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Stream: learning: questions

Topic: Commutative monoids equipped with bizarre garbage

Joshua Meyers (Apr 04 2024 at 22:38):

One would think that one-object monoidal categories are commutative monoids. Well not quite! You also get some bizarre garbage.

Let $(M,\otimes, 1)$ be a monoidal category with exactly one object. Then by the argument shown here, $\text{End}_M(1)$ is a commutative monoid under $\circ$, which coincides with $\otimes$. What happens to the associator and unitors? Since there is only one object, they are $\alpha,\lambda,\rho\in\text{End}_M(1)$. The pentagon identity reduces to $\alpha^3=\alpha^2$ which implies $\alpha=1$. The triangle identity reduces to $\lambda\alpha=\rho$, which implies $\lambda=\rho$. Let $m\coloneqq \lambda=\rho$. So, metonymically denoting $\text{End}_M(1)$ by $M$, our structure is completely determined by a pair $(M:\text{CommMon},m\in M)$.

Now, what is a strong monoidal functor between such structures $(M,m)$ and $(N,n)$? First of all, it is a commutative monoid homomorphism $F:M\to N$ of course. But then we also need invertible $\varepsilon,\mu\in N$ satisfying some equations, see nLab. These equations reduce to $\varepsilon\mu F(m)=n$. This means that the extra garbage, namely the $m$ and $n$, might prevent an otherwise perfectly fine monoid homomorphism from being a strong monoidal functor, if it turns out that there is no invertible $x\in N$ such that $xF(m)=n$.

Finally, what is a monoidal natural transformation between two such functors $(F,\eta,\varepsilon)$ and $(F',\eta',\varepsilon')$? It is an element $f\in N$ such that the diagrams in the nLab commute --- one of these is $f\varepsilon=\varepsilon'$, which forces $f$ to be invertible, and the other is $f\mu=\mu'f^2$, which, since $f$ is invertible, forces $f\mu'=\mu$. Thus $\varepsilon\mu=\varepsilon'\mu'$. Moreover, all monoidal transformations between strong monoidal functors between one-object monoidal categories are invertible.

Thus strong monoidal functors $(M,m)\to (N,n)$ modulo isomorphism are essentially homomorphisms $F:M\to N$ and invertible $x\in N$ such that $xF(m)=n$.

Does anyone have any intuition as to what this extra garbage is doing? Is there a good reason that the "one-object monoidal category = commutative monoid" folklore fails? Or does it mean that our elementary definitions are somehow wrong?

Mike Shulman (Apr 04 2024 at 23:08):

The correct slogan is not "one-object monoidal categories are commutative monoids" but "pointed connected monoidal categories are commutative monoids".

Mike Shulman (Apr 04 2024 at 23:11):

In other words, the category of commutative monoids should not be expected to be equivalent to a full sub-2-category of MonCat, but of the coslice 2-category $1 / \mathrm{MonCat}$.

Mike Shulman (Apr 04 2024 at 23:12):

This gets rid of the extra garbage.

Joshua Meyers (Apr 05 2024 at 01:07):

So I just realized that I made a mistake, "one-object monoidal category = commutative monoid" actually works. The following is incorrect:

These equations reduce to $\varepsilon\mu F(m)=n$. This means that the extra garbage, namely the $m$ and $n$, might prevent an otherwise perfectly fine monoid homomorphism from being a strong monoidal functor, if it turns out that there is no invertible $x\in N$ such that $xF(m)=n$.

since both $m$ and $n$ are invertible, so there will always be a unique $x$ such that $xF(m)=n$. Thus, up to isomorphism, we can ignore $m$. Similarly, a morphism of one-object monoidal categories decomposes $x$ into $\varepsilon\mu$. But the particular decomposition is not preserved by 2-isomorphism: up to 2-isomorphism, a morphism does not decompose $x$ --- it is simply a morphism $F:M\to N$.

Joshua Meyers (Apr 05 2024 at 01:08):

I did start working out the category that you suggested @Mike Shulman and it seems to work too but I have not finished. Why did you intuit that it was better?

John Baez (Apr 05 2024 at 01:08):

This, and its generalizations, is a well-known issue.

John Baez (Apr 05 2024 at 01:09):

I've read Mike talk about this before, when Cheng and Gurski first released their papers on the "extra garbage" problem.

John Baez (Apr 05 2024 at 01:09):

So Mike can answer how he first intuited it.

Mike Shulman (Apr 05 2024 at 01:10):

E.g. it's discussed in section 5.6 of Lectures on n-categories and cohomology.

John Baez (Apr 05 2024 at 01:11):

And, for Cheng and Gurski's work, try The periodic table of n-categories for low dimensions II: degenerate tricategories and the previous paper.

Mike Shulman (Apr 05 2024 at 01:12):

That was so long ago I don't know if I could definitely reconstruct my thought processes, but probably it came from homotopy theory, where pointed spaces and their loopings and deloopings are a standard object to study.

Mike Shulman (Apr 05 2024 at 01:13):

I didn't remember that the extra garbage does actually disappear for one-object monoidal categories, but as you can see from Cheng and Gurski's papers, there are other cases where it doesn't disappear even up to equivalence, unless you take the pointed perspective.

Jonas Frey (Apr 05 2024 at 07:41):

Another incarnation of this phenomenon: monoids don't precisely correspond to 1-object categories since the former form a 1-category and the latter form a 2-category. Again, considering "pointed" 1-object categories, however redundant that may seem, gets rid of the 2-cells, and we get a 1-category.

Joshua Meyers (Apr 05 2024 at 12:06):

Jonas Frey said:

Another incarnation of this phenomenon: monoids don't precisely correspond to 1-object categories since the former form a 1-category and the latter form a 2-category. Again, considering "pointed" 1-object categories, however redundant that may seem, gets rid of the 2-cells, and we get a 1-category.

But this discrepancy could also be viewed in a positive light, as revealing that monoids are actually a 2-category, with a 2-morphism $\eta:f\Rightarrow g:M\to N$ as an element of $N$ such that $(\forall m\in M)(\eta f(m)=g(m)\eta)$.

Tim Hosgood (Apr 05 2024 at 13:23):

I guess that this is like the monoid version of the group story: there's an equivalence of $(2,1)$-categories between groups and pointed connected groupoids (loop and deloop: $\Omega$ and $\mathbb{B}$). Or there's a similar $(2,1)$-equivalence between $G$-actions on sets and $0$-truncated groupoids over $\mathbb{B}G$.

Mike Shulman (Apr 05 2024 at 14:59):

Joshua Meyers said:

But this discrepancy could also be viewed in a positive light, as revealing that monoids are actually a 2-category, with a 2-morphism $\eta:f\Rightarrow g:M\to N$ as an element of $N$ such that $(\forall m\in M)(\eta f(m)=g(m)\eta)$.

I don't think that's a particularly useful tack to take. Most of the time when we study monoids we are interested in them as a 1-category. I think it's better to say that when we want to consider that 2-category, we replace monoids by their deloopings.

Martti Karvonen (Apr 07 2024 at 11:25):

Perhaps that's true for monoids, but for groups the associated 2-category does explain some group-theoretic constructions. For example, HNN-extensions are (certain) coinserters in the 2-category of groups.

Mike Shulman (Apr 07 2024 at 15:10):

I said most of the time! (-: