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Stream: learning: questions

Topic: metric monads : a paper by Jiri Rosicky

Jan Pax (Nov 10 2023 at 20:00):

I would like to understand here on the page 9, why there is unique $t:A\to B$ with $t\varepsilon_A=\varphi_{\varepsilon_A}$ as well as the 1st and 3rd equations $=$ below it. Also, I do not follow why there is such a canonical representation $FUFUA\to FUA\to A$ and why it commutes as I suppose.

Ralph Sarkis (Nov 10 2023 at 20:44):

I have been meaning to carefully read this paper for a long time but it is sooo hard. Maybe it's because it's too late but the whole of Observation 3.11 makes no sense to me. If you'd like a reading buddy, we could schedule a call or something.

In the meantime I am pinging @Jason Parker who corrected one of the results in this paper (5.1 I think), and hence might be able to help you. (Sorry if that is not appreciated Jason.)

Reid Barton (Nov 12 2023 at 17:55):

Whatever is being explained in the last paragraph of Observation 3.11 is something standard that must be explained more clearly somewhere else (maybe in Linton's original paper, I have not read it).

Reid Barton (Nov 12 2023 at 17:56):

For what it's worth, I think the "canonical diagram" being referred to is the diagram of all maps from free algebras to $A$ (because this is what is involved in proving that a subcategory is dense, if dense means what I think it means). I actually do not understand what the role of the canonical presentation is in the argument at all, except to introduce $\varepsilon_A$.

Reid Barton (Nov 12 2023 at 17:58):

The claim about $t$ and equations like $a = \varepsilon_A F \widetilde{a}$ are just facts about what happens whenever you have an adjunction $F \dashv U$, and consider morphisms $a : FX \to A$ and $\widetilde{a} : X \to UA$ that correspond under the adjunction.

Reid Barton (Nov 12 2023 at 18:02):

I'm confused by "Observe that $\tilde \Omega(X,Y) = \Omega^{X,Y}$" -- it seems to me that they are not equal (since $\tilde \Omega$ consists of some kind of words, while $\Omega^{-,-}$ already has a composition operation) but I could well be misunderstanding something.

Jan Pax (Nov 14 2023 at 14:22):

This is the reply of Jiri Rosicky to our problem: The presentation is a standard split coequalizer (see, for example, MacLane's book). Then, $t$ exists because the presentation is a coequalizer. My remark: If nothing else, it is such a brief explanation!

Jan Pax (Nov 16 2023 at 18:27):

Though my comment may be considered as a joke, I would still like to see an explanation of it. I would like to know why it is a coequalizer and step by step explanation of the uniqueness and existence of $t$ as given here for $t$ being $u$ there. In particular, what is $u '$ ?