Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: The usual distributive law of products over sums

fosco (Feb 10 2025 at 10:28):

All references that I found online seem to skirt over the precise definition of the distributive law between monoid and free abelian groups that we learn in school, saying that $(a+b)(c+d)=ac+bc+ad+bd$.

If I try to write down explicitly the component of the natural transformation
$\mathbb{Z}[X]^* \to \mathbb{Z}[X^*] : \Big(\sum n_{i_1} x^1_{i_1},\sum n_{i_2} x^2_{i_2},\dots,\sum n_{i_k} x^k_{i_k}\Big)\mapsto\,\, ?$
that motivated Beck, however, I have a hard time: I could probably try by induction, but I'd like to know if it appears explicitly written somewhere, so that I can avoid the pain. Any help?

Ralph Sarkis (Feb 10 2025 at 11:42):

Without proof in Alexandre Goy's thesis:
image.png

fosco (Feb 10 2025 at 13:53):

I can't really parse what I'm reading, and the lack of proof doesn't make it better than other refs...

Ralph Sarkis (Feb 10 2025 at 14:22):

Again with no proof :sad: but maybe easier to parse in Maaike Zwart's thesis:
image.png

Ralph Sarkis (Feb 11 2025 at 10:16):

Yet another approximate answer to your question. In Ernie Manes and Philip Mulry's paper:
They prove a somewhat general theorem (4.2.20) that allows them to conclude that the list monad commutes over any commutative monad. It may be more painful to instantiate that result rather than proving it for rings directly.
image.png

fosco (Feb 11 2025 at 14:59):

hmmm... I really didn't think this is so difficult to write down. I mean: someone had to, at some point, no?