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Stream: theory: category theory

Topic: categorical differential equations

fosco (May 05 2021 at 09:04):

Many of you who had to tolerate my stubborness lately, know that I've been interested in "differential 2-rigs" https://arxiv.org/abs/2103.00938; now, mostly thanks to the fact that @Todd Trimble joined @Fabrizio Genovese and me, the work has almost doubled in length, and many other interesting ideas emerged.

As you can see, the paper ends with an account of "future prospects"; one the ideas is to study differential equations in a 2-rig. Let me sketch the main lines of the discussion.

Given a 2-rig $\mathcal C$ with a derivation, it is a natural question to investigate the fixed points of a derivation $\partial$ on $\mathcal{C}$, when $\partial$ is regarded as an endofunctor, if by 'fixed point' we mean an initial algebra or a terminal coalgebra of $\partial$.

One could legitimately call an $X\in \mathcal{C}$ such that $\partial X \cong X$ a 'Napier object', (for the obvious reason that 'exponential object' already has a different meaning).

Now, given that $\partial$ preserves coproducts, initial $\partial$-algebras are seldom an interesting object; the initial chain

$\varnothing \to \partial \varnothing \to \partial\partial \varnothing \to \partial\partial\partial \varnothing \to \dots$

needed to build the initial $\partial$-algebra stops after a single step.

On the other hand, terminal colgebras turn out to be linked to the behaviour of the object $\partial 1$: the terminal cochain is

$1\leftarrow \partial 1 \leftarrow \partial\partial 1 \leftarrow \partial\partial\partial 1 \leftarrow \dots$

thus the first ordinal $\lambda$ for which the transition morphism $v : \partial^\lambda 1 \leftarrow \partial^{\lambda + 1} 1$ is invertible realises the terminal coalgebra. This is related to an old question of mine, where I was asking what's up with $\partial 1$, and whether I can expect it to be zero just as a consequence of the axioms.

Now, it takes just a little bit of imagination to zoom-out from this particular instance of a fixed point to speak of the following in a 2-rig with duals: if we interpret the word 'solution' to mean 'terminal coalgebra', then we can define also

1. hyperbolic sines, i.e. 'solutions' to 'equations' of the form $\partial\partial X \cong X$; and in a category with duals $X^L, X^R$ for every object,
2. left (resp., right) logarithms, i.e. 'solutions' to 'equations' of the form $\partial X \cong X^L$ (resp., $\partial X\cong X^R$).
3. left (resp., right) sine objects in a differential 2-rig, i.e. 'solutions' to 'equations' of the form $\partial\partial X \cong X^L$ (resp., $\partial\partial X \cong X^R$);

This paves the way to the following

Definition (Differential-polynomial equation). A (ordinary) differential-polynomial equation (DPE for short) is (slightly informally, the paper gives a better definition) a functor that can be obtained from iterate composition of 'polynomial operations' (sum and product) and derivation.

An example of a DPEs 'with constant coefficients' is

$X\mapsto A\otimes X^{(2)} \cup B\otimes X^{(1)} \cup C;$

a solution for a given DPE $F : \mathcal{C} \to \mathcal{C}$ consists of a terminal coalgebra for $F$, regarded as an endofunctor of $\mathcal{C}$. Clearly, a Napier object is a solution for the DPE $FX = X^{(1)}$, whereas a hyperbolic sine object is a solution for $GX=X^{(2)}$.

Now, let's switch to the theory of differential equations in the category of species; it has a long history; first introduced in the initial works by Joyal, it was mostly developed by Leroux and Viennot, Labelle, Rajan...

My claim now is that the general theory of combinatorial differential equations studied in these papers might be framed into a more general theory of DPEs and their solutions. As far as I understand, no one attempted to develop this general theory of DPEs in a differential 2-rig. And determining the 'solutions' of a DPE in species has always been a matter of hard group theory (see e.g. the work of Rajan on the solutions of $\partial^k Y = X^n$), hard combinatorics (see various papers by Labelle) or clever manipulations only valid in special cases.

In fact, a remarkable theorem of Labelle says that a given ODE in species can have $n$ distinct solutions for every $1 < n \le 2^{\aleph_0}$, except $\aleph_0$. Now, among these solutions, by definition, there must be the terminal coalgebra for the endofunctor that defines the ODE, and my claim is that that universal object deserves the name of "the solution" of the ODE.

A remarkable example of this multiplicity of solutions lives in the category of virtual species (more or less, formal differences of species, like one would do in a cancellative monoid to obtain its Grothendieck group); there, we can solve all combinatorial differential equations $\partial X = G$ where $G$ is a given species: its general solution is of the form $A + \int G$ where $A$ is a solution of $\partial A =\varnothing$, and

${\textstyle \int G} := \sum_{k=1}^\infty (-1)^{k+1} E_k \partial^{k-1} G$

where $E_k$ is the species of $k$-element sets, and $\partial E_k \cong E_{k-1}$; it just takes the Leibniz rule to prove that in fact $\partial(A+\int G) \cong G$.

An enticing piece of the theory of combinatorial species that might turn out to be useful in order to attack the study of DPEs relies on the results outlined by Rajan in "The adjoints to the derivative functor on species".

The upshot of the paper is that the good behaviour of the derivative functor on species can be at least partly motivated with the fact that $\partial$ sits in a triple of adjoints

$L\dashv \partial \dashv R.$

fosco (May 05 2021 at 09:12):

This is somewhat a cheap observation for a category theorist: $\partial$ happens to be defined as pre-composition with the 'add 1' functor $T : A \mapsto A+1$ on the category $\mathbf{B}$ of sets and bijections; thus, its left (resp., right) adjoint is nothing more than the left (resp., right) Kan extension along $T$:

$\text{Lan}_T \dashv \partial \dashv \text{Ran}_T.$

The observation has, however, deep consequences.

More conceptually, one can see that $\partial$ results as the functor $F\mapsto \{y[1], F\}$, where $\{\_,\_\}$ is the internal hom of the Day convolution monoidal structure; thus

• $\partial$ admits a left adjoint, given by $y[1]\ast \_$; this is the functor $L=\text{Lan}_T$ above;

• since $y[1]$ is representable, hence a tiny object of $[\mathbf{B},\mathsf{Set}]$, the functor $\{y[1],\_\}$ is also cocontinuous, hence a left adjoint by the adjoint functor theorem; its right adjoint is a functor that we denote $y[1]\lhd \_$, and that coincides with $\text{Ran}_T$.

  Explicit formulas for $y[1]\ast \_$ and $y[1] \lhd \_$ can be easily computed:

$\displaystyle\text{Lan}_T : F \mapsto \lambda A.\sum_{i\in A} F(A\setminus\{i\})\qquad \text{Ran}_T : F \mapsto \lambda A.\prod_{i\in A}F(A\setminus\{i\}).$

Being able to jump between one of these descriptions and another turns out to be useful in computations, and provides explicit formulas for $\partial F, LF, RF$ in case $F$ is the characteristic species of $n$-tuples, a representable species, the terminal species, the species of powerset...

This suggests that the study of differential equations in the category of species can be developed to a certain extent in order to obtain combinatorial equations in terms of polynomial expressions in the endofunctors $\partial,L,R$.

The only missing piece in order to carry on this analysis is a precise description of how the monads and the comonads generated by the triple of adjoints above interact with each other and split as composition of simpler functors:

Proposition. Let $U : \mathsf{Fin} \to \mathsf{Set}$ be the 'tautological' species sending a finite set $A$ to itself. Let $\text{id}$ be the identity functor on the category of species. Then,

1. $L\circ \partial \cong U \times \text{id}$, as it follows from the computation

$\displaystyle L(\partial F) : A\mapsto \sum_{a\in A}(\partial F)(A\setminus\{a\}) \cong \sum_{a\in A} FA \cong A\times FA$

2. $\partial\circ L\cong (U + {\bf 1})\times \text{id}$ sending a species $F$ to the species $A\mapsto (A+1)\times FA$ (proof: a straightforward, similar manipulation); similarly,
3. $R\circ \partial\cong \text{id}^U$, sending a species $F$ to the species $A\mapsto F(A)^A$;
4. $\partial \circ R \cong \text{id}^U\times \text{id}$, sending a species $F$ to the species $A\mapsto FA^A\times FA$.

This formalism is rather powerful: for example, it can prove by completely formal means the isomorphism $\wp\cong \mathbf{t} \ast \mathbf{t}$ between the species of powersets and the two-fold convolution of the terminal species.

Another remarkable result on the structure of the category of species was due to Yeh (afaicu, a student of Labelle) whose doctoral thesis "On the Combinatorial Species of Joyal" proved the following: let's call a species a molecule if it is indecomposable with respect to coproduct, and an atom if it is indecomposable with respect to product.

It's easy to see that a molecule is concentrated at a single cardinality $n_M=|A|$ of those $A$'s for which $MA$ is nonempty, i.e. $MB\cong\varnothing$ for every other $B$ whose cardinality is not $n_M$; the number so determined is the degree of the molecule.

Every species can be uniquely written as a sum of molecules (in fact, as a sum of its molecular sub-species), and every molecule can be uniquely written as a product of atoms.

Then, we have the following

Theorem (Yeh decomposition). The rig of isomorphism classes of species is isomorphic to the rig of polynomials with natural coefficients, on a countably infinite set of indeterminates that can be taken to form an ordered enumeration of its atoms:

$\mathsf{Spc} \cong \mathbb N\llbracket A_1 \sqcup A_2 \sqcup A_3 \sqcup \dots\rrbracket$

Evidently, the definition of molecule and atom transports in a straightforward way to the case of general 2-rigs. This leads to the following definition.

Definition (Scopic 2-rig). A 2-rig $\mathcal{C}$ is called scopic if it has a countable set of atoms $\mathcal{A}$, and

$\mathcal{C} \cong \mathsf{Fin}\llbracket \zeta_1,\zeta_2,\dots\rrbracket$

where $\underline{\zeta} = \{\zeta_1,\zeta_2,\dots\}$ is an enumeration of $\mathcal{A}$, and $\mathsf{Fin}\llbracket\underline{\zeta}\rrbracket$ is a suitably define 'free 2-rig' with coefficients in $\mathsf{Fin}$.

In addition, in $\mathsf{Spc}$ there is just a finite set of molecules of a given degree; in fact, Yeh proves the following: $d_i$ is equal to the number of conjugacy classes of subgroups of the symmetric group on $i$ elements, and there is an explicit procedure to build all the $d_i$ molecules of a given degree. As a side remark, this suggests that there exists a very tight connection between the theory of differential equations over $\mathsf{Spc}$ and the representation theory of symmetric groups; this tight connection reflects on the theory of differential equations, and a remarkable example of this connection is the main result in Rajan.

In a general 2-rig the notion of degree does not seem to make sense.

Whew, this was a long trip.

Now, my question is: is it possible to find nontrivial examples of scopic 2-rigs, other than the 2-rig of species?

John Baez (May 05 2021 at 14:35):

That was great, @fosco! I love species so it's nice to see more about them.

John Baez (May 05 2021 at 15:59):

Now, my question is: is it possible to find nontrivial examples of scopic 2-rigs, other than the 2-rig of species?

Species are presheaves on the groupoid of finite sets, and the category of species has a few interesting 2-rig structures, at least in my favorite definition of '2-rig': a symmetric monoidal cocomplete category where tensor product distributes over colimits.

John Baez (May 05 2021 at 16:00):

My favorite 2-rig structure on species is the one you're calling $\ast$: Day convolution with respect to the "disjoint union" monoidal structure on the groupoid of finite sets.

Joe Moeller (May 05 2021 at 16:00):

I wish there was an briefer way to distinguish different sorts of 2-rigs in a consistent and clear way.

John Baez (May 05 2021 at 16:01):

But the definition of "atomic" species uses products and coproducts, so maybe you (Fosco) are talking about 2-rigs where the monoidal structure is cartesian product!

John Baez (May 05 2021 at 16:02):

And of course species also becomes a 2-rig in this way, where the monoidal structure is cartesian product.

John Baez (May 05 2021 at 16:03):

Anyway, while I'm confused about your definition of 2-rig, I can try to give some examples of scopic 2-rigs.

John Baez (May 05 2021 at 16:04):

What features of a category $X$ make the category of presheaves on $X$ into a scopic 2-rig?

John Baez (May 05 2021 at 16:05):

If we use the 2-rig structure where the monoidal structure is the cartesian product, then of course the category of presheaves on any category becomes a 2-rig.

John Baez (May 05 2021 at 16:05):

So then the question is just when this is "scopic".

John Baez (May 05 2021 at 16:07):

Here's my guess about some sufficient conditions on $X$: it may suffice for $X$ to be a groupoid that's a countable coproduct of 1-object groupoids, each coming from a finite group.

John Baez (May 05 2021 at 16:08):

After all, that seemed to be the key to proving that when $X$ is the groupoid of finite sets, presheaves on $X$ form a scopic 2-rig.

John Baez (May 05 2021 at 16:10):

If my guess is right, there are tons of nice examples of $X$. For example, the groupoid of finite-dimensional vector spaces over a finite field. Or the groupoid of finite graphs!

fosco (May 05 2021 at 17:52):

@John Baez I remember there was a bit of confusion some time ago, when I started studying this stuff, about the def of "2-rig"

Let me dispel your confusion: I want to study a "differential 2-rig": a category endowed with two monoidal structures, one of which is cocartesian, and over which another monoidal structure distributes; and moreover, there is a functor D : C -> C that preserves coproducts and satisfies the Leibniz rule.

This notion of 2-rig is less general in some respect (the additive structure is cocartesian, not a generic monoidal structure): this makes easier to work without all the coherence Laplaza needs
It is also more general: I don't want the multiplicative structure to distribute over all colimits; "being linear" means "commuting with coproducts"

John Baez (May 05 2021 at 18:37):

Okay, so I'd say your 2-rigs are monoidal categories with finite coproducts, where the tensor product distributes over coproducts.

fosco (May 05 2021 at 18:38):

precisely

John Baez (May 05 2021 at 18:38):

These are a bit more general than my favorite definition of 2-rig: a monoidal category with small colimits, where the tensor product distributes over colimits.

John Baez (May 05 2021 at 18:39):

But I actually consider many different variants for different reasons, so if finite coproducts is all you need, that's fine with me.

fosco (May 05 2021 at 18:39):

it makes things work; I've been open for a long time to the possibility that some nontrivial examples of derivation live outside this framework.

fosco (May 05 2021 at 18:39):

I still am

John Baez (May 05 2021 at 18:40):

My other big question concerns this: your definition of "scopic" seemed to make use of products.

John Baez (May 05 2021 at 18:40):

I.e., cartesian products.

John Baez (May 05 2021 at 18:40):

So I was a bit confused, since your definition of 2-rig doesn't seem to require products.

John Baez (May 05 2021 at 18:40):

(There's something we could call a "cartesian 2-rig" where the monoidal structure is cartesian product.)

fosco (May 05 2021 at 18:41):

uh, no; that part was a bit confused (not only because of this): the operation that I called product is Cauchy product of species, not Hadamard product of species

fosco (May 05 2021 at 18:41):

Hadamard product = cartesian
Cauchy = Day convolution

fosco (May 05 2021 at 18:41):

at least judging from the literature...

John Baez (May 05 2021 at 18:43):

That's indeed how the terminology goes.

John Baez (May 05 2021 at 18:43):

So a species is molecular if it's not a coproduct in a nontrivial way, and then atomic if it's not a Cauchy product in a nontrivial way? I've read about this but I tend to forget...

John Baez (May 05 2021 at 19:12):

Anyway, if atomicity is defined with respect to the Cauchy product (in the case of species), I'll revise my conjecture about how to get piles of "scopic 2-rigs", though the two concrete examples I mentioned will still be on my list!

fosco (May 05 2021 at 19:26):

it's gettin late in Europe, I'll be back tomorrow. But I'm glad I generated some discussion

John Baez (May 05 2021 at 19:27):

Okay, great. If you answer my question tomorrow I'll try to give you piles of scopic 2-rigs.

Joachim Kock (May 05 2021 at 22:57):

Yeh was a student of Steve Schanuel in Buffalo, but then he went on to a postdoc position in Montréal.

John Baez (May 05 2021 at 22:58):

Who is "he"? This has been a long conversation.

Joachim Kock (May 05 2021 at 23:07):

Yeh :-)

Joachim Kock (May 05 2021 at 23:08):

(pronouns: ye/yim)

Joachim Kock (May 05 2021 at 23:08):

[typing...]

Joachim Kock (May 05 2021 at 23:10):

I am a bit confused about Yeh's decomposition theorem, and I suspect it has to do with some representation theory of the symmetric groups, and that maybe it is an artifact of working over sets.

Joachim Kock (May 05 2021 at 23:11):

Over groupoids I think it would look like this, and it is not nearly as interesting as Yeh's theorem. (And maybe I am just misunderstanding everything.)

John Baez (May 05 2021 at 23:11):

Likke most mathematics it's an artifact of working over sets.

John Baez (May 05 2021 at 23:12):

Every G-set is a coproduct of transitive G-sets, and any transitive G-set is of the form G/H for some subgroup H of G.

John Baez (May 05 2021 at 23:13):

This lets you decompose any presheaf on a groupoid as a coproduct of pieces that can't be further decomposed this way.

Namely, every groupoid X is equivalent to a coproduct of connected groupoids, which come from groups $G_\alpha$, and then every presheaf on X is just a coproduct of ones of the form $G_\alpha/H_\alpha$.

I believe this is what they call the "molecular decomposition" of a species, when X is the groupoid of finite sets.

Joachim Kock (May 05 2021 at 23:14):

Groupoid-valued species (also called stuff types) are the same thing as finitary polynomial functors over groupoids, meaning functors represented by a map of groupoids E -> B with finite discrete fibres. The equivalence goes like this: there is a classifier for finite sets, given by BB' -> BB where BB is the groupoid of finite sets and bijections, and BB' is the groupoid of finite pointed sets and basepoint preserving bijections. Since it is a classifier, any map E -> B induces a map B -> BB. That's essentially a species: you can extract a species by taking F: BB -> Grpd to be the functor sending n to the homotopy fibre over n. Conversely, given a species F: BB -> Grpd, take the Grothendieck construction to get something over BB, say B -> BB and then pullback the universal family to get something over B.

Joachim Kock (May 05 2021 at 23:15):

This is nice, because now the polynomial functor defined by E -> B is the analytic functor defined by the corresponding species F. The point is that over groupoids there is no distinction between finitary polynomial and analytic. The bad group-action quotients that screwed up this equivalence over the category of sets become well behaved over groupoids, where it is as if every group action is free.

Joachim Kock (May 05 2021 at 23:16):

Now the Cauchy product is just product of functors, and sums are sums. And polynomial functors obviously form a polynomial ring...

Joachim Kock (May 05 2021 at 23:19):

But I guess the point is that the sums involved in the definition of polynomial functors over groupoids are rather homotopy sums. That just means colimits over groupoids (just like ordinary sums are colimits over discrete sets). If these fancier sums are allowed, then

Joachim Kock (May 05 2021 at 23:23):

OK, meanwhile John explained it, now I think I understand: if you allow homotopy sums instead of just discrete sums, then it seems that the molecules are the monomials and the only atom is X, because X^n is obviously the product of n copies of X. So the subtlety is that only discrete sums are allowed in the definition of molecule, and then there are many more of them, namely all the G/H John mentioned.

John Baez (May 05 2021 at 23:24):

I think the definition of atom in species theory is different than the one you're using, because I think there's not just one atom.

John Baez (May 05 2021 at 23:25):

I just seem to recall (from the Big Red Book of Species) that there are lots of atomic species.

John Baez (May 05 2021 at 23:25):

But I don't really understand the definition of atomic species, even though Fosco just tried to explain it a while back.

Joachim Kock (May 05 2021 at 23:26):

Yes, I agree that most mathematics is an artifact of working over sets :-)

Joachim Kock (May 05 2021 at 23:27):

Yes, the notion of atom I was using is too simplistic. It just shifts all the representation theory into the notion of sums, and does probably not help at all.

John Baez (May 05 2021 at 23:34):

Okay, I looked it up, though Fosco actually said it earlier:

• Labelle and Yeh, The relation between Burnside rings and combinatorial species.

John Baez (May 05 2021 at 23:35):

A species is molecular if it can't be written as a coproduct of other species in a nontrivial way, and it's atomic if it's molecular and it can't be written as a Cauchy product of other species in a nontrivial way.

John Baez (May 05 2021 at 23:36):

(The Cauchy product is the Day convolution product with respect to the "disjoint union" monoidal structure on the groupoid of finite sets.)

John Baez (May 05 2021 at 23:39):

So what you were calling X, "being a one-element set", is atomic, but there are other atomic species.

Joachim Kock (May 05 2021 at 23:39):

Ah, the Burnside ring!

John Baez (May 05 2021 at 23:40):

It looks like after Proposition 5.1 they claim that whenever $H$ is a transitive subgroup of $S_n$, i.e. one whose obvious action on the n-element set is transitive, it gives an atomic species.

John Baez (May 05 2021 at 23:41):

Unfortunately I bet that's not an "if and only if".

Joachim Kock (May 05 2021 at 23:42):

Yes, I was being silly: the other atomic species become homotopy sums of the one-element-set species, since any group action quotient counts as a homotopy sum. So with homotopy sums there is no interesting notion of atom.

fosco (May 06 2021 at 07:33):

John Baez said:

Unfortunately I bet that's not an "if and only if".

I'm back.

It's not, but we know what we're identifying: two conjugate transitive subgroups give the same molecule

fosco (May 06 2021 at 07:47):

I see you have already mentioned Yeh's theorem about the Burnside ring

Today I have a question I've been discussing with @Todd Trimble and others, a purely combinatorial curiosity.

The way in which you build the terminal coalgebra for an endofunctor is "apply the functor to 1 a sufficient amount of times". If we do this for seemingly innocuous differential equations we find interesting recurrence relations:

1. Consider the differential operator $D:y\mapsto y'+by$; interpreted in the ring of formal power series, the solution we're looking at is an element $y$ such that $y'+by=y$, which can be easily found to be $\exp(1-b)$ (+ a constant determined by the coefficient of $b$ at 0). Regarding $D$ as an endofunction/endofunctor, and looking at its iterates starting from $D1=B$, we get

$\begin{array}{l} 1 \\ B \\ B'+B^2 \\ B''+3 B B'+B^3 \\ B^{(3)}+4 B B''+6 B^2 B'+3 B'^2+B^4 \\ B^{(4)}+5 B^{(3)} B+10 B^2 B''+10 B^3 B'+15 B B'^2+10 B' B''+B^5 \\ B^{(5)}+6 B^{(4)} B+15 B^{(3)} B^2+20 B^3 B''+10 B''^2+15 B^4 B'+45 B^2 B'^2+15 B'^3+15 B^{(3)} B'+60 B B' B''+B^6 \\ B^{(6)}+7 B^{(5)} B+21 B^{(4)} B^2+35 B^{(3)} B^3+35 B^4 B''+70 B B''^2+21 B^5 B'+105 B^3 B'^2+105 B B'^3+21 B^{(4)} B'+35 B^{(3)} B''+105 B^{(3)} B B'+210 B^2 B' B''+105 B'^2 B''+B^7 \\ \end{array}$

the coefficients of these expressions seem to be some multinomials (35, 70, 105.. appear in https://oeis.org/A080575)

fosco (May 06 2021 at 07:47):

If you play the same game with the differential operator $DX = AX'+B$, starting from $X=1$, you get $D1=B$, and $DD1=DB=AB'+B$, and $D(AB'+B)=A(AB'+B)' + B$, etc.:

$\begin{array}{l} 1 \\ B \\ A B'+B \\ A A' B'+A^2 B''+A B'+B \\ A^2 A'' B'+3 A^2 A' B''+A A'^2 B'+A A' B'+A^3 B^{(3)}+A^2 B''+A B'+B \\ A^{(3)} A^3 B'+4 A^3 A'' B''+A^2 A'' B'+6 A^3 B^{(3)} A'+7 A^2 A'^2 B''+3 A^2 A' B''+A A'^3 B'+A A'^2 B'+A A' B'+4 A^2 A' A'' B'+A^4 B^{(4)}+A^3 B^{(3)}+A^2 B''+A B'+B \\ \end{array}$

fosco (May 06 2021 at 07:50):

The solution of the differential equation $ay'+b=y$ is not so immediate to find, but with some Weyl-algebra sorcery you get that

$y = (1-a\frac{d}{dt})^{-1}b = \displaystyle\sum_{k=0}^\infty (aD)^kb$

in the ring of power series, taking into account the fact that $[a, \frac{d}{dt}]\ne 0$, and in fact finding a "nice" closed form for the iterated powers of $a\frac{d}{dt}$

Morgan Rogers (he/him) (May 06 2021 at 09:02):

Joe Moeller said:

I wish there was an briefer way to distinguish different sorts of 2-rigs in a consistent and clear way.

A naming or notation system would be handy.

Nathanael Arkor (May 06 2021 at 09:53):

Morgan Rogers (he/him) said:

Joe Moeller said:

I wish there was an briefer way to distinguish different sorts of 2-rigs in a consistent and clear way.

A naming or notation system would be handy.

This was discussed extensively in the other thread. There are consistent names for each of these concepts, but "2-rig" is shorter, so people tend to overload it.

fosco (May 06 2021 at 10:09):

[xkcd_standards.jpeg]

fosco (May 06 2021 at 10:09):

I'm guilty, alas!

John Baez (May 06 2021 at 17:03):

Now that I know what "scopic 2-rig", I think I can give @fosco a shitload of scopic 2-rigs. But I'm not going to prove anything here, just conjecture stuff.

John Baez (May 06 2021 at 17:04):

My claimed scopic 2-rigs will be of this form: the category $\hat{X}$ of presheaves on a groupoid $X$, made monoidal via Day convolution with respect to some monoidal structure on $X$.

John Baez (May 06 2021 at 17:05):

The classic example to keep in mind is when $X$ is the groupoid of finite sets and bijections, made monoidal using disjoint union. By the way, disjoint union is not coproduct in $X$, but it is the coproduct in a larger category $X'$ containing $X$, namely the category of finite sets and functions. We can take the coproduct monoidal structure on $X'$ and restrict it to $X$ to get a monoidal structure on $X$.

John Baez (May 06 2021 at 17:08):

All my other claimed examples of scopic 2-rig will have the same general flavor; in particular the groupoid $X$ will always be the core of some category $X'$ with finite coproducts, and $X$ will get its monoidal structure that way.

John Baez (May 06 2021 at 17:13):

Here are some examples of $X'$:

• the category of finite simple graphs
• the category of finite directed graphs
• the category of finite directed multigraphs (or "quivers" - the graphs that category theorists like)
• the category of finite-dimensional vector spaces over some finite field (there's one finite field for each prime power)
• the category of finite posets

John Baez (May 06 2021 at 17:14):

What do these have in common, that makes $\hat{X}$ with Day convolution be scopic?

John Baez (May 06 2021 at 17:14):

First of all, every object of $X'$ has a finite group of automorphisms.

John Baez (May 06 2021 at 17:16):

This guarantees that $X$ is equivalent to a coproduct of finite groups - or, more pedantically, the coproduct of 1-object groupoids that are deloopings of finite groups.

John Baez (May 06 2021 at 17:18):

Second of all, there are countably many isomorphism classes of objects in $X'$. This is because @fosco had a countability condition in his definition of "scopic".

John Baez (May 06 2021 at 17:20):

Third, $X'$ is $\mathbb{N}$-graded in the following sense: each object $x \in X'$ has a grade $d(x)$, such that

$d(x+x') = d(x)+d(x')$

John Baez (May 06 2021 at 17:21):

And fourth, $X'$ has finitely many isomorphism classes of objects in each grade. (The third and fourth conditions actually imply the second.)

John Baez (May 06 2021 at 17:22):

In my categories of graphs, the grade of a graph is its number of vertices. In the category of vector spaces, the grade of a vector space is its dimension.

John Baez (May 06 2021 at 17:23):

We don't actually need $X'$ to be $\mathbb{N}$-graded; it could be $\mathbb{N}^k$-graded, but there are already lots of $\mathbb{N}$-graded examples.

John Baez (May 06 2021 at 17:24):

So that's my conjecture: if $X'$ is a category with finite coproducts obeying conditions 1)-4), and $X$ is its core, then $\hat{X}$ is scopic.

John Baez (May 06 2021 at 17:29):

If this conjecture is wrong, it could easily be because I don't fully grok @fosco's definition of "scopic", or Yeh's theorem. What my conditions 1)-4) are really supposed to imply is this:

Every object of $\hat{X}$ can be uniquely written as a coproduct of molecules (objects that cannot be further decomposed as coproducts in a nontrivial way), and every molecule can be uniquely written as a tensor product of atoms (molecules that cannot be further decomposed as tensor products in a nontrivial way).

fosco (May 06 2021 at 17:36):

@John Baez : I will soon get back at you. Unfortunately the different timezones we're in make the conversation a bit asyncronous.

John Baez (May 06 2021 at 18:13):

Yup!

Notification Bot (May 06 2021 at 23:21):

This topic was moved by Nathanael Arkor to #learning: questions > All about 2-rigs: should they be called 2-rigs?