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Stream: learning: questions

Topic: can you make a "polynomials" category from a category?

David Egolf (Mar 04 2024 at 23:39):

I've recently been learning about a way to modify a ring to create a new one. First, you start with a ring $R$, and then create $R[x]$, the ring of polynomials in $x$ with coefficients in $R$. Then we can compute various quotients of $R[x]$ and get back rings that are like $R$, but with some additional stuff added in.

For example, starting with a field $k$, one can form the quotient ring $k[\epsilon]/(\epsilon^2)$. This quotient ring has elements that can be written in the the form $a+b \epsilon$, with $a,b \in k$. This is like "$k$ with some extra stuff": setting $b=0$ in $a + b \epsilon$ shows that we can nicely associate each element of $k$ to an element of $k[\epsilon]/(\epsilon^2)$. And we've acquired an extra element $\epsilon$ now, which has the really cool property of being nonzero, but squaring to zero!

I'm thinking of this process that moves from $R$ to $R[x]$ to $R[x]/I$ (where $I$ is some ideal of $R[x]$) as intuitively "add in a 'degree of freedom' to $R$, and then harness that degree of freedom to add in additional stuff to $R$ which obeys certain equations".

Can this intuition be generalized to categories in general? I'd like to start out with a category $C$, add in some "free" structure, and then take some kind of quotient to get back our starting category with a bit of extra stuff that obeys some equations I specify. For example, I could imagine adding in a "free endomorphism $f$" at some object, which obeys a minimum of equations. Then we could try to create a new category from this category $C[f]$ by considering functors from $C[f]$ which "collapse" certain expressions involving $f$ down to an identity morphism.

David Egolf (Mar 04 2024 at 23:42):

I wanted to say that we would "collapse" certain polynomials in our free morphism $f$ to zero. But I suppose we'd need a way to both "add" and "multiply" $f$ in that case (and a notion of "zero"). If each hom set in $C[f]$ has the structure of a commutative group, though, maybe one could consider trying to do this!

I'm also unconvinced that "adding a free morphism" to a category is really analogous to adding in a "free symbol" $x$ to a ring $R$. Any thoughts on what might be a closer analogy would be welcome!

John Baez (Mar 05 2024 at 01:02):

You can certainly freely throw in an endomorphism into a category and get a new bigger category. And in a sense this includes rings $R[x]$ as a special case!

To see this you need to work with Ab-categories, or "categories enriched over abelian groups". These are categories where the homsets are abelian groups and composition of morphisms distributes over addition. A one-object Ab-category is the same as a ring!

John Baez (Mar 05 2024 at 01:24):

You can probably work out the rest: if $R$ is a ring, and we think of it as a one-object Ab-category, freely throwing in a new endomorphism $x$ gives a new one-object Ab-category that corresponds to the ring $R[x]$.

John Baez (Mar 05 2024 at 01:28):

Furthermore you should realize that traditional algebra textbooks are missing a lot of important material on monoids. They talk a lot about monoids where the monoid operation distributes over some operation, namely addition in some abelian group. These are called rings. But they don't talk enough about "rings without addition", namely monoids. Given any monoid $M$ it's perfectly interesting to look at the monoid $M[x]$ where we freely throw in an extra element $x$. It's just a deficiency in our education that we don't learn the basic theorems about this idea.

John Baez (Mar 05 2024 at 01:29):

So you were left analogizing rings and categories, which is not a perfect analogy. The really good analogies are:

• a monoid is a one-object category,
• a ring is a one-object Ab-category.

John Baez (Mar 05 2024 at 01:30):

Further let me point out that you can take a category $C$ and throw in an extra morphism between any two objects, and let it freely generate a new category! It doesn't need to be an endomorphism.

John Baez (Mar 05 2024 at 01:30):

Indeed you can take a category, throw in any set of new objects and new morphisms, and let those freely generate a new larger category.

John Baez (Mar 05 2024 at 01:36):

Oh, and one more thing: just as it can be good sometimes to think of a category as a "monoid with many objects", it can be good to think of an Ab-category as a "ring with many objects". So, Ab-categories are sometimes called [[ringoids]]. They're pretty interesting because there are so many important examples. For example, the category of modules of any ring is a ringoid!

John Baez (Mar 05 2024 at 01:37):

The point of calling Ab-categories "ringoids" is that it helps you realize that many things we like to do with rings, you can also do with ringoids.

Chris Grossack (they/them) (Mar 05 2024 at 06:21):

If you're willing to work strictly (that is, to consider categories as algebraic objects defined up to isomorphism, instead of defined up to equivalence) then you can make this analogy EXTREMELY precise. The theory of rings is an algebraic theory, which gives us access to lots of great things like free rings, presentations, etc. The theory of (strict) categories is essentially algebraic, and it turns out these have basically all the nice properties of algebraic theories!

Chris Grossack (they/them) (Mar 05 2024 at 06:23):

In particular, just like you can take a ring $R$ and freely adjoin an element $x$, and then (if you like) impose some relations on how $x$ should interact with the rest of the ring, you can similarly take a (strict) category $\mathcal{C}$ and freely adjoin either an object $X$ or an arrow $f : A \to B$. If you like, you can also impose some relations on how these new objects/arrows should interact with the objects and arrows already in $\mathcal{C}$.

Chris Grossack (they/them) (Mar 05 2024 at 06:26):

I suspect there's a less strict version of this story, where the relevant universal properties for freely adjoining stuff and quotienting stuff are all defined up-to-equivalence in the bicategory of categories (rather than the category of strict categories). Unfortunately, I don't have a reference for that. Maybe someone else does? Certainly things will be a bit more complicated than the nice world of (essentially) algebraic theories!

John Baez (Mar 05 2024 at 07:51):

There's another variation on your theme, Chris: categories with a fixed set $S$ of objects are algebras of a multisorted Lawvere theory where the set of sorts is $S \times S$.

John Baez (Mar 05 2024 at 07:52):

Here we think of a category as a bunch of homsets $\mathrm{hom}(x,y)$, one for each $x,y \in S \times S$, and a bunch of composition and identity-assigning operations obeying the associative, left and right units.

John Baez (Mar 05 2024 at 07:54):

There's also a multisorted Lawvere theory whose algebras are categories with set $S$ of objects, and ones whose algebras are PROPs, and one whose algebras are... multisorted Lawvere theories with a fixed set $X$ of sorts!

David Egolf (Mar 05 2024 at 22:18):

Thanks, both of you! That's all really cool to learn about!

Philip Saville (Mar 07 2024 at 09:31):

I suspect it's a special case of the perspectives mentioned above, but Lambek and Scott also talk about "polynomial categories" in Section 5 of their book. They show how to universally add a new arrow $x$ to a (cartesian / cartesian closed) category $C$ to get a category $C[x]$ of polynomials over $C$. As I understand it, this corresponds to adding a new constant to the type theory corresponding to $C$, and then taking terms over the language extended with that constant. There's a rather nice characterisation of $C[x]$ as the Kleisli category of a comonad, too

John Baez (Mar 07 2024 at 16:44):

That's nice. I'm a bit surprised at how little this "adding a new arrow" construction is used in category theory compared to in ring theory, where it's ubiquitous and important. Maybe because it's most important in commutative ring theory?

A ring is a one-object Ab-category, but a commutative ring is a one-object monoidal Ab-category, or equivalently a one-object symmetric monoidal Ab-category.

So maybe adjoining a morphism is most useful in the symmetric monoidal case... which indeed is the case for Lambek and Scott.

John Baez (Mar 07 2024 at 17:31):

And indeed a cartesian category is a kind of categorified commutative monoid, or categorified commutative ring if it also has sums and products distribute over sums (as they do in the cartesian closed case). So the construction Lambek and Scott describe is really close to taking a commutative ring $R$ and putting in a new variable $x$ to get the polynomial ring $R[x]$.

John Baez (Mar 07 2024 at 17:33):

And I claim it even has similar applications.

Eric M Downes (Mar 07 2024 at 17:45):

What would the zero-divisors be? It seems like a CCC ringoid (finite products always exists and distribute over always existing coproducts) is already an "integral domainoid"?

As per similar applications...

• can we form a projective spaceoid from the unit groupoid of the product?
• can we factor polynomials in $\mathsf{C}[x]$ via the Euclidean algorithm(oid)?

John Baez (Mar 07 2024 at 18:01):

These are great questions... or at least the kind of questions I love. I think you're saying ringoid where I'd say [[rig category]]: a rig category is one with two monoidal structures, $\oplus$ and $\otimes$, where the first is symmetric monoidal and the second distributes over the first in a coherent way. A [[ringoid]] is a category enriched over abelian groups. So I'd call your "CCC ringoid" a "CCC rig category".

One way to think about these questions is that you can take the set of isomorphism classes of objects in any rig category and get a rig.

For a CCC rig category the addition $\oplus$ is coproduct so

$x \oplus y \cong 0 \implies x,y \cong 0$

Thus, this rig has "characteristic zero" in a reasonable sense.

I believe we can take any rig and throw in additive inverses to make it into a ring. However addition in a rig may not be cancellative

$x + y = x' = y \nRightarrow x = x' ,$

so this process may be destructive.

John Baez (Mar 07 2024 at 18:03):

Q: Can the rig of isomorphism classes of objects in a CCC rig category fail to be cancellative?

John Baez (Mar 07 2024 at 18:06):

Anyway, what I'm trying to say is that from a CCC rig category you can get a commutative rig and then a commutative ring, functorially. So it makes sense to ask questions about those, like:

Q: is the commutative ring we get from a CCC rig category a field?

John Baez (Mar 07 2024 at 18:07):

And if the answers to those questions are affirmative we should feel emboldened to 'lift' those questions to the CCC 2-rig itself, e.g. trying to define '2-fields' and work with those, etc.

Eric M Downes (Mar 07 2024 at 18:09):

JB clearly means $x+y=x'+y\not\!\!\implies x=x'$... and I'm not sure; there may be a difference between working up to equivalence and working up to isomorphism here. Something something injective modules. I think you've defined a good direction here, and if I find anything I'll post progress here. What a great forum!

Chris Grossack (they/them) (Mar 07 2024 at 18:24):

Instead of saying "CCC rig category", would people be happy to say bicartesian closed category? That's language I'm more familiar with, and says it's a CCC with coproducts. The linked nlab page says it's a kind of 2-rig...

Eric M Downes (Mar 07 2024 at 18:49):

Fine by me if it helps you prove anything! (and we can prove/disprove equivalence of the different concepts at some point if there's anything solid) The broad approaches seem to be

• For what categories $\mathsf{C}$ does adjoining an arrow to create $\mathsf{C}[x]$ allow one to form "polynomials" to which some given concept from ring theory lifts?
• In what ways are bicartesian closed categories ring-like; that is, what are the minimal restrictions needed to get various constructions to lift? (seems like there's more literature here; https://youtu.be/wG5MZqj_JK8?si=oiXg1WVnokjB7Ss1 on rings ~ topoi)
  
• are any of the above equivalent or isomorphic to one another.

Also there's an implicit thread concerning fields. The category of fields is so poorly behaved, and there are no field objects in $\mathsf{Ab}$ (mult. inverse not an iso of the abelian group object)... it might be more productive to think in terms of Zariski Topology : Field :: Etale Topology : ?; but I'm getting well out of my depth here. I just mean that in $\mathsf{Set}$ fields are special rings, but that basic analogy doesn't really lift well, so thinking beyond just algebra about what makes a field a field seems less cursed an endeavor.

James Deikun (Mar 08 2024 at 01:39):

Instead of defining a field as a ring with an extra (almost) algebraic operation of division, you can define a field as a "complete ring" in the sense that any equation you add to the ring that isn't already true makes the ring inconsistent ($0=1$).

Nathanael Arkor (Mar 08 2024 at 09:56):

What's a reference for the characterisation of fields as "complete rings"?

James Deikun (Mar 08 2024 at 13:02):

I don't have a reference, though I'm sure it must be a known fact. The equations in a ring are characterized by which ideals they annihilate, and a field is well-known to be precisely a commutative ring that has exactly two ideals: the zero ideal and the whole ring/field.