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Stream: learning: questions

Topic: free-forgetful adjoint for Sets and Monoids question

Olli (Jul 01 2021 at 11:52):

Screenshot_20210701-185002_Xodo Docs.jpg

Morgan Rogers (he/him) (Jul 01 2021 at 12:48):

What's the question?

Olli (Jul 01 2021 at 13:47):

Whops, I meant to write the question with the image included but I didn't realize it posted the image separately and then got distracted. I'll write it out a bit later

Olli (Jul 01 2021 at 15:03):

I was trying to work out this example of the adjoint functor pair mentioned in the topic by looking at the left-adjunct of the morphism $m: 1 \to M$ e.g. $m \in M$.

I think my $Fm$ is correct, i.e. it sends each $n \in \N$ to a list of length $n$ with every element of the list as $m$.

I am a little bit uncertain about my $\epsilon_M$, which as I wrote in my note I think it just forgets the indexing set $n$ and therefore it loses the order and count properties of the list, leaving just a set of elements in the list.

So $f$ as a monoid morphism is a constant map sending everything except the unit to $m$.

However when I try to calculate $f$ as the composition of the above two, I can get both $f(a+b) = m$ and $f(a+b) = m \oplus m$ which doesn't seem right, and I feel like I'm just missing something really obvious but at this point I've stared at this thing for so long that I'm running out of my wits.

I'd also like to confirm my understanding that the free functor always adds a point, and that the forgetful functor does not "remove" the point, but just the structure of which point was special, e.g. $UF(\empty) = \{*\}$.

If so, I think then my understanding of what $Uf$ does should be correct, i.e. it works exactly as $f$ except operating on sets.

John Baez (Jul 05 2021 at 02:59):

Are you thinking free-forgetful adjoint between sets and monoids? Your title says "monads".

John Baez (Jul 05 2021 at 03:01):

It'd be better if you explained your terminology a bit. I'll guess that by $U$ you mean the forgetful functor from monoids to sets. Then your last understanding is correct: the function $Uf: UM \to UN$ is just the underlying function of the monoid homomorphism $f: M \to N$

Olli (Jul 05 2021 at 08:51):

Sorry, yes indeed I meant monoids

Olli (Jul 05 2021 at 08:52):

And you are correct that my $U$ is the forgetful functor and $F$ the free one.

Morgan Rogers (he/him) (Jul 05 2021 at 09:19):

Corrected the topic.
I think I understand the question, too. You were right to ask, since it sounds like you've misunderstood what the counit of the adjunction, $\epsilon_M$, does here. Observe that for a monoid $M$, the monoid $FU(M)$ consists of finite (possibly empty) ordered lists of elements of $M$. Can you work out how one might recover a canonical element of $M$ from such a list?

Morgan Rogers (he/him) (Jul 05 2021 at 09:21):

I think that understanding this might resolve your other questions.

Olli (Jul 06 2021 at 08:18):

At first I thought it would be to pick the first element of the list, but that's not a valid monoid morphism.

I think I get it now though, it is to combine all elements of the list using the monoid operation, so $[a, b, c] \in \sum_{n}M^n$ becomes $(a \oplus b \oplus c) \in M$.

And yes this answers the other question I had regarding $f$. Thanks!