Added a reference on cohomological viewpoint on entropy,

- P. Baudot, D. Bennequin,
*The homological nature of entropy*, Entropy, 17(5):3253–3318, 2015 doi (open access)

added pointer to

- Ted Jacobson,
*Entropy from Carnot to Bekenstein*(arXiv:1810.07839)

Entropy considerations from thermodynamics generalize to many PDEs. I have added a note in the references section of entropy:

]]>Entropy-like quantities appear in the study of many PDEs, with entropy estimates. For an intro see

I think I asked a similar question back then. I had glanced at the article, but not gotten any clear idea of what’s going on.

I had that phase for a while: wondering what all those “categorical” discussions of entropy actually do or aim for. I still don’t have a clear picture. Wish somebody would add a paragraph to the entry that explains it.

]]>Somebody wrote within entropy

A discussion of entropy with an eye towards the presheaf topos over the site of finite measure spaces is in

- Mikhail Gromov,
In a Search for a Structure, Part I: On Entropy(2012) (pdf)

There are some functors in the game, but as I did not read in detail, I do not see where really the properties of the whole category of such functors is used. Where is really in this paper the view toward topos point of view (more than just a definition) ? (this is not a complaint, but a question toward appreciation of the topos theoretic aspect there if any!)

I added the following reference to entropy

- William Lawvere,
*State categories, closed categories, and the existence*(subtitle: Semi-continuous entropy functions), IMA reprint 86, pdf

I removed the discussion started by David in #36 from the page.

]]>I have added two references to entropy (on entropy of states on operator algebras).

]]>Comment at entropy.

]]>What happened to the discussion of entropy? I was really curious to see what category theorists had to say on the topic. ;)

I’ve never understood whether we’re supposed to think of every appearance in maths of entropy as a manifestation of a common idea.

Me neither until I got working with my colleague Steve Shea. He did his PhD on entropy under Michael Keane. After discussing the mathematical, information theoretic, medical (yes, there are medical uses of the term!) and physics-related definitions of entropy, we concluded that they were all variations on the same thing - at least all the ones we were aware of. Certainly Kolmogorov entropy is a generalization of Gibbs-Boltzmann-Shannon entropy and there were several other “entropies” out there that we (and my student) found that all turned out to be variations on the same idea.

]]>I had to write rng too.

]]>@ Urs #31

I think not; I’m afraid that somebody might well make a link to unit that should be moved to unit of an adjunction in the future. In fact, I’m surprised that there was only one such link to begin with! (I also briefly moved the page to units to check for links to that name; there were none.)

]]>@ Urs #29

Well, I think that it’s perfectly reasonable to note that torsor trivialisations correspond precisely to elements of the torsor, that oriented $\mathbb{R}$-lines (or $\mathbb{R}_{\geq0}$-lines period) correspond precisely to $\mathbb{R}_{\gt0}$-torsors), and apply these insights to this situation. I just didn’t write that in.

]]>So should we remove the leading sentence “not to be confused with unit of an adjunction” now?

]]>There is now a very general definition at unit.

]]>Well, I can fit that in!

Okay, thanks.

By the way: I was going to complain that units of measurements are not defined as torsor trivializations, but maybe you and Mike have convinced me that the way the entry now presents it is good.

]]>I’ve written unit of an adjunction; next I’ll fix links to unit. (The only one seems to be at adjunction itself.)

]]>Ah, right. Well, I can fit that in!

]]>Yeah, I was a little surprised when I found myself writing it and not covering the unit of an adjunction. (Why is that called “unit”?

because it

$i : Id \to R L$is the unit (= “identity element”) of a monoid $A = R L$ (here: monad)

$i : \mathbb{1} \to A$ $\mu : A \otimes A \to A$ ]]>@ Urs

On the other hand, it seems that in the “Avogadro project” one approach is to

defineit to be some natural number, and then define the kilogram by this.

Yes, but even there, there’s no reason in principle why this should be a natural number. (In practice, of course, it’s easiest if it’s not only a natural number but a multiple of a power of as large a power of $10$ as possible.)

I would have thought we need a genuine disambiguation page here

Yeah, I was a little surprised when I found myself writing it and *not* covering the unit of an adjunction. (Why is that called “unit”? What does it have to do with the other units?) I don’t know if it’s a coincidence that a unit of a ring and a unit of measurement are both special cases of a general concept, but it’s too good to pass up!

@ Mike

Is there a name for a rig in which every nonzero element is a unit? A “semifield”?

Apparently (2nd version). Note that the rig of tropical numbers is an important example.

]]>[edit: Isaid something here but I should better comment later when I have more leisure to be distracted]

]]>Crossing wires with Urs #22, I added a bit to Toby’s unit. I do like this point of view that unifies two or three of the concepts, but I’m also not sure that we want the page called unit to be only about it.

Is there a name for a rig in which every nonzero element is a unit? A “semifield”?

]]>Created unit.

I would have thought we need a genuine disambiguation page here, redirecting to *unit of an adjunction* , *unit in measurement* , *unit of a ring* . I see that you phrased a definition such that the last two are special cases of it, but I am not sure yet if I find this the best explanation:

I would have thought that the “right” definition for “unit of measurement” is: an isomorphism that identified an $\mathbb{R}$-torsor with $\mathbb{R}$.

]]>In principle, Avogadro’s number is not a natural number (merely a positive real number), because the ratio of the masses of a carbon-12 atom and the reference kilogramme in Paris is not is an exact multiple of 1000/12.

That’s true. On the other hand, it seems that in the “Avogadro project” one approach is to *define* it to be some natural number, and then define the kilogram by this.

This should be better said in the entries. I can’t do it now. Maybe later.

]]>Created unit.

]]>