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Stream: deprecated: algebra & CT

Topic: The category of fields

Joshua Meyers (Feb 27 2021 at 21:15):

People seem to have a bad opinion of the category of fields $\mathbf{Fld}$, but I think it's actually quite nice. Sure you don't find the usual structures (initial object, pushouts, etc.) but there are other structures instead which are just as nice.

First of all, instead of an initial object you have a family of partially initial objects $I=\{\mathbb{Q}\}\cup\{\mathbb{F}_p|p\text{ prime}\}$. By this I mean that for every field $F$ there is a unique pair $(k \in I,f:k \to F)$. Another way to say this is that $\sum_{k\in I}\mathbf{Fld}(k,-)$ is isomorphic to the functor $c_1:\mathbf{Fld}\to\mathbf{Set}$ sending every field to the singleton $1$. (Maybe we could say that this functor is represented by $I$?)

Secondly, suppose we have a diagram $D=(E\leftarrow k \rightarrow F)$ in $\mathbf{Fld}$. This diagram does not have a pushout in general, but it does have a family of partial pushouts. By this I mean there is a family $EF$ of cocones under $D$ such that for all cocones $L$ under $D$ there is a unique pair $(G\in EF, f:G\to L)$, where $f$ is a cocone-homomorphism. Another way to say this is that $\sum_{G\in EF}\mathbf{Fld}(tG,-)\cong \text{Cocone}(D,-)$, where $tG$ is the tip of the cocone $G$.

As a simple example of this family of partial pushouts, take $D=(\mathbb{Q}(x)\leftarrow \mathbb{Q}\rightarrow \mathbb{Q}(y))$. Then the partial pushouts of this diagram include $\mathbb{Q}(x)\rightarrow \mathbb{Q}(x,y)\leftarrow \mathbb{Q}(y)$, where $x\mapsto x$ and $y\mapsto y$, $\mathbb{Q}(x)\rightarrow \mathbb{Q}(x)\leftarrow \mathbb{Q}(y)$, where $x,y$ both $\mapsto x$, and $\mathbb{Q}(x)\rightarrow \mathbb{Q}(x)\leftarrow \mathbb{Q}(y)$, where $x\mapsto x$ and $y\mapsto x^2$.

If you know what a "compositum" is in field theory, you'll recognize the family of partial pushouts as all possible composita of $E$ and $F$ relative to some ambient field in which $E\cap F=k$.

If the field theory here is too much for you, you can see the same phenomenon in the category $\mathbf{Inj}$ of sets and injections. There, a diagram $X\xleftarrow{f} S\xrightarrow{g} Y$ has a family of partial pushouts $(X\sqcup Y) / \sim$, where $\sim$ runs over all equivalence relations such that $f(s)\sim g(s)$ for $s\in S$ and $x\not\sim x',y\not\sim y'$ for $x,x'\in X,y,y'\in Y$ distinct.

If you really miss an initial object in the category of fields, you could decide to just look at fields of a certain characteristic, but in my view that's just kicking the can down the road, because even in that characteristic, you have families of partial pushouts rather than usual pushouts.

John Baez (Feb 27 2021 at 21:23):

Nice stuff! I like the idea of the forgetful functor $\mathsf{Fld} \to \mathsf{Set}$ being represented by a collection of objects.

some might call these "cocones" but I think that the word "under" suffices

:fear: Not some but everyone in category theory distinguishes cones and cocones, so please talk like us if you want to communicate. In my opinion language exists for communication, and talking in a way that people easily understand almost always trumps talking in the way that seems logical to me... unless the way people easily understand embodies serious misconceptions.

John Baez (Feb 27 2021 at 21:24):

(And see? - you got me to think about language instead of your actual ideas.)

Joshua Meyers (Feb 27 2021 at 21:24):

Well not everyone...I learned from Emily Riehl's book, which uses "cone over" and "cone under". But I'll change it if you insist lol

John Baez (Feb 27 2021 at 21:24):

I'll have to send her a nasty email.

John Baez (Feb 27 2021 at 21:26):

Maybe I'll just send her a corrected latex file of her book. :upside_down:

Joshua Meyers (Feb 27 2021 at 21:28):

From her book (p.74):

Cones under a diagram are also called cocones---a cone under $F:J\to C$ is precisely a cone over $F:J^\text{op}\to C^\text{op}$---but we find the terminology "under" and "over" to be more evocative.

Joshua Meyers (Feb 27 2021 at 21:30):

She also speaks of the "summit" or "apex" of a "cone over" and the "nadir" of a "cone under". I like that, so I changed "vertex" in my post to "nadir".

John Baez (Feb 27 2021 at 21:31):

Apex is standard, I've never heard "summit". I don't mind "nadir" but I bet 90% of people won't know what that means... and a bunch will think you're talking about Ralph.

"Tip" would be a fine word for both the apex of a cone and the nadir of a cocone... I don't think people use it, but if you said "tip of the cone" people would understand it.

Joshua Meyers (Feb 27 2021 at 21:32):

Back to the actual topic: The family of partial pushouts of $D$ can also be expressed by first taking the tensor product $E\otimes_k F$ of $E$ and $F$ as $k$-algebras (this is also the pushout of $D$ in the category of $k$-algebras, and then quotienting that by a maximal ideal $m$ such that $m\cap E= m \cap F=(0)$.

Joshua Meyers (Feb 27 2021 at 21:33):

Alright, changed to "tip".

Joshua Meyers (Feb 27 2021 at 21:39):

John Baez said:

Nice stuff! I like the idea of the forgetful functor $\mathsf{Fld} \to \mathsf{Set}$ being represented by a collection of objects.

Well, in the OP I only represented the constant functor at $1$ by a collection of objects. I'm not sure that you could do that same for the forgetful functor...the closest I can think of is the family $\{\mathbb{Q}(x)\}\cup\{\mathbb{F}_p(x)|p\text{ prime}\}$, which will represent the functor $\mathbf{Fld}\to\mathbf{Set}$ sending $F$ to the set of elements of $F$ that are not in the smallest subfield of $F$ (that is, $\mathbb{Q}$ if $F$ has characteristic $0$ or $\mathbb{F}_p$ if $F$ has characteristic $p$).

Nathanael Arkor (Feb 27 2021 at 21:58):

If you're looking for terminology to describe this phenomenon, these structures are "multicolimits". In fact, $\mathsf{Fld}$ is a locally finitely multipresentable category.

Nathanael Arkor (Feb 27 2021 at 22:00):

I agree that it's interesting structure to consider – and changes one's perspective from "fields aren't algebraic" to "fields are almost algebraic from the right perspective".

Joshua Meyers (Feb 27 2021 at 22:05):

Thanks! Here's the definition for others who might be following this thread:

Definition 5.2. For μ a regular cardinal, we say that a category K is locally μ-[multi/poly]presentable if it is μ-accessible and has all [multi/poly]colimits. Locally [multi/poly]presentable means locally μ′-[multi/poly] presentable for some μ′.

In this definition, a "multicolimit" is the same thing I am calling "a family of partial colimits" above, and a "polycolimit" is a weaker notion defined in the paper.

Joshua Meyers (Feb 27 2021 at 22:08):

Later:

As noted above, the category of all fields with field homomorphisms is locally $\aleph_0$-multipresentable. It is not locally $\aleph_0$-presentable as it cannot have an initial object—a consideration of characteristics makes clear that no single field can map into all of the others. There is, however, an initial object relative to the fields of each fixed characteristic, which together form the multiinitial family ([AR94, 4.25(2), 4.29])

At first I thought fields wouldn't satisfy this, because a diagram like $\mathbb{C}\xrightarrow{\text{conj}}\mathbb{C}$ (the field of complex numbers with the conjugation automorphism) has no cocones whatsoever! But then I realized that this diagram does have a multicolimit: the empty family!

Joshua Meyers (Feb 27 2021 at 22:10):

What is the point of the part of the definition that says the category is $\mu$-accessible?

Joshua Meyers (Feb 27 2021 at 22:12):

Later:

Let K be a μ-accessible category.
[...]
K is locally μ-multipresentable if and only if K has all connected limits.

What a great theorem!

Nathanael Arkor (Feb 27 2021 at 22:14):

Joshua Meyers said:

What is the point of the part of the definition that says the category is $\mu$-accessible?

It means that the category has $\mu$-filtered colimits, and that these are nicely behaved in some sense. The intention is for the definition to act like an analogue of locally presentable category (which is an accessible category that has all small colimits, rather than all small multicolimits).

Nathanael Arkor (Feb 27 2021 at 22:14):

I imagine this endows locally multipresentable categories with a good (multi)adjoint functor theorem, but I'm not sure if that appears anywhere in the literature.

Martti Karvonen (Feb 27 2021 at 22:15):

I've also heard the phrase "familially representable" being used for this notion

Joshua Meyers (Feb 27 2021 at 22:18):

Nathanael Arkor said:

I agree that it's interesting structure to consider – and changes one's perspective from "fields aren't algebraic" to "fields are almost algebraic from the right perspective".

What do you mean by "algebraic" here? My idea of algebraic is "equationally presented"

Nathanael Arkor (Feb 27 2021 at 22:22):

Algebraic varieties are locally (strongly) presentable categories. Fields don't form a variety, so aren't algebraic, but they do form a locally multipresentable category, which still has a lot of nice structure, but from a familial perspective. So if you're happy to move from structure to "multistructure", much of the same intuition for algebraic varieties will carry across to fields.

Joshua Meyers (Feb 27 2021 at 22:23):

I see! Great!

Joshua Meyers (Feb 27 2021 at 22:28):

Another thing that happens in $\mathbf{Fld}$ (and in $\mathbf{Inj}$): every slice category is posetal. This doesn't happen in all locally multipresentable categories, for example $\mathbf{Set}$ is locally multipresentable but it doesn't happen there.

John Baez (Feb 27 2021 at 22:38):

That last fact reminds me of Galois theory, which is about the poset of subfields of a field. But you're saying the category of fields over a given field is a poset? Are all fields over a given field actually subfields of it?

Fawzi Hreiki (Feb 27 2021 at 22:48):

Well it's just because all field homomorphisms are monomorphisms.

John Baez (Feb 27 2021 at 22:49):

Right, okay. So when Joshua said every slice category is "posetal" he means it's a preorder, not actually a poset... that's part of what confused me.

Fawzi Hreiki (Feb 27 2021 at 22:52):

Another way in which fields are almost algebraic is that the theory of fields is disjunctive: i.e. you can define a field internal to any extensive category with finite limits.

Fawzi Hreiki (Feb 27 2021 at 22:54):

Which a priori is a 'logic free' environment

John Baez (Feb 27 2021 at 22:55):

Can you give a fun example or two of an extensive category with finite limits, and the fields in it?

Fawzi Hreiki (Feb 27 2021 at 22:58):

Well the first example I can think of is topological fields, although you have to be a bit careful about how you define it

Fawzi Hreiki (Feb 27 2021 at 23:00):

Basically every reasonable enough category of spaces is lextensive so that's the main family of examples.

Fawzi Hreiki (Feb 27 2021 at 23:00):

$\text{Cat}$ is lextensive so there should be a notion of a 2-field but I have no idea what that would look like.

Joshua Meyers (Feb 27 2021 at 23:01):

John Baez said:

That last fact reminds me of Galois theory, which is about the poset of subfields of a field. But you're saying the category of fields over a given field is a poset? Are all fields over a given field actually subfields of it?

The slice category over a field $F$ is the poset of subfields of $F$.

Fawzi Hreiki (Feb 27 2021 at 23:01):

Well, modulo automorphisms, which is what John meant.

Joshua Meyers (Feb 27 2021 at 23:02):

What's confusing is that "over" usually refers to a picture where the arrows point down, but when we draw diagrams of fields the arrows usually point up.

Joshua Meyers (Feb 27 2021 at 23:02):

Fawzi Hreiki said:

Well, modulo automorphisms, which is what John meant.

Right, the slice category over $F$ is equivalent to the poset of subfields of $F$.

John Baez (Feb 27 2021 at 23:03):

Okay... I wouldn't call that "modulo automorphisms": I'd call that "taking a skeleton". Right? The poset of subfields of F is a skeleton of the slice category of F.

Fawzi Hreiki (Feb 27 2021 at 23:04):

Yes you're right. I should have said modulo isomorphisms

John Baez (Feb 27 2021 at 23:04):

The objects in the slice category of a field F don't have nontrivial automorphisms, unless I'm really confused.

John Baez (Feb 27 2021 at 23:04):

Okay, good.

Joshua Meyers (Feb 27 2021 at 23:04):

Yes

Fawzi Hreiki (Feb 27 2021 at 23:05):

Joshua Meyers said:

What's confusing is that "over" usually refers to a picture where the arrows point down, but when we draw diagrams of fields the arrows usually point up.

That's because usually 'over' is in the sense of fibre bundles etc.. where the bigger thing varies over the little thing via a covering map

John Baez (Feb 27 2021 at 23:05):

Joshua Meyers said:

What's confusing is that "over" usually refers to a picture where the arrows point down, but when we draw diagrams of fields the arrows usually point up.

Yet again I'm glad I never took a course on Galois theory. :upside_down:

But there's another flip going on around here: a "commutative algebra over the field k" is the same as a "commutative ring under the field k".

Joshua Meyers (Feb 27 2021 at 23:08):

Well it's not Galois theory that's so special, I'm sure that if you were going to draw the Hasse diagram of the powerset of a set, you would also draw inclusions pointing up.

Fawzi Hreiki (Feb 27 2021 at 23:08):

It's inclusions in general which get drawn up

Joshua Meyers (Feb 27 2021 at 23:08):

I think that somehow when we consider "an extension $E$ of a field $k$" we want to put the $E$ above the $k$, because it's an extension.

John Baez (Feb 27 2021 at 23:14):

Okay, true. When I was a kid, I used to find it confusing that category theorists prefer to use $x < y$ to mean there's an arrow $x \to y$... the arrowhead seems to turn around!

Fawzi Hreiki (Feb 27 2021 at 23:15):

I think the reason for that is just to keep the left-right directionality consistent

John Baez (Feb 27 2021 at 23:16):

I thought the main reason is that $2 < 3$ means there's an injection from the 2-element set to the 3-element set, $2 \to 3$.

Matteo Capucci (he/him) (Feb 28 2021 at 08:54):

I wonder if the 'multi' structure of the category of fields can help to do things like 'working at every $p$ and then passing to the limit'. In general I see there's a pattern in <name of the subject I don't know, I guess algebraic number theory?> where you look at things parametrized by primes (thinking of them as characteristics of finite fields) and then try to come up with their description at $\mathrm{char}=0$.
It's also very possible that I'm getting the causality wrong: the cat of fields is locally finitely multipresentable exactly because you can do stuff in that way.

Matteo Capucci (he/him) (Feb 28 2021 at 08:55):

Also if this was a good idea I bet intelligent people like Scholze wouldn't have let them slipped past their eyes

Morgan Rogers (he/him) (Feb 28 2021 at 10:38):

John Baez said:

The objects in the slice category of a field F don't have nontrivial automorphisms, unless I'm really confused.

Huh? Aren't the automorphisms of objects in slices exactly the elements of Galois groups? Oh nope, I was thinking of the diagram upside down too :upside_down:

David Michael Roberts (Feb 28 2021 at 11:04):

I never liked the old fashioned notation of drawing field extensions (or subgroups) via vertical lines. I very much prefer just drawing them as they are, good old fashioned $$\hookrightarrow$$s, horizontally. This accords nicely with the idea of (spectra of) fields as base objects in algebraic geometry, and extending coefficients means pullbacks, clearly along a horizontal arrow. :-)

David Michael Roberts (Feb 28 2021 at 11:35):

I don't know how old that vertical line notation is, but it feels very 19th century to me. Like Burnside era old.

David Michael Roberts (Feb 28 2021 at 11:35):

But, to be fair, the very term "descent" probably originates with it!

John Baez (Feb 28 2021 at 17:08):

Matteo Capucci (he/him) said:

I wonder if the 'multi' structure of the category of fields can help to do things like 'working at every $p$ and then passing to the limit'. In general I see there's a pattern in <name of the subject I don't know, I guess algebraic number theory?> where you look at things parametrized by primes (thinking of them as characteristics of finite fields) and then try to come up with their description at $\mathrm{char}=0$.

It's also very possible that I'm getting the causality wrong: the cat of fields is locally finitely multipresentable exactly because you can do stuff in that way.

I think the locally finite multirepresentability of the category of fields is a chunk of why this strategy works. I wouldn't call it "working at every $p$ and then passing to the limit". I'd call it "solving problems by solving them over each $p$ and separately in characteristic zero".

Sometimes the results at primes determine what's going on in characteristic zero - for some reason I'm just remembering a rather quirky example, namely how a certain scattering amplitude for strings is the inverse of the product of the scattering amplitudes in "p-adic string theory". (This is quirky because doing string theory using p-adic numbers is a pretty wild idea.)

Sometimes you have to do solve a problem at each prime and also for the real numbers to solve the problem for the rationals. This is called the Hasse principle:

In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.

And often you have to separately figure things out for each prime and also over the rationals. An example is figuring out a finitely generated abelian group $A$. The part of $A$ that survives when you tensor with $\mathbb{Q}$ is just the part isomorphic to $\mathbb{Z}^n$. $A$ is a product of this part and a bunch of p-power torsion parts, one for each prime. This idea generalizes; this kind of idea is called a fracture theorem:

A basic fact in number theory is that the natural numbers may be decomposed into the rational numbers and the p-adic integers for all prime numbers p. Dually in arithmetic geometry this says that Spec(Z) has a cover by all its formal disks and the complements of finitely many points, a fact that is crucial in the geometric interpretation of the function field analogy and which motivates for instance the geometric Langlands correspondence.

Lifting this statement to stable homotopy theory and “higher arithmetic geometry” the arithmetic fracture theorem says that stable homotopy types (and suitably tame plain homotopy types) canonically decompose into their rationalization and their p-completion for all primes p, hence into their images in rational homotopy theory and p-adic homotopy theory. Since these images are typically simpler than the original homotopy type itself, this decomposition is a fundamental computational tool in stable homotopy theory, often known under the slogan of “working one prime at a time”.

John Baez (Feb 28 2021 at 17:09):

I don't understand any of this stuff very well... but @Joshua Meyers can probably connect his stuff to some of these ideas.

John Baez (Feb 28 2021 at 17:27):

Maybe you could try to relate that "local finite multirepresentability" idea for fields and this idea, @Joshua Meyers:

For each prime p we define the p-power torsion subgroup of A to be

$\{a\in A \;|\; \exists n\in \mathbb{N}\;, p^n a = 0\}$

Any homomorphism between abelian groups sends each p-power torsion subgroup into the corresponding p-power torsion subgroup.

For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups. The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups. In a sense, this means that studying p-torsion groups in isolation tells us everything about torsion groups in general.

Joshua Meyers (Mar 01 2021 at 22:27):

Nathanael Arkor said:

Algebraic varieties are locally (strongly) presentable categories. Fields don't form a variety, so aren't algebraic, but they do form a locally multipresentable category, which still has a lot of nice structure, but from a familial perspective. So if you're happy to move from structure to "multistructure", much of the same intuition for algebraic varieties will carry across to fields.

What do you mean by "multistructure" btw?

Nathanael Arkor (Mar 01 2021 at 22:30):

Informally, structure in the free coproduct completion $\mathbf{Fam}$ of the category.

Joshua Meyers (Mar 01 2021 at 22:33):

By "structure" do you just mean object?

Nathanael Arkor (Mar 01 2021 at 22:34):

I mean colimit/left adjoint/etc.

Joshua Meyers (Mar 01 2021 at 22:36):

So you mean that the formal coproduct of all prime fields would be initial in $\mathbf{Fam}$? What would be a morphism from that to $\mathbb{Q}$?

Joshua Meyers (Mar 01 2021 at 22:37):

Maybe the free product completion would work better?

Nathanael Arkor (Mar 01 2021 at 22:37):

I'm sure this is made precise somewhere, though I'm not quite sure where. Perhaps Section 2 of Osmond's On Diers theory of Spectrum I : Stable functors and right multi-adjoints covers some of the intuition, but I haven't looked closely. I think it's also the dual case: e.g. free product completion and right adjoints.

Nathanael Arkor (Mar 01 2021 at 22:38):

Or maybe I'm getting mixed up and it's all the product completion instead of coproduct :)

Nathanael Arkor (Mar 01 2021 at 22:39):

I think there was also a MathOverflow post that had some insight; let me try to find it.

Nathanael Arkor (Mar 01 2021 at 22:39):

Are multicolimits suitable colimits?