Niles Johnson

I can't believe, all this time, I thought Krita was *only* for digital painting, and wouldn't be good for the incredibly basic image editing that I sometimes want to do. I just tried Krita for the first time! I had to spend about 30 seconds coming to terms with a slightly different menu structure, and I'll probably have to look up some things later, but this will clearly work just fine for me!! So, thanks for mentioning it! @wigglytuffitout@elekk.xyz@Craigp@mastodon.social

Oh, and what journal is this for? It's Applied Categorical Structures https://link.springer.com/journal/10485 The automated emails we get say "Straive is a service provider of Springer Nature". I already didn't love publishing in Springer journals; perhaps this will be the thing that finally makes me avoid them altogether.

@MartinEscardo@mathstodon.xyz This popped back into my mind this morning because I was thinking about a discussion with another friend about some art. There's a long history of people trying to exclude certain styles, or media, or themes, or etc., by calling them non-art, or (even worse) not *true* art. Those kinds of separations always seem to be more about the motivations of the people calling for them, rather than something intrinsic about the nature of art. I'm not enough of an art historian to say more about that, but the parallels between the questions "what is art?" and "what is math?" are interesting, and maybe useful. The answer you started with---"It's something that people do." is one I like a lot, both for what it says, and what it doesn't say.

@caten@mathstodon.xyz Thanks for writing that up, and sharing it. The line about "wealthy white men with tenure complaining" hits hard, and rightly so. I hope things will be so much better for you after this change!

@matthras@aus.social thanks for the offer, but I am unfortunately nowhere close to the decision-making about this stuff :/

@christianp oof; this made *me* tear up. Why? The social pressure on emotional detachment---*especially* in academia---is bullshit. But it's real. I don't know what else to say. Good luck? (Also, we wouldn't be talking about this if you weren't (a) giving this talk and (b) saying how you feel. That matters too. Thank you.)

To explain the main ideas in our paper, consider a symmetric monoidal category (A,+,0,β). So, the monoidal sum is denoted +, the monoidal unit is denoted 0, and the braiding (a.k.a. symmetry) is denoted β. We assume that the unit and associativity isomorphisms are identities, so the monoidal structure is strict. An invertible object x in A has a weak inverse, x', with morphisms ε:x+x' ≅ 0 (cancel) and η:0 ≅ x'+x (decancel) satisfying triangle identities that make the functors x+(-) and x'+(-) adjoint inverse equivalences. Using the braiding, β, each invertible x gives us an automorphism of the unit 0 -η-> x'+x -β-> x+x' -ε-> 0 This composite is sometimes called the _trace_ of 1ₓ or the _Euler characteristic_ of x. We call it the _figure eight on x_ and write 8ₓ because the string diagram looks like a figure eight. (2/9)

Before that, here's an example of two other composites that might appear different but are actually equal to each other and to the figure eight. We call one Cₓ, the *figure C*, and the other Hₓ, the *figure H*, since the string diagrams sort of look like those letters. So, Cₓ and Hₓ also have *odd* parity, because they're each equal to one instance of 8ₓ. Not pictured: There is also a "reverse C" that uses the braiding of x' with itself, and a figure eight on x' that reverses the roles of x and x'. Both of these are also equal to 8ₓ = Cₓ = Hₓ, and therefore have odd parity. (4/9)

As often happens, our main theorem is a little easier to state with some additional background. I'll add that now. In Pₓ, the free symmetric monoidal category on the invertible object x, the morphisms are generated as sums and composites of six basic morphisms, with the following parities: There are two de/cancel morphisms, and they have even parity: ηₓ: 0 → x'+x εₓ: x+x' → 0 Then, there are four basic braiding morphisms βₚ,ₛ: p+s → s+p where s and p are each either x or x'; each of these has odd parity. These parities follow from the parities of the "figure morphisms", 8, C, and H above. Then, parity for any other morphisms in Pₓ are computed from these: parity is additive on sums or composites of morphisms. Our main theorem says that any parallel morphisms with the same parity are equal. (5/9)

With even more background, I can give an even easier statement of our main theorem, and finally an explanation of how it's proved. The required background involves a cute little category that we call the *Super Integers*. This is a symmetric monoidal category, Z, whose objects are the integers, and where each object has two automorphisms called "odd" and "even" or denoted ±1; that's the "super" part. There are no morphisms between non-equal objects. [Aside: yes, this name is too hip, but I've come to terms with it.] You can think of the Super Integers like the integers with "virtual permutations": it's symmetric monoidal, so you can make sums and permute summands, but each permutation is characterized only by its *sign*. (These generating objects could also be denoted ±1, but then I get confused by having the same notation for objects and morphisms, so I'll avoid that here!!) (6/9)

The shortest version of our main theorem is that there is an equivalence of symmetric monoidal categories K: Pₓ → Z, where Pₓ is the free symmetric monoidal category on one invertible object x. Moreover, this equivalence K does the following on generating morphisms: The de/cancel morphisms ηₓ and εₓ are sent to identities. The four braidings βₚ,ₛ (for p,s ∈ {x, x'}) are sent to *odd* morphisms in Z. This is the version we prove, and it's the one that isn't part of the previous literature. It's also the one with our favorite conceptual interpretation: in Pₓ you have a formally constructed object that, by design, has a free universal property. So, Pₓ is easy to work with in abstract or general terms. But—as often happens with universal constructions—Pₓ is a big complicated mess of objects and morphisms. So, it's hard to tell whether two morphisms (such as two ways around a diagram) are equal or not. On the other hand, Z is so simple it can be explained in a couple of paragraphs. Coherence in Z is so easy you don't even have to think about it. But—because Z is so simple—it's not something that appears "in nature". The examples that made people want to know about invertibility, like invertible modules over a ring or virtual vector spaces, almost never have *identities* for their de/cancel (i.e., unit/counit) morphisms. So, the equivalence K explains how to take interesting diagrams in Pₓ and convert them to easy diagrams in Z. Then you can use parity of morphisms there to determine whether the diagrams commute. (7/9)

The way we prove our main theorem uses some abstract 2-monad theory going back to Blackwell-Kelly-Power (flexibility of monads), and also Lack's model structure on 2-monads. I'll certainly leave those details to the paper, but they're not *that* hard. We've structured it so that you just need to understand the statements we've extracted, and then apply them as black boxes. This isn't the first time some wildly general 2-monadic algebra has been applied for concrete, computational applications; I think those applications are how people got into abstract 2-monad theory in the first place! But I do think ours is another neat one for those who are interested in such things. (8/9)

Finally, I want to conclude with a mention of the related literature. Monoidal categories where every morphism and every object is assumed to be invertible are sometimes called *2-groups*. When the monoidal structure is symmetric, they're called *symmetric 2-groups* or *Picard categories*. These have been studied *a lot*, for a long time. There are various coherence theorems in the literature, for both the non-symmetric and symmetric cases. Highlights include work of Laplaza [1], Baez-Lauda [2], Kelly-Laplaza[3], and Dugger [4]. (More detail in our "Relation to literature" subsection.) So, why do we need another version some decade(s) later?? Well, one honest reason is that we had a hard time understanding the older versions. Even the more recent ones depend crucially on the early Laplaza and Kelly-Laplaza work. We tried to explain them in a way we could understand, and wound up with the independent (2-monadic) approach I mentioned above. So here we are. Yes, serious people have known all about the essential computational facts for decades, but our version adds a nontrivial and (we think) useful perspective! [1]: Laplaza, Coherence for categories with group structure: An alternative approach (1983) https://dx.doi.org/10.1016/0021-8693(83)90081-9 [2]: Baez-Lauda, HDA V: 2-groups (2004) http://tac.mta.ca/tac/volumes/12/14/12-14abs.html [3]: Kelly-Laplaza, Coherence for compact closed categories (1980) https://dx.doi.org/10.1016/0022-4049(80)90101-2 [4]: Dugger, Coherence for invertible objects and multigraded homotopy rings (2014) https://dx.doi.org/10.2140/agt.2014.14.1055 (11/11)

@prismika might be interested in this too! @dantheclamman

@11011110@mathstodon.xyz It's confusing; maybe I have the wrong url; a different email from them says their name is "Straive" at straive.com, with an AI in the middle. That website is somehow even worse.

oof; if you're confused about why the company name is sometimes "Strave", and someties "Straive", so am I! I just realized they use both spellings, and straive.com (with an AI in the middle) is even more embarrassing.

@maxsnew wow that's AWESOME; congrats!!!

One version of our main theorem can be explained in Pₓ, the free symmetric monoidal category on an invertible object x. It says that morphisms in Pₓ are characterized by the *parity* of how many instances of 8ₓ they have. In particular, the composite or sum of two figure eights is the identity! Our fantastic choice of notation expresses this fact as follows: 8ₓ∘8ₓ = 8ₓ+8ₓ = 1₀. So, this fact implies that any composite or sum of figure eights can be reduced to just *odd* or *even*. The main theorem says, moreover, that *every* morphism in Pₓ boils down to some composite or sum of figure eights. A little later in this thread I'll give some more precise (more comprehensible) versions of the same result. [Aside: I want to pause and note that this might sound familiar to some readers, because these facts have been known in some form or other for a *long time*. They are very well studied! Our paper has a "Relation to Literature" subsection that addresses some of this, and I'll make some further comments below, but this thread is mostly for people who haven't seen it before, or have seen it but would like to see an alternative explanation because it's neat.] (3/9)

We put a bunch of examples in the last section of our paper, starting with some of those figure morphisms and gradually building up to more complex examples. Here, I'll just give the final one, because it illustrates a diagram that *doesn't* (generally) commute, but looks at first like it ought to. To start, suppose a is an invertible object in a symmetric monoidal category A, with inverse a'. Then there is a conjugation functor Gₐ: A → A given by z ↦ zᵃ = a' + z + a You can show (using our coherence stuff) that this is a symmetric monoidal functor. Furthermore, you can show (again using coherence) that Gₐ is isomorphic to the identity on A. So, this is a categorification of the fact that conjugation in an abelian group is the identity homomorphism. Of course, conjugation by a' is also a symmetric monoidal functor, and also isomorphic to the identity. The example gets going when you realize that there is a natural isomorphism between these two, with components given by an isomorphism a' + z + a ≅ a + z + a' permuting the summands by a (1 3) permutation. So, is this a *monoidal* natural isomorphism? How could it not be??! (9/11; there are two bonus posts!)

Continuing the previous post, here is the monoidal naturality diagram for two objects z and w: Checking the a-parity, one composite is even but the other is odd. So, the two composites around the diagram are not generally equal. In particular, they are not equal when z and w are the unit object, 0, and a is a free invertible generator. The paragraph after the diagram gives this explanation: conjugation by a and a' are both symmetric monoidal functors, and are both monoidal naturally isomorphic to the identity. So, they are monoidal naturally isomorphic to each other, but the (1 3) permutation above is *not* that isomorphism. Instead, that isomorphism factors through the identity functor, so it involves just de/cancellation morphisms with no permutations of the object a past its inverse a'. I think that makes sense in retrospect, but also could be a source of confusion. (It definitely was for me!! One day while we were working on this I sent Nick a sequence of increasingly frantic/confused emails, followed the next morning by a long explanation of how useful it is to get a good night sleep.) (10/11)

@brocolie@tech.lgbt @luna@tech.lgbt that's 1.5 hours per miles; those babies are moving at a crawl

@Daojoan@mastodon.social Thanks for this article. It is thoughtful, and it made me think. One of those thoughts is about these lines: "Plenty of people stop and produce nothing. The graveyard of failed comebacks is large, and wintering is dangerous as a strategy because most attempts at it collapse into actual stagnation. The difference between the two is invisible from the outside, until the end." I realized that there's an implicit *goal* of wintering---at least in the description of it here---that can fail, or succeed. I wondered what that implicit goal could be, and I think the answer is in the comparisons with fallow fields and hibernating bears: to eventually produce *more* or *better*. I'm not sure that's what you meant, or what you wanted to mean. Some of the examples in the article align with this view (people who intentionally withdraw, to produce something), and some don't (people who fall out of fashion). All of this made me think more about what kind of "breaks" I want to take at various times. So, thanks.