### 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

$p: 1 \to \Delta_a$

sending the monoid in $1$ to the monoid $1 \in \Delta_a$. 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.

## 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.