May 25, 2018

Laxification

Posted by John Baez

Talking to my student Joe Moeller today, I bumped into a little question that seems fun. If I’ve got a monoidal category $A$, is there some bigger monoidal category $\hat{A}$ such that lax monoidal functors out of $A$ are the same as strict monoidal functors out of $\hat{A}$?

Someone should know the answer already, but I’ll expound on it a little…

Here’s an example where this works beautifully.

Let $\Delta_a$, the augmented simplex category, be the category of finite ordinals 0, 1, 2, … and order-preserving maps between these. A famous fact is that $\Delta_a$ is the ‘walking monoid’. In other words, for any strict monoidal category $X$, strict monoidal functors

$f : \Delta_a \to X$

are the same as monoids in $X$.

But another famous fact is that if $1$ is the terminal category made strict monoidal in the only way possible, lax monoidal functors

$g : 1 \to X$

are the same as monoids in $X$.

So, somehow $\Delta_a$ is the “laxification” of $1$: a puffed-up version of the terminal category such that lax monoidal functors out of $1$ can be reinterpreted as strict monoidal functors out of $\Delta_a$.

Indeed, combining these two facts we get a lax monoidal functor

 p: 1 \to \Deltaa$sending the monoid in$1$to the monoid$1 \in \Deltaa$. We then have$$g = f \circ p .$$So, I'm thinking this should be an example of a general pattern. The idea is roughly that for any strict monoidal category$A$, there should be a strict monoidal category$\hat{A}$and a lax monoidal functor$p: A \to \hat{A}$such that every lax monoidal functor$g: A \to X$is of the form$f \circ p : \hat{A} \to X$for some strict monoidal functor$f: \hat{A} \to X$. Or more precisely, precomposition with$p$gives an equivalence of categories$$p^* : StrictMon(\hat{A}, X) \to LaxMon(A, X )$$I imagine that the augmented simplex category will play a big role in the construction of$\hat{A}\$. I’m also imagining that some words like “monoidal nerve” and maybe “fibrant replacement” will show up.

Posted at May 25, 2018 9:40 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3042

Re: Laxification

If I’m not mistaken, your claim is a special case of Lemma B-1.1.6 in the Elephant, for which Johnstone doesn’t give a proof, “which is rather messy to write out in full generality”.

Johnstone talks about 2-categories rather than monoidal categories, but the construction for monoidal categories is a special case, since the general construction keeps the set of objects fixed, thus the 2-category $\hat{A}$ associated to a monoidal category $A$ viewed as a 2-category is in fact a monoidal category again.

I think in general, the 1-cells in $\hat A$ are paths in $A$, and one intruces new 2-cells of the sort of $(f,g)\to (gf)$ and $()\to\mathrm{id}$ – thus relating paths to their composites.

The 2-cells of $\hat A$ are then generated by the 2-cells of $A$ and the new 2-cells, subject to relations to enforce that $\hat A$ is well-defined, and that $A\to\hat A$ is a lax functor.

The augmented simplex category is certainly a special case, maybe the constrcution can also be formulated purely in terms of this category rather than by generators and relations.

And I see a possible link to homotopy theory through the Thomason model structure on $\mathrm{Cat}$, since the construction of $\hat A$ is used to reduce lax limits to 2-limits, and on the other hand homotopy limits of diagrams in $\mathrm{Cat}$ wrt the Thomason model structure are are precisely lax limits, where the index category is viewed as a 2-category. But I don’t really see how this fits together.

Posted by: Jonas Frey on May 26, 2018 12:04 AM | Permalink | Reply to this

Re: Laxification

This is called the lax morphism classifier and exists for algebras over any sufficiently well-behaved 2-monad. I believe it was first constructed in Blackwell-Kelly-Power Two dimensional monad theory, and there are expositions in Lack’s Codescent objects and coherence and 2-categories companion (section 4).

Posted by: Mike Shulman on May 26, 2018 1:35 PM | Permalink | Reply to this

Re: Laxification

As Mike Shulman has already pointed out, this is explained in Steve Lack’s “Codescent Objects and Coherence” paper. It was explained quite well here, during the first Kan Extension Seminar.

Here is informally how the construction works in the strict case:

Let $C$ be a strict 2-category and $(T,\mu,\eta)$ a strict 2-monad on $C$. Given a strict $T$-algebra $a:T A\to A$, we know that $a$ is the (strict) coequalizer of the two arrows $\mu,T(a):T T A\to T A$, or even the strict colimit of the whole bar construction.

Instead of the strict colimit, we can take a particular lax colimit. Intuitively, if $C$ has 1-categories as objects (“2-concrete”), the coequalizer $a$ is such that for every $x\in T T A$, $a\circ\mu(x)=a\circ(T a)(x)$, and universally so. We can instead try to form an object (a 1-category) such that there will only be an arrow $a\circ\mu(x)\to a\circ(T a)(x)$, with a universal property, and satisfying certain coherence conditions.

Such a colimit is called by Lack the lax codescent object of $A$, and it can be obtained via coinserters and coequifiers. Let’s call such an object $A'$. When it exists, it is automatically a strict $T$-algebra.

A slogan could be “what is identified in $A$ is only related by an arrow in $A'$”.

Now comes the main result: let $B$ be a strict $T$-algebra. Then lax morphisms $A\to B$ correspond to strict morphisms $A'\to B$.

This should be somehow expected: lax functors preserve operations only “laxly”, that is, they put arrows whenever there “should” be an identification. In $A'$, identifications are in a way already replaced by arrows, and so, preserving the arrows of $A'$ is the same as laxly-preserving the identifications of $A$.

The whole construction actually works even if $A$ is a lax algebra, however $B$ has to be strict.

This is of course an intuitive explanation, for the actual rigorous and fully general result, you can look at Section 2 of Lack’s paper.

Posted by: Paolo Perrone on May 26, 2018 3:00 PM | Permalink | Reply to this

Re: Laxification

Thanks! It makes a lot of sense, at least intuitively. I’m too lazy to penetrate the details of the generalizations described in Alex Corner’s nice post, especially when we hit the regular cardinals and enhanced factorization systems. I’m more curious about what this construction gives in some other examples.

It seems that if $X$ is a strict symmetric monoidal category, a lax symmetric monoidal functor $g: 1 \to X$ is a commutative monoid in $X$. Is that right? A commutative monoid in $X$ is also the same as a strict symmetric monoidal functor $f: FinSet \to X$. So it looks like we can build $FinSet$ as a lax codescent object of $1$ using the 2-monad for strict symmetric monoidal categories.

The braided case seems more interesting. It seems we should be able to build the “free braided monoidal category on a commutative monoid object” in a similar way, as a the lax codescent object of $1$ using the 2-monad for strict braided monoidal categories. If I’m not mixed up, this is an interesting entity whose morphisms look like graphs embedded in a 3d cube where edges go downwards, and edges can merge but not split:

It’s sort of halfway between $\Delta_a$ and $FinSet$. Click on the picture for more details!

Posted by: John Baez on May 26, 2018 4:24 PM | Permalink | Reply to this

Re: Laxification

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 $\hat{\hat{1}}$.

Posted by: Pavol Severa on May 27, 2018 2:22 PM | Permalink | Reply to this

Re: Laxification

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?

Posted by: John Baez on May 28, 2018 8:22 PM | Permalink | Reply to this

Re: Laxification

Re: #2, a symmetric lax monoidal functor $F:\text{FinSet}\to C$ is the same as an (ordinary) functor from FinSet to the category of commutative monoids in $C$. In fact, every object in FinSet is a commutative monoid (being a tensor power of the generating commutative monoid (1-element set)), and $F$ sends commutative monoids to commutative monoids. A better understanding of this thing, and especially any understanding of what happens in the braided case (when those tensor powers are no longer commutative) would be very welcome.

Posted by: Pavol Severa on May 29, 2018 2:51 PM | Permalink | Reply to this

Re: Laxification

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.

Posted by: Mike Shulman on May 28, 2018 11:23 PM | Permalink | Reply to this

Re: Laxification

Mike wrote:

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.

Not yet a “lax model structure”, but it is at least the cofibrant replacement for the awfs of “$U$-lalis” on $\mathbf{MonCat}$ (Theorem 17 of Bourke–Garner “Algebraic Weak Factorisation Systems II”).

Posted by: Richard Garner on May 29, 2018 1:58 AM | Permalink | Reply to this

Re: Laxification

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”.

Posted by: John Baez on May 29, 2018 12:02 AM | Permalink | Reply to this

Re: Laxification

John wrote:

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?

This is indeed true. In fact, the lax morphism classifier 2-comonad is the “lax-idempotent reflection” of the 2-comonad $F U$ generated by the free-forgetful adjunction $F \dashv U \colon \mathbf{MonCat} \rightarrow \mathbf{Cat}$.

The construction of the lax-idempotent reflection of a 2-comonad is a generalisation of the idempotent reflection of a comonad due to Fakir. The latter involves coequalisers; the two-dimensional generalisation involves codescent objects, hence the appearance of these in Steve’s paper. In general the coreflection process requires not just codescent objects but also transfinite iteration but a little bit of magic (Theorem 4.2 in Blackwell–Kelly–Power) makes it converge after one step.

Similarly, the pseudomorphism classifier is the “pseudo-idempotent reflection” of $F U$.

Posted by: Richard Garner on May 29, 2018 1:51 AM | Permalink | Reply to this

Re: Laxification

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

You might be confusing lax-idempotent with pseudo-idempotent. Lax-idempotence doesn’t imply that $T T A \simeq T A$, only that they are related by an adjunction. (So the “lax” in “lax-idempotent” is a red herring adjective: a lax-idempotent monad is not idempotent. Of course, “lax” often behaves this way.) So doing it twice is really different from doing it once.

Posted by: Mike Shulman on May 29, 2018 8:41 AM | Permalink | Reply to this

Re: Laxification

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.)

Posted by: John Bourke on May 30, 2018 12:41 PM | Permalink | Reply to this

Post a New Comment