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Stream: theory: category theory

Topic: Virtual equipments as categorified operads?

Nathaniel Virgo (Jul 28 2024 at 02:11):

I sort of already asked this at #theory: category theory > (ID my structure) operad in a monoidal bicategory, but the question was buried among the specifics, so I thought I'd ask it more directly.

Virtual equipments are usually characterised as virtual double categories such that all units and restrictions exist. But I have an intuition that they can also be seen as categorified operads in some sense.

Does that intuition resonate with anyone and is there a result that makes it formal? I sketched out one possible version in the other thread, but it's weirdly complicated and I'm wondering if there might be a different way to flesh the intuition out.

John Baez (Jul 28 2024 at 07:56):

I had trouble following that thread, but a common viewpoint is that a multicategory is a virtual version of a monoidal category, because we don't have a tensor product on objects, just a notion of morphisms out of the (virtual) tensor products, which are multimorphisms

$f: (x_1, \dots, x_n) \to y$

Similarly an operad is a virtual symmetric monoidal category. Is this the kind of sense in which you want a virtual equipment to be like a categorified operad?

Nathan Corbyn (Jul 28 2024 at 08:13):

This paper by @Geoff Cruttwell and @Mike Shulman likely has all the answers you are looking for

Nathaniel Virgo (Jul 29 2024 at 02:58):

John Baez said:

I had trouble following that thread, but a common viewpoint is that a multicategory is a virtual version of a monoidal category, because we don't have a tensor product on objects, just a notion of morphisms out of the (virtual) tensor products, which are multimorphisms

$f: (x_1, \dots, x_n) \to y$

Similarly an operad is a virtual symmetric monoidal category. Is this the kind of sense in which you want a virtual equipment to be like a categorified operad?

Yes, that's the kind of intuition I mean. I was using "operad" to mean a non-symmetric, non-coloured one - I should be more specific in future.

It seems like there's a kind of hierarchy of non-symmetric operad-like things, where you go from non-coloured operads, which have string diagrams like this

image.png

to coloured operads, where the diagrams look like this (I've coloured the wires using letters)

image.png

to what I called "virtual bicategories" in the other thread, whose string diagrams look like this - both the wires and the gaps in between the wires are coloured

image.png

Then there are virtual double categories, which are maybe a different kind of thing because they have these extra horizontal lines (both the wires and the nodes are also coloured, I just didn't draw the labels)

image.png

(see David Jaz Myers' A Yoneda-style embedding for virtual equipments for the string diagram calculus)

but then in double equipments you can bend the horizontal wires around, so you can always draw the diagrams like this if you want to

image.png

and then it starts to look like a kind of operad again. The intuition is that the gaps in between the wires are coloured with the objects of a category, and the morphisms in that category can act on the objects of the operad, from the left and the right.

The last three should also have monoidal versions, where the string diagrams become sheet diagrams.

So I'm wondering if there's a nice systematic way to understand this hierarchy of non-symmetric operad-like things, possibly excluding virtual double categories.

In the other thread we concluded/conjectured that (excluding virtual double categories) they can all be characterised as "lax monoids in a Gray monoid", for different choices of Gray monoid. But these are themselves quite complicated structures, so it seems a bit unsatisfying.

Nathaniel Virgo (Jul 29 2024 at 03:00):

Nathan Corbyn said:

This paper by Geoff Cruttwell and Mike Shulman likely has all the answers you are looking for

Maybe - I've been slowly going through that paper for a while and only understand a fraction of it, so it's possible that they cover this. My impression though is that they're mostly going the other way, using virtual equipments as a framework for thinking about operad-like things, rather than seeing virtual equipments themselves as a kind of operad.

John Baez (Jul 29 2024 at 08:39):

Thanks for the review of what you've been talking about in that other thread, @Nathaniel Virgo. It makes a lot of sense how all the structures you listed are increasingly fancy instances of the same general concept. I don't get what feature of a monoid is lax in a lax monoids, so I don't have an intuition yet for lax monoids in a Gray monoid, though I'm fairly comfortable with Gray monoids. Maybe you could say what feature(s) of a monoid are getting 'laxened'.

John Baez (Jul 29 2024 at 09:01):

I'm familiar with a lot of examples of how laxness is "creative", making situations much more interesting than the strict or pseudo analogues. For example a category is a lax map of bicategories from a chaotic (= codiscrete) category to the bicategory of sets and spans.

Nathaniel Virgo (Jul 30 2024 at 04:06):

This is a good question. The concept is from Day and Street's Lax monoids, pseudo-operads, and convolution. I was going with their unpacked definition ("a lax monoid ... can equally be described as ...") rather than trying to see why it's a lax monoid. I'm not familiar with Gray monoids at all (is there a good reference for them?) so I'm winging it a bit, but I gather I can think of them as monoidal bicategories.

The short answer to your question is that the laxness gives us the composition law for the operad-like thing. (So it's an example of "creative" laxness.)

The following all makes sense to me and I like it, though I'm a bit puzzled about why it's called a monoid.

The long answer:

Day and Street define a lax monoid as a strict monoidal lax functor $M:\Delta\to\mathcal{M}$ from the simplex category (objects $\{1,\dots,n\}$, morphisms order preserving maps, addition as monoidal product) to a Gray monoid.

$M$ being strict monoidal means that on objects $M(\mathbf{n}) = A\otimes \dots \otimes A = A^{\otimes n}$ for some fixed object $A$. It being a lax functor means that for every order-preserving map $f:\bf m\to n$ there is a 1-cell $M(f) = M(\mathbf{m})\to M(\mathbf{n}) = A^{\otimes m}\to A^{\otimes n}$. The laxness means that instead of an equation $M(f);M(g) = M(f;g)$ we instead get a 2-cell $M(f);M(g) \Rightarrow M(f;g)$.

We also have (from properties of $\Delta$ and the fact that $M$ is strict monoidal) that each 1-cell $M(f):A^{\otimes m}\to A^{\otimes n}$ factors canonically into a monoidal product of 1-cells with $A$ as their codomain,

image.png

where $s_k$ is $M$ applied to the unique map $\bf k\to 1$ in $\Delta$.

Put together, this all means that for each composable pair of maps $\bf m\to n\to r$ in $\Delta$, the laxness gives us a 2-cell that looks like

image.png

Additionally, laxness gives us a 2-cell $id_{M({\bf n}}\to M(id_{\bf n})$ for each $\bf n$ in $\Delta$, which in the end boils down to a single 2-cell

image.png

All these 2-cells end up having to obey coherence laws that look very similar to the ones for an operad.

If you instantiate all this in $\bf Span$ (as a Gray monoid) then you get exactly a coloured operad. The object $A$ is the set of colours, the 1-cells are the hom-sets (with the legs of the spans mapping a morphism to its domain and codomain) and the 2-cells are the composition law.

I conjecture that you can also get virtual equipments by instantiating this in a different Gray monoid, but it's not as simple as instantiating it in $\bf Prof$ instead of $\bf Span$ - if the conjecture is true then you have to use a different and somewhat weird-looking Gray monoid. I described it in this message. The fact that it's weird-looking suggests there's probably something else going on. It sort of feels like it wants to involve the tricategory (triple category?) of categories, functors, profunctors and spans of profunctors (which might pique @CB Wells' interest since he's written about that). But I'm not sure.

I don't really see where the term "lax monoid" comes from for this construction though, because I don't see a way to make a non-lax version that would yield monoids.

Nathaniel Virgo (Jul 30 2024 at 04:33):

Ah - I guess $\Delta$ must include the empty set - they don't explicitly say this, but it wouldn't have a monoidal unit otherwise. In that case nlab says that for a strict monoidal category $\mathcal{C}$, strict monoidal functors $\Delta\to\mathcal{C}$ are the same as monoids in $\mathcal{C}$, which makes sense. So that explains the name "lax monoid".

John Baez (Jul 30 2024 at 12:42):

Nathaniel Virgo said:

I'm not familiar with Gray monoids at all (is there a good reference for them?)

I don't know if it's good, but I'm familiar with my paper with Martin Neuchl on braided monoidal 2-categories, which starts by explaining Gray monoids. We call them "semistrict monoidal 2-categories" since they are monoidal 2-categories where the only thing weakened is the interchange law between tensor product and composition of morphisms:

$( f \otimes g) \circ (h \otimes k) \cong (f \circ h) \otimes (g \circ k)$

But more conceptually they are monoids in the monoidal category $\mathsf{Gray}$.

Nathaniel Virgo (Jul 30 2024 at 12:43):

Thanks, that looks very useful.

John Baez (Jul 30 2024 at 12:45):

You may recall that $\mathsf{Cat}$ has just two symmetric monoidal closed structures: the cartesian product, where we have

$( f \times g) \circ (h \times k) = (f \circ h) \times (g \circ k),$

and the [[funny tensor product]] where we don't make the squares coming from a pair of morphisms commute:

$( f \otimes g) \circ (h \otimes k) \ne (f \circ h) \otimes (g \circ k)$

(in general). The category of 2-categories and strict 2-functors has a third symmetric monoidal structure, the Gray tensor product, where we make these squares commute up to coherent isomorphism. And this gives the monoidal category $\mathsf{Gray}$.

John Baez (Jul 30 2024 at 12:49):

The main virtue of $\mathsf{Gray}$ is that every tricategory is equivalent to a $\mathsf{Gray}$-enriched category - this is the main strictification theorem for tricategories. As a spinoff, every monoidal bicategory is equivalent to a $\mathsf{Gray}$-monoid.

So, $\mathsf{Gray}$-monoids are often used as a convenient substitute for fully general monoidal bicategories.

John Baez (Jul 30 2024 at 12:49):

That's the main stuff to know, until you get into all the detailed diagrams for the coherence laws!

John Baez (Jul 30 2024 at 12:54):

But thanks for explaining lax monoids:

Day and Street define a lax monoid as a strict monoidal lax functor $M:\Delta\to\mathcal{M}$ from the simplex category (objects $\{1,\dots,n\}$, morphisms order preserving maps, addition as monoidal product) to a Gray monoid.

Okay, that's a nice snappy definition.

I don't really see where the term "lax monoid" comes from for this construction though, because I don't see a way to make a non-lax version that would yield monoids.

A strict monoidal functor from $\Delta$ to any monoidal category $\mathsf{M}$ is the same as a monoid in $\mathsf{M}$. This is one reason simplicial stuff is so great.

So when we go to a Gray-monoid, aka semistrict monoidal 2-category $\mathcal{M}$, it makes a lot of sense to look at a strict monoidal lax functor from $\Delta$ to $\mathcal{M}$.

I'm not saying I instantly see what this laxening achieves, but it's definitely a concept that God would have looked into.

John Baez (Jul 30 2024 at 12:57):

Oh, now I see you said all this - I tend to respond as I read through someone's comments. Yes, $\Delta$ here must be the augmented simplex category - it's probably more wise to call it $\Delta_a$.