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Stream: learning: questions

Topic: Fields categorically

Fabrizio Genovese (Apr 04 2020 at 15:55):

What is the right categorical setting to talk about fields (as in rings with inverses)? For instance, we can define algebraic theories as Lavwere theories, and generalize to props. Fields are notoriously problematic in classical universal algebra, what is the right categorical tool to formalize them? Is there a way to define a field an "take models of field theory" in a category via functors like we do with Lawvere theories?

Nathanael Arkor (Apr 04 2020 at 16:50):

you can talk about fields as models of finite product + finite coproduct sketches (https://ncatlab.org/nlab/show/sketch)

Nathanael Arkor (Apr 04 2020 at 16:50):

see example 4.3 of On sifted colimits and generalized varieties (http://www.tac.mta.ca/tac/volumes/8/n3/n3.pdf)

Fabrizio Genovese (Apr 04 2020 at 16:54):

Very interesting.

Fabrizio Genovese (Apr 04 2020 at 16:54):

I guess there's no way to have a field if you just have a monoidal category, right?

Fabrizio Genovese (Apr 04 2020 at 16:54):

(I see no way to do this, but I hope to be mistaken)

Fabrizio Genovese (Apr 04 2020 at 16:56):

Also I'm not sure I understand the commutativity condition on page 50

Fabrizio Genovese (Apr 04 2020 at 16:57):

It's embedding Y in X on one hand, and taking the multiplicative inverse on the other, and then multiplies them. This should give back 1, right? Why does it give back the embedding morphism instead?

Nathanael Arkor (Apr 04 2020 at 17:03):

I don't see how you would define something like a field in a monoidal category, because you need something like a diagonal to describe the conditions for an inverse

Fabrizio Genovese (Apr 04 2020 at 17:05):

Yes, you need a copy operation. And at that point you have a product, that we know doesn't cut it

Fabrizio Genovese (Apr 04 2020 at 17:05):

So I guess you really need a product and a coproduct, that's unavoidable T_T

Nathanael Arkor (Apr 04 2020 at 17:07):

This should give back 1, right?

yes, I would have thought so — I'm not quite sure what's going on with that diagram either

Reid Barton (Apr 04 2020 at 17:07):

Fabrizio Genovese said:

It's embedding Y in X on one hand, and taking the multiplicative inverse on the other, and then multiplies them. This should give back 1, right? Why does it give back the embedding morphism instead?

I think that's a typo.

Fabrizio Genovese (Apr 04 2020 at 17:09):

Ok, then i guess i get this now

Fabrizio Genovese (Apr 04 2020 at 17:09):

well, it complicates what I was working on tenfold, but at least my eyes are open now :P

Fabrizio Genovese (Apr 04 2020 at 17:10):

Thanks!

Paolo Capriotti (Apr 04 2020 at 17:16):

I'm not sure about fields, but at least for (abelian) groups you can play the game where you make the comonoid operations part of the group structure, and this you can do in a (symmetric) monoidal category. This way you get things like bimonoids, Hopf monoids and Frobenius monoids.

Fabrizio Genovese (Apr 04 2020 at 17:17):

Yes, absolutely

Fabrizio Genovese (Apr 04 2020 at 17:17):

You just put the copy morphism in the signature

Nathanael Arkor (Apr 04 2020 at 17:39):

maybe meadows are appropriate for your context, @Fabrizio Genovese?
they're actually algebraic, so they're much better behaved

Fabrizio Genovese (Apr 04 2020 at 17:41):

I have to check, but I am not sure of that. I'm doing cryptography and finite fields are heavily used there

Fabrizio Genovese (Apr 04 2020 at 17:43):

Hm. ok this really is an interesting concept

Fabrizio Genovese (Apr 04 2020 at 17:43):

I did not know of that

Fabrizio Genovese (Apr 04 2020 at 18:51):

Woa. This meadow thing really is a rabbit hole. Turns out people have problems with 0 not having an inverse in finite field crypto. It's too soon to tell but it may be that using meadows instead of fields could be just better.

John Baez (Apr 04 2020 at 19:44):

Digression: there's another generalizations of fields called "pastures".

Joe Moeller (Apr 04 2020 at 20:49):

Pastures are generalizations of meadows, and are specifically designed to have nicer categorical properties than meadows, fields, and hyperfields.

sarahzrf (Apr 04 2020 at 20:49):

jdskl

John Baez (Apr 04 2020 at 20:50):

jdskl?

sarahzrf (Apr 04 2020 at 20:54):

just keysmashing

Nathanael Arkor (Apr 04 2020 at 20:58):

@Joe Moeller: do you have a reference?

Fabrizio Genovese (Apr 04 2020 at 20:58):

Joe Moeller said:

Pastures are generalizations of meadows, and are specifically designed to have nicer categorical properties than meadows, fields, and hyperfields.

Do you have any reference? Google is not giving me anything. Anyway meadows are already waaaay better than fields for what I have to do, I'd like to check if pastures can improve the situation even more

Joe Moeller (Apr 04 2020 at 21:01):

I have to look through some notes, I'll check.

John Baez (Apr 04 2020 at 21:02):

I think Matt Baker wrote about pastures, right?

Joe Moeller (Apr 04 2020 at 21:03):

Right, I'm trying to figure out which one it is.

Joe Moeller (Apr 04 2020 at 21:08):

Well, here's a talk at least:
https://jh.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7babd930-de74-4eba-8e77-aaea010462ca

Fabrizio Genovese (Apr 04 2020 at 21:23):

Ok, these seems too general for what I need, but interesting anyway

Sam Tenka (Apr 04 2020 at 23:48):

@Fabrizio Genovese Some perhaps very off-track thoughts:
a. As you mention, fields are a bit tricky since they don't form a universal variety, as can be seen by the HSP theorem. That said, it would be cool to "formally close the class of fields" by HSP and work with these "ghost fields". I don't know what I mean by this, but it sounds like a fun thing to try.
b. Perhaps fields are useful as special rings? That is, as rings for which every module is free? (This is just one characterization; I bet there are great viewpoints on fields orthogonal to this one).

Nathanael Arkor (Apr 05 2020 at 00:02):

@Sam T (naive student): The structure of finite meadows (https://arxiv.org/pdf/0903.1196v1.pdf) shows that the class of finite meadows is the closure of the class of finite fields under finite products, so you could think of meadows as carrying out this idea to some extent

Sam Tenka (Apr 05 2020 at 00:07):

Great! Thanks, @Nathanael Arkor! Sounds like meadows are a neat concept.
edit: their Definition 2.4 (defn of generalized inverse) looks like a characterizing property of penrose moore inverses in linear algebra!

Nathanael Arkor (Apr 05 2020 at 00:08):

I would be interested to see if there was a nice characterisation of the closure under the other parts of HSP, though

Nathanael Arkor (Apr 05 2020 at 00:08):

and if the result also holds for infinite meadows

Joe Moeller (Apr 05 2020 at 00:09):

Baker says meadows and partial fields don't have good notions of tensor product, which is part of the motivation for extending the definition to pastures.

Nathanael Arkor (Apr 05 2020 at 00:12):

the definition of meadow in the talk you linked looks much less pleasant than the definition of meadow: it's not algebraic for one thing

Nathanael Arkor (Apr 05 2020 at 00:14):

does he give a categorical definition in the talk?
this one is very set-theoretic

Nathanael Arkor (Apr 05 2020 at 00:16):

I've just flicked through the video — I know partial fields are used in matroid theory, and matroids seem to play a prominent part in the talk — were pastures specifically defined for matroid theory?

Joe Moeller (Apr 05 2020 at 00:20):

I get the sense that matroids aren't the end goal. The real end goal is understanding $\mathbb F_1$.

Joe Moeller (Apr 05 2020 at 00:21):

Matroids play a technical role is algebraic geometry over hyperfields.

Nathanael Arkor (Apr 05 2020 at 00:25):

I would be interested to see if there was a nice characterisation of the closure under the other parts of HSP, though

ah, meadows are algebraic, so they should satisfy HS as well — which means that meadows exactly correspond to HSP-closure of fields, at least in the finite setting

Joe Moeller (Apr 05 2020 at 00:37):

this is in the wrong chat (edit: it was fixed)

Nathanael Arkor (Apr 05 2020 at 00:39):

@John Baez: I find it helps to narrow down on a specific topic by clicking on the heading, so as not to accidentally post in the wrong topic

Nathanael Arkor (Apr 05 2020 at 00:39):

(it's very hard to get the context of messages when you see everything at once, too!)

Fabrizio Genovese (Apr 05 2020 at 00:49):

Nathanael Arkor said:

I would be interested to see if there was a nice characterisation of the closure under the other parts of HSP, though

ah, meadows are algebraic, so they should satisfy HS as well — which means that meadows exactly correspond to HSP-closure of fields, at least in the finite setting

Yes, this is why I instantly loved them

Fabrizio Genovese (Apr 05 2020 at 00:50):

They have an equational presentation, so they are clearly satisfying HSP

Sam Tenka (Apr 05 2020 at 00:50):

@Nathanael Arkor Does the following sound right?

The ring {of piecewise continuous real functions (on, say, the circle) that have open support} is a meadow. But it isn't a product of fields, since it has no minimal idempotents

[edited from before, where I was considering continuous functions and hence wouldn't have gotten a meadow)

Fabrizio Genovese (Apr 05 2020 at 00:51):

Sam Tenka said:

Nathanael Arkor Does the following sound right?

The ring of continuous real functions (on our favorite infinite domain --- say, the circle) is a meadow. But it isn't a product of fields, since it has no minimal idempotents

Actually you can try to verify this. They give the equations for a meadow, and there are only two more equations to be verified if you already know you have a commutative ring

Fabrizio Genovese (Apr 05 2020 at 00:51):

Joe Moeller said:

Baker says meadows and partial fields don't have good notions of tensor product, which is part of the motivation for extending the definition to pastures.

What do you mean? If you mean tensor product as monoidal product, then meadows have products

Fabrizio Genovese (Apr 05 2020 at 00:52):

If you mean the "true" tensor product, defined with a universal property like we do for rings, then you are probably right

Fabrizio Genovese (Apr 05 2020 at 00:53):

For the applications I have in mind it is enough to be able to put some wires in parallel in a string diagram tho, so I can definitely just use the cartesian product :slight_smile:

Fabrizio Genovese (Apr 05 2020 at 00:53):

Anyway, fields can obviously be interpreted as meadows. I'm courious if this can be made more formal tho, akin to Mal'cev interpretations

Fabrizio Genovese (Apr 05 2020 at 00:54):

But then again Mal'cev theory is for varieties, so I guess even understanding what this means precisely is hard, if not impossible

Joe Moeller (Apr 05 2020 at 03:12):

Yeah, I meant like the one with the funny universal property.

Christian Williams (Apr 05 2020 at 05:07):

Fabrizio Genovese said:

What is the right categorical setting to talk about fields (as in rings with inverses)?

Y'all have been talking about interesting stuff that's probably better, but I'm mainly familiar with the notion of field as a geometric theory -- hence something people find natural in the context of topos theory.

thfld.png

Just wondering, do you want to consider "fields in a monoidal category"? If so why?

Fabrizio Genovese (Apr 05 2020 at 11:01):

I am basically investigating cryptographic protocols from a process theoretic point of view. Process theories are basically SMCs, so I need my stuff to live there. Also, I cannot assume to work in Set because I have to be able to distinguish between functions that are easy to compute and functions that are not. Recasting Set in complexity-theoretic terms is a monumental work that is very much outside of the scope of what I am doing, so that's not an option. What remains is recasting the structures I need as coloured props, and being very clever about their signatures so that some operations cannot be used. This is very ok for groups, but lately I was thinking that fields are also very much used in crypto, and those clearly don't form a prop. My fear was indeed that I had to drop props altogether and directly employ toposes. This does not pose any problem wrt what I am doing, but it just puts the important stuff out of focus, so i'd gladly avoid it :slight_smile:

Verity Scheel (Apr 05 2020 at 13:25):

Since you mentioned props, it reminded me of graphical linear algebra, which actually has a neat story for dividing by zero, although it may not be what you need: https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/

I can’t quite tell if it forms a (skew) meadow or not, hmm ...

Fabrizio Genovese (Apr 05 2020 at 14:49):

I am reasonably familiar with Pawel's work, but linear algebra is a bit too much for my applications

Fabrizio Genovese (Apr 05 2020 at 14:50):

Actually, it goes in the opposite way. It's too well-behaved!

Sam Tenka (Apr 06 2020 at 18:44):

@Fabrizio Genovese
Ah, yes, good advice. Upon checking, it was immediately clear that I was wrong! Here's something that I /did/ check:

Consider the (countably infinite) meadow $M$ of finite-support functions from $\mathbb{N}$ to $\mathbb{Z}/2\mathbb{Z}$. This is a commutative ring, and every element is idempotent and hence its own generalized inverse. But it cannot be a product of fields: for, each factor $F$ in such a product would contain some nonzero element of the meadow $M$ and thus would contain some sub-ring isomorphic to $\mathbb{Z}/2\mathbb{Z}$; consequently, $F$ would have characteristic $2$. However, since the meadow $M$ has no element that obeys $x^4=x$ without also obeying $x^2=x$, each factor $F$ must be the minimal characteristic $2$ field, namely $\mathbb{Z}/2\mathbb{Z}$. So $M$ would be a product of $\mathbb{Z}/2\mathbb{Z}$ s. But, whether this product is finitary or infinitary, it cannot have the same cardinality as $M$, since $2^{\text{finite}} < \text{{countable}} < 2^{\text{countable}}$.

@Nathanael Arkor
So the variety of meadows is not the closure of the class of fields under P! (But this doesn't resolve the question of whether they are the closure of the class of fields under HSP).

Fabrizio Genovese (Apr 06 2020 at 18:48):

Wait, I am not sure I follow. If I have two fields F, G and I consider them as meadows, then F x G is a meadow, right?

Sam Tenka (Apr 06 2020 at 18:49):

Fabrizio Genovese said:

Wait, I am not sure I follow. If I have two fields F, G and I consider them as meadows, then F x G is a meadow, right?

Yep! Every product of fields is a meadow. We show that a sort of converse doesn't hold: not every meadow is a product of fields.

Fabrizio Genovese (Apr 06 2020 at 18:49):

Oh, now I follow

Fabrizio Genovese (Apr 06 2020 at 18:50):

Yes, this makes sense, and actually would be a really really strong condition

Fabrizio Genovese (Apr 06 2020 at 18:50):

Now I get your analogy with linear algebra. A (finite dimensional) vector space can be seen, squinting your eyes a bit, as a product of fields

Fabrizio Genovese (Apr 06 2020 at 18:51):

And indeed, vector spaces are such a strong algebraic theory that they are all free, which is increadible if you think about it

Fabrizio Genovese (Apr 06 2020 at 18:53):

I always thought this could be related to the fact that we are defining a universally quantified theory on something that is not universally quantified, and somehow the existential quantifier of fields plays some tricks in the definition that gives you freeness, but I don't know if there is a deep nice formal logical reason for this, nor if someone studied this in detail from this point of view

Sam Tenka (Apr 06 2020 at 18:59):

Fabrizio Genovese said:

I always thought this could be related to the fact that we are defining a universally quantified theory on something that is not universally quantified, and somehow the existential quantifier of fields plays some tricks in the definition that gives you freeness, but I don't know if there is a deep nice formal logical reason for this, nor if someone studied this in detail from this point of view

I like the shape of this question / remark! This is implicit in your remark, but the non-equationality of fields comes not only from its existential but also from its partial function (same thing, though, if the quantifier has a domain restricted to non-zeros). It's either interesting or trivial to note that if we insist on total functions, then we collapse to just the zero ring, which also has all modules free. (I forgot about that when claiming that fields are exactly the rings with all modules free!) So there's something I don't yet know how to talk about or isolate that makes fields special, even from the module perspective.

Nathanael Arkor (Apr 06 2020 at 19:00):

@Sam Tenka (naive student): ah, sorry, I forgot to reply yesterday

Sam Tenka (Apr 06 2020 at 19:01):

Nathanael Arkor said:

Sam Tenka (naive student): ah, sorry, I forgot to reply yesterday

But this doesn't resolve the question of whether they are the closure of the class of fields under HSP

doesn't this demonstrate that they are not?

I think this just shows that when we close the class of fields F under product, we get some bigger class G that is strictly between the class F and the variety M. I might be missing something, but isn't it conceivable that once we close G under H and S as well, that we get M exactly?

Fabrizio Genovese (Apr 06 2020 at 19:01):

Yes, as I said for now this is just a vague gut feeling for me. But I also have the feeling that the proper way to say this goes much, much deeper in logic than I'm comfortable with

Nathanael Arkor (Apr 06 2020 at 19:02):

yes, it is conceivable: I realised as soon as I posted it, which I why I removed that part of my message :sweat_smile:

Sam Tenka (Apr 06 2020 at 19:02):

Nathanael Arkor said:

Sam Tenka (naive student): ah, sorry, I forgot to reply yesterday

No worries! As far as I've gathered, this is a very asynchronous, no-obligation, fun-driven forum. That's how I've been treating it anyway. :-)

Fabrizio Genovese (Apr 06 2020 at 19:05):

With meadows you get genuinely new stuff. I think that your idea could be rephrased as follows:

1. There are meadows that are "indecomposable", that is, they cannot be written as a non-trivial product of meadows;
2. There is an indecomposable meadow that is not a field, call it M
3. For each homorphism of meadows $f:F \to M'$, with F field, $im(f) \neq M$

Fabrizio Genovese (Apr 06 2020 at 19:06):

Hm, actually it's more difficult than that, point 3 can be reiterated at libitum

Fabrizio Genovese (Apr 06 2020 at 19:07):

Ok, actually there's another very interesting question we can ask: How does the lattice of congruences of a meadow look like?

Sam Tenka (Apr 06 2020 at 19:07):

Fabrizio Genovese said:

Ok, actually there's another very interesting question we can ask: How does the lattice of congruences of a meadow look like?

Ooh What's a congruence?

Fabrizio Genovese (Apr 06 2020 at 19:07):

For instance, we know that the lattice of congruences of a lattice is distributive, and that the congruences of a group permute with each other. These facts are at the basis of Mal'Cev theory

Fabrizio Genovese (Apr 06 2020 at 19:08):

Sam Tenka (naive student) said:

Fabrizio Genovese said:

Ok, actually there's another very interesting question we can ask: How does the lattice of congruences of a meadow look like?

Ooh What's a congruence?

It's an equivalence relation that respects the operations of your underlying algebra

Fabrizio Genovese (Apr 06 2020 at 19:09):

Essentially, this is what modern universal algebra studies. For instance, we have ways to say that if a variety has permutable congruences, then it shares many similarities with groups

Fabrizio Genovese (Apr 06 2020 at 19:09):

Similarly for distributivity

Fabrizio Genovese (Apr 06 2020 at 19:09):

Studying the properties of the congruences of meadows seems to me the best strategy to identify all those universal structures that are "field like".

Fabrizio Genovese (Apr 06 2020 at 19:11):

I study applied stuff, but this strikes me as a genuinely interesting theoretical question, if someone wants to invest some time in it :slight_smile:

Nathanael Arkor (Apr 06 2020 at 19:15):

But this doesn't resolve the question of whether they are the closure of the class of fields under HSP

it's also not clear to me that closing a "non-algebraic variety" under HSP is even enough to get a variety of an algebraic theory, at least in general

Fabrizio Genovese (Apr 06 2020 at 19:18):

I think it does, because of the HSP theorem. Like, the product of fields is not a field. We include it in anyway, and we recursively close against H and S as well. What we get in the end is a variety, which will admit an equational theory

Fabrizio Genovese (Apr 06 2020 at 19:19):

But for what I was able to see HSP has a highly non-constructive proof, so I don't know if we have any sort of procedure to get the equational theory out of the variety

Nathanael Arkor (Apr 06 2020 at 19:22):

Birkhoff's HSP theorem says that if you have a class of algebraic structures for a given signature, then the class is a variety iff it's closed under HSP

Nathanael Arkor (Apr 06 2020 at 19:22):

but if you have a class of non-algebraic structures, I don't see how you can conclude anything from it

Fabrizio Genovese (Apr 06 2020 at 19:24):

Yes, I see what you mean, the axioms of a generic field are non-universal anyway, so the question is ill-posed as it is

Fabrizio Genovese (Apr 06 2020 at 19:25):

I think "closure" is a very misleading word for HSP by the way

Fabrizio Genovese (Apr 06 2020 at 19:26):

Closure in a variety and closure in a topological space are nothing alike. In a topological space you have an "outher part" you can borrow points from until your open set is closed, there's nothing like this in universal algebra

Fabrizio Genovese (Apr 06 2020 at 19:26):

Either your structure is defined equationally, and then you get the variety straight away, or it's not, and in that case there's no way to "close" HSP, at least not in the naive way of saying "I'll squeeze in all the stuff that is being left out"

Nathanael Arkor (Apr 06 2020 at 19:29):

sure there is: you can take the big union of the class, the collection of binary products of algebras in the class, etc.

Fabrizio Genovese (Apr 06 2020 at 19:32):

Yes, this is a different thing tho: You are saying: If I have a class of ->algebras<-, I can close it and get a variety. What I'm saying is: If I have a class of "things", what is the "smallest" variety I can embed this into?

Fabrizio Genovese (Apr 06 2020 at 19:33):

And as we were saying I don't know if there's even a real way to formalize this question

Sam Tenka (Apr 06 2020 at 19:34):

Hmm... that's a subtle (to me!) and such an important point!
Is it reasonable to take as our working space the ambient class of rings? And find a smallest subvariety that contains the class of fields? Maybe something about this doesn't work since we'll be committed to the signature of rings?

Fabrizio Genovese (Apr 06 2020 at 19:34):

Probably an insightful way to understand this better would be by asking the following question: Are meadows "canonical" wrt fields in some sense?

Nathanael Arkor (Apr 06 2020 at 19:34):

What I'm saying is: If I have a class of "things", what is the "smallest" variety I can embed this into?

ah, I see

Nathanael Arkor (Apr 06 2020 at 19:35):

yes, it only becomes well-formed when you define the new notion of structure, along with homomorphisms, substructures and products

Nathanael Arkor (Apr 06 2020 at 19:35):

so you could talk about the HSP closure of a class of sketches, for instance (which would include fields)

Fabrizio Genovese (Apr 06 2020 at 19:36):

So you are saying this: Consider all fields. They form a subclass of $\mathcal{R}$, the class of rings. This is the class you obtain if you take fields and strip away the "bad equations", those having $\exists$. Now I want to find the smallest $\mathcal{V} \subseteq \mathcal{R}$ such that every field is in $\mathcal{V}$

Fabrizio Genovese (Apr 06 2020 at 19:48):

Actually a way simpler way to rephrase this is: I consider $\bigcap \mathcal{V}$ for each $\mathcal{V}$ containing $F$, the class of fields T_T

Oscar Cunningham (Apr 08 2020 at 09:55):

Here's something that might be useful:

Oscar Cunningham (Apr 08 2020 at 09:57):

Consider the monoidal category of pointed sets with the smash product. A monoid internal to this category is just an ordinary monoid equipped with a $0$ (a two-sided absorbing element).

Oscar Cunningham (Apr 08 2020 at 09:59):

The Hopf-algebras internal to this category are groups with a $0$ adjoined to them; which are precisely the multiplicative structures we want fields to have.

Oscar Cunningham (Apr 08 2020 at 10:00):

So you could define fields by demanding a Hopf-algebra structure on their underlying pointed sets.

Amar Hadzihasanovic (Apr 08 2020 at 10:43):

That's quite nice, actually. You could think of defining a “relative theory” to be a morphism of PROPs $P \to Q$, and a “relative model” to be a commutative square of monoidal functors from $P \to Q$ to a functor $F: \mathcal{C} \to \mathcal{D}$.

If I'm not wrong, this would make fields relative models of $\mathrm{CMon} \to \mathrm{CHopf}$ in $U: \mathrm{Ab} \to \mathrm{Set}_*$.

Amar Hadzihasanovic (Apr 08 2020 at 10:44):

You would recover the usual notion by considering relative theories over the terminal PROP and models in $\mathcal{C} \to 1$...

Fabrizio Genovese (Apr 08 2020 at 10:45):

This is interesting indeed. I thought about meadows more in the meantime, and I realized they are most often fine if you need to do just algebra, but things can explode if your algebra has to interact with other stuff such as, say, topology. For instance, if you see the Reals as a meadow then multiplication is not anymore continous in the standard topology of the reals.

Oscar Cunningham (Apr 08 2020 at 10:50):

Right. But I have no idea if such 'relative theories' are as nice as ordinary theories.

Oscar Cunningham (Apr 08 2020 at 10:55):

@Fabrizio Genovese How does that work out? The multiplication map hasn't changed, right? Do you mean that the inversion map isn't continuous?

Oscar Cunningham (Apr 08 2020 at 10:56):

I should mention that Hopf-algebras in pointed sets also have $0^{-1}=0$, so inversion also won't be contiuous there.

Fabrizio Genovese (Apr 08 2020 at 10:57):

Yes, it's the inversion map that it's not continuous, I stated it this way because it's easy to see it when you compose it with multiplication, but you are 100% right. What I mean is that it becomes apparent when you do
$\lim_{x \to 0^+} \frac{1}{x} = \infty$, but $\frac{1}{0} = 1 \cdot 0^{-1} = 1 \cdot 0 = 0$

Fabrizio Genovese (Apr 08 2020 at 10:58):

Oscar Cunningham said:

I should mention that Hopf-algebras in pointed sets also have $0^{-1}=0$, so inversion also won't be contiuous there.

I see. There was another definition of meadow that I found where you actually add a new element as the inverse of 0, but I don't think this will save the topology of the reals anyway

Morgan Rogers (he/him) (Apr 08 2020 at 14:01):

Fabrizio Genovese said:

Oscar Cunningham said:

I should mention that Hopf-algebras in pointed sets also have $0^{-1}=0$, so inversion also won't be contiuous there.

I see. There was another definition of meadow that I found where you actually add a new element as the inverse of 0, but I don't think this will save the topology of the reals anyway

Wouldn't it? If you add a point at infinity as the inverse of zero your last argument stops working, doesn't it?

Fabrizio Genovese (Apr 08 2020 at 14:10):

Yes, but then do you put this extra point "on the left" or "on the right" of all real numbers? Or you do both, wrapping them in a circle? I didn't look into this seriously at all, but I'd guess there would be quite a few details to fiddle with that could make the continuity of things explode in your face in a way or another :slight_smile:

Joe Moeller (Apr 08 2020 at 15:40):

People definitely do exactly this in the $\mathbb F_1$ gang. For instance here: https://arxiv.org/abs/1308.0042

Joe Moeller (Apr 08 2020 at 15:42):

This ties into the whole meadows/hyperfields/pastures stuff too.

Sam Tenka (Apr 11 2020 at 19:44):

Joe Moeller said:

People definitely do exactly this in the $\mathbb F_1$ gang. For instance here: https://arxiv.org/abs/1308.0042

Ooh what is $\mathbb{F}_1$?!

Fabrizio Genovese (Apr 11 2020 at 19:46):

It's the "field" with one element!

Fabrizio Genovese (Apr 11 2020 at 19:47):

Which doesn't exists in the strict sense but it makes a lot of sense to consider in many situations, so people are generalizing the teory of fields in a way that makes it possible

Sam Tenka (Apr 11 2020 at 19:54):

Fabrizio Genovese said:

Which doesn't exists in the strict sense but it makes a lot of sense to consider in many situations, so people are generalizing the teory of fields in a way that makes it possible

That's really cool! Are there accessible readings about this (for someone who just dabbles in math)?

Joe Moeller (Apr 11 2020 at 20:24):

https://arxiv.org/abs/1801.05337
$\mathbb F_1$ for everyone by Oliver Lorscheid

John Baez (Apr 12 2020 at 01:05):

I always find Wikipedia articles useful when learning about math:

• Wikipedia, Field with one element.

They usually have lots of references, and one can try to pick a nice one.

Morgan Rogers (he/him) (Apr 12 2020 at 09:45):

Joe Moeller said:

https://arxiv.org/abs/1801.05337
$\mathbb F_1$ for everyone by Oliver Lorscheid

I find this quote from the introduction, "However, many approaches contain an explicit definition of $\mathbb{F}_1$, and in most cases, the field with one element is not a field and has two elements. Namely, the common answer of many theories is that $\mathbb{F}_1$ is the multiplicative monoid $\{0,1\}$, lacking any additive structure," rather strange. Having seen Connes talk on this subject, he talks about $\mathbb{F}_1$ as a semifield (or at least refers to the semifield $\mathbb{B}$ whose multiplicative structure is that mentioned above, with $\max$ as addition), so there is an additive structure, there's just no subtraction.

Joe Moeller (Apr 12 2020 at 16:24):

Right, many people actually do consider that additive structure at times. I think a good example of what he's referring to can be found in this paper: https://arxiv.org/pdf/1308.0042.pdf Equations of tropical varieties by Giansiracusa and Giansiracusa.

Rather than defining an object $\mathbb F_1$, one starts by defining the category of modules, $\mathbb F_1$-Mod, to be the category of pointed sets. The basepoint of an $\mathbb F_1$-module M is denoted 0M and is called the zero element of M. This category has a closed symmetric monoidal tensor product ⊗ given by the smash product of pointed sets (take the cartesian product and then collapse the subset M × {0N} ∪ {0M} ×N to the basepoint). The two-point set {0,1} is a unit for this tensor product. An $\mathbb F_1$-algebra is an $\mathbb F_1$-module A equipped with a commutative and unital product map A⊗A→ A (i.e., it is a commutative monoid in $\mathbb F_1$-Mod). Concretely, an $\mathbb F_1$-algebra is a commutative and unital monoid with a (necessarily unique) element 0A such that 0A · x = 0A for all x; thus $\mathbb F_1$-algebras, as defined here, are sometimes called monoids-with-zero. The two-point set {0,1} admits a multiplication making it an $\mathbb F_1$-algebra, and it is clearly an initial object, so we can denote it by $\mathbb F_1$ and speak of $\mathbb F_1$-algebras without ambiguity.

Morgan Rogers (he/him) (Apr 13 2020 at 09:37):

They mention the Boolean semiring $\mathbb{B}$ afterwards, which has the structure I described. I guess the convention is the way that the earlier article describes, strange as that seems. Thanks for the extra source!