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Stream: learning: questions

Topic: Composing monoids

Asad Saeeduddin (Apr 24 2020 at 15:51):

Is it correct to say that monoid objects compose precisely when there is a duoidal structure formed by the monoidal structure in which they are being composed and the monoidal structure in which they are monoids?

Reid Barton (Apr 24 2020 at 16:05):

What do you mean by "monoid objects compose"?

Asad Saeeduddin (Apr 24 2020 at 16:05):

For example, monoidal functors are monoids with respect to Day convolution. The composition of two monoidal functors is another monoidal functor. Coincidentally (or perhaps not so coincidentally) Day convolution and functor composition together form a duoidal structure on the functor category.

There's a couple other examples I've found so far. Day convolution also forms a duoidal structure with the "pointwise" lifting of a monoidal structure from a functor category's codomain, and again we find that the pointwise product of monoidal functors is monoidal. Another simple one is the cartesian product of monoids in Set.

Does this hold in general?

Asad Saeeduddin (Apr 24 2020 at 16:09):

@Reid Barton I mean that given some monoidal structure $\otimes : C \times C \to C$ and two monoids $M, N$ with respect to another monoidal structure $\oplus : C \times C \to C$, $M$ and $N$ "compose" along $\otimes$ if $M \otimes N$ is also a monoid with respect to $\oplus$

Asad Saeeduddin (Apr 24 2020 at 16:10):

If we take $C = Set$ and $\oplus = \otimes$ to be the cartesian product in $Set$, this denegerates to the statement that the product of two monoids is a monoid

Asad Saeeduddin (Apr 24 2020 at 16:19):

The intuitive explanation for the phenomenon (assuming it's not an illusion) is that monoidal functors map monoids to monoids. Thus in a duoidal structure $\star, \diamond$ where $\star$ is a lax monoidal functor cohering $\diamond$ with itself, $\star$ will send a pair of $\diamond$-monoids to a $\diamond$-monoid

Reid Barton (Apr 24 2020 at 16:21):

It's certainly true that
(a) a specific duoidal structure relating the two given monoidal structures
gives you
(b) a specific way to make the category of monoids for one of the monoidal structures monoidal using the other monoidal structure.
This is described at https://ncatlab.org/nlab/show/duoidal+category#bimonoids_bicomonoids_and_duoids