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Stream: theory: category theory

Topic: the positive bias

Matteo Capucci (he/him) (Feb 14 2025 at 15:27):

When objectifying combinatorics (as in e.g. the paper of Lawvere and Menni I'm reading) one quickly has to come to terms to the absence of negatives. In general, 'structural'/'objective' constructions/arguments seem to invariantly leave use with negative-free structures (when decategorifying), such as monoids (rather than groups), rigs (rather than rings), etc.
As anyone explored this phenomenon?
My hunch is that this is a reflection of a bias of logic towards positive evidence (proofs, witnesses, etc) as opposed to negative (refutations, counterexamples, etc), which leave us with a clear notion of 'stuff' and a much less clear notion of 'non-stuff'. This, naively, makes me think that, in principle, the situation above could be amended by working in more balanced logical settings like linear logic.
On the other hands, at the 0-level we are perfectly fine with the positive bias, and for instance define $\Z$ as the group of formal differences in $\N$. Why doesn't this approach work in the objective setting?

David Corfield (Feb 14 2025 at 15:47):

Loads of great stuff at @John Baez 's The Mysteries of Counting. Negative cardinality makes an appearance.

Matteo Capucci (he/him) (Feb 14 2025 at 16:04):

Yeah I've been on that page before, really fascinating but I never found an answer to my question above (though, of course, I might have overlooked it!)

David Corfield (Feb 14 2025 at 16:14):

So you're not counting things like that the cardinality of the open real interval is $-1$, as it is of the space of piecewise linear maps $f: [0,1] \to \mathbb{R}$ ?

John Baez (Feb 14 2025 at 16:33):

It sounds like Matteo was wondering why it's hard to work with negative numbers in an 'objective' (= categorified) context. The first reason is that in a category with coproducts, if you have two objects $X$ and $-X$ with

$X + (-X) \cong 0$

then

$X \cong -X \cong 0$

It's a fun puzzle to prove this.

John Baez (Feb 14 2025 at 16:35):

For a classic discussion of this issue, see Schanuel's Negative sets have Euler characteristic and dimension.

John Baez (Feb 14 2025 at 16:37):

This then sets off a search for workarounds.

David Corfield (Feb 14 2025 at 17:10):

That got me wondering what makes for a productive turn to the group completion. So [[topological K-theory]] uses virtual vector bundles. And we have [[virtual permutation representations]].

What makes these good moves?

Mike Shulman (Feb 14 2025 at 17:26):

However, there are ways to obtain negative numbers by other kinds of decategorification. For instance, the Euler characteristic of a topological space or chain complex can be negative, and it is also additive and multiplicative in the sense that $\chi(X+Y) = \chi(X)+\chi(Y)$ and $\chi(X\times Y) = \chi(X) \cdot \chi(Y)$ (where in the case of chain complexes $\times$ means $\otimes$).

John Baez (Feb 14 2025 at 17:35):

Yes, I've been working with @Todd Trimble on linear species (functors from the groupoid of finite sets to the category of, let's say, finite-dimensional vector spaces), and to get negatives into this game we switch to 'derived species': functors from the groupoid of finite sets to the category of finite-dimensional bounded chain complexes. (They're 'derived' when we invert the quasi-isomorphisms.)

A linear species $F$ has a generating function

$\sum_{n \ge 0} \frac{\dim (F(n))}{n!} x^n$

while a derived species has a generating function

$\sum_{n \ge 0} \frac{\chi(F(n))}{n!} x^n$

where $\chi$ is the Euler characteristic. So, for example, one can find a derived species whose generating function is $\sin x$.

sarahzrf (Feb 14 2025 at 18:00):

John Baez said:

It sounds like Matteo was wondering why it's hard to work with negative numbers in an 'objective' (= categorified) context. The first reason is that in a category with coproducts, if you have two objects $X$ and $-X$ with

$X + (-X) \cong 0$

then

$X \cong -X \cong 0$

It's a fun puzzle to prove this.

this seems a bit glib—this certainly tells you something about universal properties not liking negatives, but it's really about universal properties in specific, not about the fact that you've categorified. the same proof shows that a lattice cannot have negatives, right?

sarahzrf (Feb 14 2025 at 18:01):

if we really want to compare corresponding concepts, then surely we should be asking about more general monoidal structures, since negatives in the 0-world arise in the monoidal case

Kevin Carlson (Feb 14 2025 at 18:38):

I’m not sure which monoidal structure you want to generalize. We presumably want at least to recover the existence of a Burnside rig, which would suggest a distributive monoidal category, where you still have the problems of coproducts. If you want the multiplication and the addition to be general monoidal structures, I don’t know how you get a rig of isomorphism classes; the closest thing I’m aware of is a linear distributive category, which doesn’t decategorify to a rig in general.

Kevin Carlson (Feb 14 2025 at 18:40):

I wonder whether anyone has a feeling about how “believable” the Schanuel-style negative sets are as an objective correlate of negative integers. In particular, what is the geometric intuition making a half-open interval coincide with $0$? The fact that you can shrink it down until the point pops off the open end into the abyss, something like that? But I’m not sure that visual intuition extends to see why the open interval is not $0$, or, indeed, $1$…it’s risky to use homotopies here for the latter reason!

Kevin Carlson (Feb 14 2025 at 18:43):

I suppose you can always just note the half-open interval solves $x=2x.$ But that’s somehow not fully satisfying to me.

Mike Shulman (Feb 14 2025 at 18:57):

Kevin Carlson said:

If you want the multiplication and the addition to be general monoidal structures, I don’t know how you get a rig of isomorphism classes

A [[rig category]] has two general monoidal structures and a distributive law, so it has a rig of isomorphism classes. And every rig is a discrete rig category, even if it's also a ring. So you can certainly have a rig category with negatives.

Kevin Carlson (Feb 14 2025 at 19:18):

Ah, thanks, I forgot that those are a thing as well as linearly distributive categories.

Kevin Carlson (Feb 14 2025 at 19:19):

Is there a reasonable rig category candidate for a categorification of the integers, then?

Mike Shulman (Feb 14 2025 at 19:43):

Yes, here's one. It's a skeletal groupoid, the objects are the integers, and the automorphism group of each object is $C_2$, which I'll write multiplicatively as $\{+1,-1\}$.

The additive monoidal structure is addition on objects and multiplication on morphisms. Its associators, unitors, and distributors are identities, while its symmetry isomorphism $m+n \cong n+m$ is $(-1)^{mn}$.

The multiplicative monoidal structure is multiplication on objects, and on morphisms sends $\alpha:m\cong m$ and $\beta:n\cong n$ to $\alpha^n \beta^m$. Its associators, unitors, and distributors are also identities, and its symmetry isomorphism $mn \cong nm$ is $(-1)^{{m\choose 2}{n\choose 2}}$.

There is a rig functor to this category from the groupoid of finite sets that takes the cardinality of every finite set and the sign of every permutation.

Now I'll let the people who haven't seen this before have some fun pondering where on Earth this comes from and why it is a "reasonable" categorification of the integers (although the last paragraph above gives a bit of a hint in that direction). (-:

John Baez (Feb 14 2025 at 21:17):

sarahzrf said:

John Baez said:

It sounds like Matteo was wondering why it's hard to work with negative numbers in an 'objective' (= categorified) context. The first reason is that in a category with coproducts, if you have two objects $X$ and $-X$ with

$X + (-X) \cong 0$

then

$X \cong -X \cong 0$

It's a fun puzzle to prove this.

this seems a bit glib.

That's why I called it the "first" reason. The next step is to investigate ways of defining addition and negatives in categories where addition is not defined as coproduct and zero is not defined as an initial object. For example, Grothendieck and Sinh and others introduced 'Picard categories' or 'Pic-categories' These are symmetric monoidal categories where every object has a tensor inverse.

John Baez (Feb 14 2025 at 21:32):

Kevin Carlson said:

I wonder whether anyone has a feeling about how “believable” the Schanuel-style negative sets are as an objective correlate of negative integers. In particular, what is the geometric intuition making a half-open interval coincide with $0$? The fact that you can shrink it down until the point pops off the open end into the abyss, something like that? But I’m not sure that visual intuition extends to see why the open interval is not $0$, or, indeed, $1$…it’s risky to use homotopies here for the latter reason!

Well, you need to assign the half-open interval a Euler-Schanuel characteristic of zero to get the good properties Schanuel wants. The Euler-Schanuel characteristic is not homotopy invariant (since the open and closed and half-open intervals have Euler-Schanuel characteristics of -1, 1 and 0 respectively). Instead, it's a well-defined invariant of bounded polyhedra, meaning bounded subsets of finite-dimensional vector spaces lying in the Boolean algebra generated by subsets $\{f > 0\}$ where $f: V \to \mathbb{R}$ is linear. Schanuel shows the rig of isomorphism classes of bounded polyhedra is $\mathbb{N}[x]/(x \sim 2 x + 1)$ where $x$ is the isomorphism class of the open interval. Euler-Schanuel characteristic is the homomorphism

$\chi: \mathbb{N}[x]/(x \sim 2x + 1) \to \mathbb{Z}$

sending $x$ to $-1$.

All this and more is in his paper.

John Baez (Feb 14 2025 at 21:37):

Someone should by now have extended the Euler-Schanuel characteristic to something more general, maybe smooth quasiprojective varieties or even motives. Schanuel studied the generalization to not-necessarily-bounded polyhedra.

Kevin Carlson (Feb 14 2025 at 21:55):

Yeah, all those technical facts are clear enough to me! I'm just not sure whether there's some angle from which I ought to really "feel" like an open interval is the opposite of a point, or like a half-open interval is "basically nothing".

John Baez (Feb 14 2025 at 22:02):

Yes, if you glue together an open interval and a point you get

$\textsf{O}$

which is nothing, nada, zip, zero.

I actually think it's not a coincidence that the Euler characteristic of the numeral 1 is 1 and the Euler characteristic of the numeral 0 is 0. You get 1 when you draw a mark - homotopy equivalent to a point, but easier to see. You get zero when you draw a blotch and then remove most of it, leaving only the frame so we can see that nothing is there. In other words,

$\textsf{O}$

is like a picture frame for a picture of nothing. I really don't think it's an accident that people chose that symbol! And when you draw a closed disk and then subtract a closed disc from the interior, it has Euler-Schanuel characteristic 1-1 = 0, or maybe I should say

$\bullet$ - $\bullet = \circ$

Evan Washington (Feb 14 2025 at 22:04):

Here are some negative-set-like things: surreal numbers/combinatorial games. Are the surreal numbers a decategorification of anything interesting? To that end, what is the "right" kind of morphism between combinatorial games

Kevin Carlson (Feb 14 2025 at 22:10):

This is somewhat compelling in a slightly Dada way, John, thanks.

John Baez (Feb 14 2025 at 22:48):

Thanks! Thinking hard about negative numbers has led me down some very strange roads. Right now I'm getting a bunch of emails from a guy who read my post on negative probabilities. Unlike most unsolicited emails I get about math and science, his ideas are not insane. But I have to decide how much I want to think about this right now....

Mike Shulman (Feb 15 2025 at 04:37):

Evan Washington said:

Here are some negative-set-like things: surreal numbers/combinatorial games. Are the surreal numbers a decategorification of anything interesting? To that end, what is the "right" kind of morphism between combinatorial games

Andre Joyal wrote a short note describing a category of Conway games whose morphisms are strategies and whose class of connected components is the surreal numbers.

Amar Hadzihasanovic (Feb 15 2025 at 08:14):

There seems to be also a sense in which, sometimes, orientation/direction is a categorification of sign, in that you can take a directed cell complex/computad-like object, which you can see as a cellular presentation of a higher category-like object, and take its cellular chain complex. Then the subdivision of the boundary of a cell into “target” and “source” cells is turned into the cellular chain

(sum of target cells) - (sum of source cells)

I have sometimes had a hunch that something like this is behind certain “negatives” in combinatorics, that is, you are seeing the shadow of a higher structure in which the things that get a negative sign in the count are “negatively oriented” with respect to it. But I cannot think of a good example right now.

David Corfield (Feb 15 2025 at 08:33):

Mike Shulman said:

Now I'll let the people who haven't seen this before have some fun pondering where on Earth this comes from and why it is a "reasonable" categorification of the integers (although the last paragraph above gives a bit of a hint in that direction). (-:

Is this something like a ring completion? The skeletal groupoid of 'virtual sets' $(A - B)$ up to equivlence, where, unlike virtual bundles/representations, etc., equivalence class representatives here are simply either $[n]$ or $-[n]$?

Kevin Carlson (Feb 15 2025 at 08:37):

I’d be surprised if that completion process killed all the permutations of non-virtual sets up to sign, though…it seems as if Mike has something a little more striking in mind, though I haven’t caught his drift yet.

David Corfield (Feb 15 2025 at 08:53):

With all of the advantages of theft over honest toil, it looks like Todd Trimble has some useful things to tell us about virtual sets here.

David Corfield (Feb 15 2025 at 09:11):

With Todd's talk of 1-cobordisms we're heading very much to @John Baez territory. Here he is categorifying the integers:

image.png

We'd often be led to right up to the sphere spectrum as the full categorification of the integers.

Mike Shulman (Feb 15 2025 at 10:01):

Indeed -- my example is the 1-type of the sphere spectrum. I got this explicit description of it from Modeling Stable One-Types by Niles Johnson and Angelica Osorno.

Mike Shulman (Feb 15 2025 at 10:06):

As for what I was hinting at, I'm not 100% sure of the precise statement, but I do think it's true that if you try to map FinSet into a riNg category all the permutations must get killed up to sign. This is because in a ring category all objects are (additively) invertible, which means that adding a fixed object is an automorphism of the category, and hence all automorphism groups are isomorphic. In particular, every automorphism group is isomorphic to the automorphism group of the (additive) unit object, and the automorphism group of the unit object in any monoidal category is abelian by the Eckmann-Hilton argument. And the sign homomorphism is the abelianization of any nontrivial symmetric group.

That doesn't explain why 0 and 1 also end up with nontrivial automorphisms, though. That probably has to do with stabilization.

David Corfield (Feb 15 2025 at 11:05):

I remember there used to be speculation about $\mathbb{Z}$-categories, with one pointer being that spectra should be some form of $\mathbb{Z}$-groupoids. I see work in this area develops:

• @David Kern, Categorical spectra as pointed $(\infty, \mathbb{Z})$-categories

image.png

John Baez (Feb 15 2025 at 22:24):

Nice. I never thought of it as speculation, more like a way of thinking, since spectra acts exactly like stable infinity-groupoids with homotopy groups indexed by $\mathbb{Z}$. But with a more general theory of $\mathbb{Z}$-categories one could turn this into a theorem, and from the title it sounds like Kern has.

John Baez (Feb 15 2025 at 22:24):

He notes they need to be pointed.

David Kern (Feb 20 2025 at 01:12):

Indeed, Naruki Masuda points out in his thesis that the co/op-duality operation on categorical spectra (and in particular its appearance in the half-central structure on the directed circle) can be thought of as a categorification of the Koszul sign rule, in the spirit of @Amar Hadzihasanovic's suggestion above. So my impression is that, while $\mathbb{Z}$-categories are probably not in themselves a categorification of counting with negative numbers (simply because I am not aware of any categorification of combinatorics where the counting numbers are directly categorified to the category level, though this can just be a flaw in my knowledge), they are maybe a more natural setting than simply higher categories in which this categorified sign rule can appear — in the same way that non-connective chain complexes do not have more “negative dimension” than connective complexes, but accommodate the Koszul sign rule just as well if not even more naturally.

David Kern (Feb 20 2025 at 01:12):

I should also point out that both the construction of the notion of (strict, and model-categorically weak) $\mathbb{Z}$-categories and the comparison with spectra are due to Paul Lessard in his thesis. (He only restricts the comparison to $\mathbb{Z}$-groupoids because the phrase “categorical spectrum” didn't exist at the time but I'm pretty sure his argument would work all the same; my paper was mainly just to give a “simpler” or clearer $\infty$-categorical argument than the model-categorical one I find unwieldy.)

John Baez (Feb 20 2025 at 01:33):

I'll have to read your paper, thanks!

Joyal (and others) regard the sphere spectrum as the "true integers", the perfect $\infty$-categorification of $\mathbb{Z}$. I gave a simple explanation of this in week102. This is an example of how I'd use spectra to $\infty$-categorify negatives. More generally the passage from $E_\infty$-spaces to connective spectra is a way of "throwing in negatives", a sophisticated generalization of group completion for commutative monoids. But notice none of this really requires spectra with negatively-indexed homotopy groups: connective spectra are enough.

Martin Brandenburg (Feb 20 2025 at 01:46):

FWIW there is a mathoverflow discussion about categorification of the integers from 2014 with interesting answers.

Martin Brandenburg (Feb 20 2025 at 01:54):

Coming back to the original question, I would say the natural categorification of a ring is a category equipped with two monoidal structures $+,\times$ (not assumed to be coproducts or products), as in a rig category, but also equipped with another functor $X \mapsto -X$ together with isomorphisms $\delta : X + (-X) \to 0$ and maybe more data (maybe we also need isomorphisms $-(X+Y) \to -X + -Y$, not sure), and of course several coherence axioms. If we denote this a ring category, then a discrete ring category is the same as a ring.

I don't know if this notion has been studied (does someone have a reference?). What seems to be much more common is to simply assert the existence of objectwise inverses (cf. the notion of a 2-group). But here the inverse operation shall belong to the data.

Then the original question becomes: why are rig categories so much more common than ring categories? At least, as of today?

John Baez (Feb 20 2025 at 02:03):

Here's my theory about that. Negative numbers were invented not by mathematicians or physicists, but by Venetian bankers, to allow people to go into debt. (They were written, not with minus signs, but in red, hence the term 'in the red' for being in debt.) So, they have a long history of being somewhat 'formal'. That is, they didn't first arise by a process of decategorication, like the natural numbers.

John Baez (Feb 20 2025 at 02:10):

Of course negative real numbers are very natural if we believe that upon fixing coordinates you can specify a point in space by listing three numbers: negative numbers just go in the opposite direction from positive numbers! But this seems quite different than 'counting', which was originally a form of decategorification (identifying elements of a finite set with elements of a standard set 1, 2, 3, ..., as a method of determining the isomorphism class of that finite set).

David Kern (Feb 20 2025 at 02:11):

John Baez said:

Joyal (and others) regard the sphere spectrum as the "true integers", the perfect $\infty$-categorification of $\mathbb{Z}$. I gave a simple explanation of this in week102. This is an example of how I'd use spectra to $\infty$-categorify negatives. More generally the passage from $E_\infty$-spaces to connective spectra is a way of "throwing in negatives", a sophisticated generalization of group completion for commutative monoids. But notice none of this really requires spectra with negatively-indexed homotopy groups: connective spectra are enough.

Right, turning this analogy with group completion into an actual “commutative diagram” is precisely what categorical spectra/$\mathbb{Z}$-categories are doing: the usual triple adjunction of groupoidal core and localisation between $\infty$-categories and $\infty$-groupoids extends to a triple adjunction giving the same tools between $\mathbb{Z}$-categories and $\mathbb{Z}$-groupoids. In particular, the sphere spectrum is the groupoid completion of the categorical spectrum coming from the (non-grouplike) symmetric monoidal groupoid $\mathsf{Fin}^\simeq$ — which is indeed a connective categorical spectrum (this groupoid is commonly seen by homotopy theorists as the “true natural integers”, and Maruki suggests, following Connes–Consani, that we should think of the corresponding categorical spectrum as the initial $\mathcal{E}_\infty$-rig categorifying or deriving $\mathbb{N}$). So I suppose that the moral of all this is indeed that connective $\mathbb{Z}$-groupoids are “more negative” than even non-connective $\mathbb{Z}$-categories.

David Kern (Feb 20 2025 at 02:25):

(Incidentally, I've just realised that there has been discussion of $\mathbb{Z}$-categories before in this very place, all coming from your week53.)

Matteo Capucci (he/him) (Feb 21 2025 at 14:25):

John Baez said:

Here's my theory about that. Negative numbers were invented not by mathematicians or physicists, but by Venetian bankers, to allow people to go into debt. (They were written, not with minus signs, but in red, hence the term 'in the red' for being in debt.) So, they have a long history of being somewhat 'formal'. That is, they didn't first arise by a process of decategorication, like the natural numbers.

They stand-in for transactions though. Like $n$ now decategorifies: you gave me $n$ things, and $-n$ decategorifies: I gave you $n$ things. The weird thing now is, if you give me one thing I have you, it's like nothing happened...

James Deikun (Feb 21 2025 at 16:18):

I think to categorify the thing with transactions in a really nice way you need to weaken that to "it's like nothing happened up to homotopy" and that line of thought leads you to the sphere spectrum rather than something (1,1)-categorical.

John Baez (Feb 21 2025 at 21:00):

Matteo Capucci (he/him) said:

They stand-in for transactions though. Like $n$ now decategorifies: you gave me $n$ things, and $-n$ decategorifies: I gave you $n$ things. The weird thing now is, if you give me one thing I have [given] you, it's like nothing happened...

Great! This sort of analysis is a good way to categorify negative numbers: pick an example where they are used (here commerce, or finance), and think hard about what's actually going on. We should expect that the existing description of what's going on (e.g. a single real number or integer describing the net amount you gave me or I gave you) is a useful summary that deliberately leaves out a lot of the information. Then we should try to restore that information.

Matteo Capucci (he/him) (Feb 25 2025 at 13:25):

James Deikun said:

I think to categorify the thing with transactions in a really nice way you need to weaken that to "it's like nothing happened up to homotopy" and that line of thought leads you to the sphere spectrum rather than something (1,1)-categorical.

Uh that sounds plausible, but how so?

David Corfield (Feb 25 2025 at 13:28):

John has the answer, I think.

John Baez said:

Joyal (and others) regard the sphere spectrum as the "true integers", the perfect $\infty$-categorification of $\mathbb{Z}$. I gave a simple explanation of this in week102.

Matteo Capucci (he/him) (Feb 25 2025 at 13:38):

oh well :upside_down:

David Corfield (Feb 25 2025 at 22:09):

Matteo Capucci (he/him) said:

This, naively, makes me think that, in principle, the situation above could be amended by working in more balanced logical settings like linear logic.

Linear HoTT, perhaps. It seems we may have a use for the negatives of the sphere spectrum in evading the dagger category approach to QM.

image.png

Quantum and Reality

Formalizations of quantum information theory in category theory and type theory, for the design of verifiable quantum programming languages, need to express its two fundamental characteristics: (1) parameterized linearity and (2) metricity. The first is naturally addressed by dependent-linearly typed languages such as Proto-Quipper or, following our recent observations: Linear Homotopy Type Theory (LHoTT). The second point has received much attention (only) in the form of semantics in "dagger-categories", where operator adjoints are axiomatized but their specification to Hermitian adjoints still needs to be imposed by hand.

We describe a natural emergence of Hermiticity which is rooted in principles of equivariant homotopy theory, lends itself to homotopically-typed languages and naturally connects to topological quantum states classified by twisted equivariant KR-theory. Namely, we observe that when the complex numbers are considered as a monoid internal to Z/2-equivariant real linear types, via complex conjugation, then (finite-dimensional) Hilbert spaces do become self-dual objects among internally-complex Real modules.

The point is that this construction of Hermitian forms requires of the ambient linear type theory nothing further than a negative unit term of tensor unit type. We observe that just such a term is constructible in LHoTT, where it interprets as an element of the infinity-group of units of the sphere spectrum, tying the foundations of quantum theory to homotopy theory. We close by indicating how this allows for encoding (and verifying) the unitarity of quantum gates and of quantum channels in quantum languages embedded into LHoTT.