor from FinSet to the category of commutative algebras in $C$, i.e. to a symmetric cosimplicial algebra in $C$, which should be seen as “the algebra of functions on the nerve of something”. If FinSet is replaced with BrSet (and symmetric with braided) the algebras become noncommutative but the general structure is not clear to us (as I wrote, so far it works for Hopf algebras and (some) Hopf algebroids).

One amusing thing is that since $\text{BrSet}=\hat{1}$ and we want braided lax functors from BrSet, we are really interested in $\hat{\hat{1}}$.

Wow, that is really cool! I hadn’t been bold enough to think about iterated laxification. Should this called “relaxation”? It’s great to see that it has applications in physics.

It’s also great to see the free braided monoidal category on a commutative monoid showing up in your work… with a typical morphism looking like this:

So now I have to ask:

1) If $C$ is a monoidal category, what does a lax monoidal functor $f: \Delta_a \to C$ look like? I want an answer that describes it as sort of structured object in $C$: a monoid object with some extra bells and whistles. (In case anyone forgets, $\Delta_a$ is the monoidal category of finite ordinals and order-preserving maps, with addition as the monoidal structure.)

2) If $C$ is a symmetric monoidal category, what does a lax monoidal functor $f: FinSet \to C$ look like? Again, I want to see some sort of structured object in $C$: a commutative monoid object with some extra bells and whistles.

3) Is this laxification process $A \mapsto \hat{A}$ some sort of monad?

Re: #3, for any 2-monad $T$, the lax morphism classifier is a comonad on the 2-category $T Alg_{strict}$ of $T$-algebras and strict maps, and a monad on the 2-category $T Alg_{lax}$ of $T$-algebras and lax maps. This monad/comonad pair are precisely the ones induced by the (identity-on-objects) inclusion functor $T Alg_{strict} \to T Alg_{lax}$ and its left adjoint.

The coalgebras for this comonad played an important role in my paper Enhanced 2-categories and limits for lax morphisms with Steve Lack. In the pseudo rather than lax case, this comonad is a cofibrant replacement for a model structure on $T Alg_{strict}$ whose homotopy 2-category is $T Alg_{pseudo}$; thus its coalgebras are a sort of algebraically cofibrant object. In particular, if $T$ is the 2-monad whose algebras are $Cat$-valued diagrams on some small 2-category, then the cofibrant algebras are the flexible weights, while the algebraically cofibrant ones are the PIE weights.

This suggests that the lax morphism classifier could be regarded as a sort of “cofibrant replacement” for a “lax model structure”, although I don’t think anyone has tried to make that precise.

Mike wrote:

Re: #3, for any 2-monad $T$, the lax morphism classifier is a comonad on the 2-category $T Alg_{strict}$ of $T$-algebras and strict maps, and a monad on the 2-category $T Alg_{lax}$ of $T$-algebras and lax maps.

Great! I have a vague intuition that this should be lax-idempotent monad, or something like that. It’s just the idea that laxifying and then relaxifying isn’t much different than laxifying — I mean, how relaxed can you get?

On the other hand, if laxifying were lax-idempotent, why would Pavol Severa get mileage out of doing it twice?

So I’m a bit confused. Maybe my vague intuition only applies to the pseudo case:

In the pseudo rather than lax case, this comonad is a cofibrant replacement for a model structure on $T Alg_{strict}$ whose homotopy 2-category is $T Alg_{pseudo}$.

Great, so my question about fibrant replacement wasn’t completely off base, though I seem to have lost a “co-“, perhaps because there’s both a monad and a comonad running around here.

A cofibrant replacement functor that’s also a comonad feels like it must be lax-idempotent. After all, you don’t make something more cofibrant by doing cofibrant replacement twice!

Oh, wait… cofibrant replacement will only be “lax-idempotent up to homotopy”.

The lax morphism classifier of an algebra is, as was shown in Steve Lack’s paper Codescent and coherence the codescent object of the resolution of the algebra. You can be sure that these colimits exist just by general cocompleteness arguments.

But there is still the question of computing these colimits explicitly, so that you can see pretty structures like $\Delta_+$ pop out. Fortunately, in many cases, these simplicial objects whose codescent objects we want to compute have many nice properties - for instance if the 2-monad $T$ in question is cartesian (e.g. the 2-monad for strict monoidal categories) then the simplicial object is, in fact, an internal category – which is a good start. So if the 2-monad is on Cat - as usual - the simplicial object is then a double category.

Recently Mark Weber in Internal Algebra Classifiers showed that if this double category is what he called a crossed double category, meaning that the source map is a (split) opfibration compatible with the composition and identity structure of the double category then you can get a neat construction of its codescent object. The objects are those of the double category and the morphisms are equivalence classes of maps $a \to b \to c$ whose first leg is a vertical map and whose second leg is horizontal. The opfibration structure enables you to compose them.

In Section 6 of his paper he uses this to calculate the free strict monoidal category containing a monoid (though, as explained there, this case was understood before), the free braided strict monoidal category containing a commutative monoid and many other nice things explicitly. (I suspect that his work, possibly with a little a bit of adaptation, can be used to calculate the lax morphism classifier of a general monoidal category – not just the terminal one.)

(Incidentally these crossed double categories also pop up in the theory of (algebraic) weak factorisation systems as double categories of left maps. See Section 3 of Categories of weak maps.)