Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Properads

Joe Moeller (Mar 25 2020 at 18:58):

@Jonathan Beardsley or anyone else, can you tell me what a properad is?

Jonathan Beardsley (Mar 25 2020 at 18:59):

Sure, so let's think about one way to define an operad: as a functor from a category of trees to the category of sets (or spaces, if you want topological operads or something)

Jonathan Beardsley (Mar 25 2020 at 18:59):

And by tree here, I mean a rooted tree.

Jonathan Beardsley (Mar 25 2020 at 19:00):

Like this: Y

Jonathan Beardsley (Mar 25 2020 at 19:00):

So one "downward root" and then some collection of leaves.

Jonathan Beardsley (Mar 25 2020 at 19:01):

Let's see... to actually give this in any reasonable detail I'd have to probably think/read for a little bit...

Jonathan Beardsley (Mar 25 2020 at 19:02):

I'd also have to tell you about the morphisms in the category of trees.

Jonathan Beardsley (Mar 25 2020 at 19:02):

I guess one important thing to know is that the category of trees actually extends the classical simplex category $\Delta$

Jonathan Beardsley (Mar 25 2020 at 19:02):

Hm, that didn't work.

Jonathan Beardsley (Mar 25 2020 at 19:02):

$\Delta$

Jonathan Beardsley (Mar 25 2020 at 19:03):

Where we can think of $\Delta$ as being the category of linear trees.

Jonathan Beardsley (Mar 25 2020 at 19:04):

There are a lot of morphisms between the trees, but they're generated by a nice collection of morphisms, just like you have with $\Delta$

Jonathan Beardsley (Mar 25 2020 at 19:04):

Basically, you can "contract a branch" and you can "insert a degenerate node" into a branch.

Jonathan Beardsley (Mar 25 2020 at 19:06):

(the only place I know that this stuff is written down is in Moerdijk's stuff on dendroidal sets)

Jonathan Beardsley (Mar 25 2020 at 19:07):

And so if we call this category $\Omega$ then an operad is a presheaf on $\Omega$ satisfying certain conditions.

Joe Moeller (Mar 25 2020 at 19:08):

Reading along as I make a salad :salad:

Jonathan Beardsley (Mar 25 2020 at 19:08):

Haha, I think you're also reading a (word) salad.

Jonathan Beardsley (Mar 25 2020 at 19:08):

Are you familiar with this way of describing operads?

Jonathan Beardsley (Mar 25 2020 at 19:09):

Like, basically, your presheaf takes the one-vertex tree with n leaves, call it $T_n$, to your set of n-ary operations.

Joe Moeller (Mar 25 2020 at 19:09):

Not exactly, but I'm liking it so far.

Jonathan Beardsley (Mar 25 2020 at 19:10):

But then you have to know what to do with all the other trees, and the maps between them.

Jonathan Beardsley (Mar 25 2020 at 19:10):

But so let's take a simple example.

Jonathan Beardsley (Mar 25 2020 at 19:11):

Let's say I've got the binary tree Y, and then I've got another tree which is the binary tree Y with its root stuck into its top right leaf

Jonathan Beardsley (Mar 25 2020 at 19:11):

I guess I should say $Y=T_2$

Jonathan Beardsley (Mar 25 2020 at 19:12):

And so then I have another tree which looks like $T_2$ plugged in to itself. Let's call it $T_2\circ_2 T_2$ maybe for obvious reasons.

Jonathan Beardsley (Mar 25 2020 at 19:13):

Well it's hard to say without drawing a picture, but by "contracting" the branch on $T_2\circ_2 T_2$ which was originally the root of the upper $T_2$, we can create $T_3$

Joe Moeller (Mar 25 2020 at 19:14):

This all makes sense. I think this formalizes the basic way I describe operads to people pictorially.

Jonathan Beardsley (Mar 25 2020 at 19:14):

So, in $\Omega$ there's this morphism $T_3\to T_2\circ_2 T_2$, which when we take a functor out of $\Omega^{op}$ turns into "composition."

Jonathan Beardsley (Mar 25 2020 at 19:15):

I.e. the set of operations of "shape" $T_2\circ T_2$ needs to come equipped with some kind of map to the set of operations of shape $T_3$

Jonathan Beardsley (Mar 25 2020 at 19:19):

Sorry, my daughter just came in and started pushing buttons on my computer.

Jonathan Beardsley (Mar 25 2020 at 19:20):

So right, maybe you can see how the above goes. There are "face" and "degeneracy" maps in $\Omega$ and then you can talk about a "dendroidal set" or "dendroidal object" wherever you like.

Jonathan Beardsley (Mar 25 2020 at 19:21):

(it turns out that $\Omega$ is a generalized Reedy category, so functors out of it into, e.g., Quillen model categories, have nice properties)

Jonathan Beardsley (Mar 25 2020 at 19:22):

Okay so anyway, the EXTREMELY brief description of properads is: do all of that again, but with graphs instead of trees.

Jonathan Beardsley (Mar 25 2020 at 19:22):

Now, you have to be careful about what you mean by "graphs" of course, just as we had to be careful about what we meant by "trees."

Joe Moeller (Mar 25 2020 at 19:22):

So the way that it extends $\Delta$ is that you can think of an finite ordinal as a line graph?

Jonathan Beardsley (Mar 25 2020 at 19:22):

Ah right but we have to be careful I think.

Jonathan Beardsley (Mar 25 2020 at 19:23):

Let me see here.

Jonathan Beardsley (Mar 25 2020 at 19:23):

There's some subtlety here that I usually mess up.

Jonathan Beardsley (Mar 25 2020 at 19:24):

So in this framework, the "branches" are the "colors" I think, for instance if you wanted to do a colored operad.

Jonathan Beardsley (Mar 25 2020 at 19:25):

And the vertices of the trees are the "operations"

Jonathan Beardsley (Mar 25 2020 at 19:27):

So I think, for instance, we want to think of, in $\Delta$, the object $[1]$ as the "tree" that looks like $-\cdot-$

Jonathan Beardsley (Mar 25 2020 at 19:27):

And $[0]$ as the "tree" that looks like $-$

Jonathan Beardsley (Mar 25 2020 at 19:28):

Now there are two injections of $[0]$ into $[1]$ which become "face maps" in simplicial sets

Jonathan Beardsley (Mar 25 2020 at 19:29):

And in the context of simplicial sets we think of them as "projecting down onto a face of the 1-simplex," i.e. telling us which of all the 0-simplices in a simplicial set "belong" to that particular 1-simplex.

Jonathan Beardsley (Mar 25 2020 at 19:31):

But here, we want to think of the morphism (in $\Omega^{op}$) as being something like "the two ways to contract a branch to get $-$ from $-\cdot -$, if I'm recalling correctly.

Jonathan Beardsley (Mar 25 2020 at 19:31):

So there are sort of... ALMOST dual geometric interpretations

Jonathan Beardsley (Mar 25 2020 at 19:31):

As a tree, the face maps are contracting branches, as a simplicial set, they're projecting to a face.

Jonathan Beardsley (Mar 25 2020 at 19:34):

But so anyway, you can see in $\Omega$ that there's something special about the trees $T_n$.

Jonathan Beardsley (Mar 25 2020 at 19:35):

And for something to actually be an operad, you need it to satisfy a sort of "Segal condition." This corresponds to the fact that a functor out of $\Delta^{op}$ satisfying the Segal condition is a monoid. I.e. it has a reasonable notion of "composition."

Jonathan Beardsley (Mar 25 2020 at 19:36):

In that case, the Segal condition says that the image of $[n]$ needs to be the n-fold Cartesian product of the image of $[1]$.

Jonathan Beardsley (Mar 25 2020 at 19:37):

And then magically the face and degeneracy maps turn into the structure maps of a monoid.

Jonathan Beardsley (Mar 25 2020 at 19:39):

(this is, of course, assuming that $[0]=\{\ast\}$, without which you just get a category!)

Jonathan Beardsley (Mar 25 2020 at 19:40):

So yeah, maybe a better way to say this is that with the Segal condition, a simplicial set is a category.

Jonathan Beardsley (Mar 25 2020 at 19:40):

(and a Simplicial space is an $\infty$-category)

Jonathan Beardsley (Mar 25 2020 at 19:40):

(and a category with one object is a monoid)

Jonathan Beardsley (Mar 25 2020 at 19:41):

But so now, okay, in that case, the object $[0]$ goes to the "set of objects of the category."

Jonathan Beardsley (Mar 25 2020 at 19:41):

In the case that we're thinking of $[0]$ as being in $\Omega$, we want $[0]$ to go to the "set of colors of the operad."

Jonathan Beardsley (Mar 25 2020 at 19:41):

(or equivalently, the set of objects of the multicategory)

Jonathan Beardsley (Mar 25 2020 at 19:42):

And then you have a whole bunch of trees that can, in many different ways, "contract their branches" to end up just giving you a single branch, i.e. a single "color."

Jonathan Beardsley (Mar 25 2020 at 19:43):

So given a tree, there are a whole bunch of "face maps" that tell me what its output and input colors are.

Jonathan Beardsley (Mar 25 2020 at 19:44):

And basically what the Segal condition is doing here is saying that the image of some arbitrary tree needs to be able to be constructed by "gluing together" the basic trees $T_n$.

Jonathan Beardsley (Mar 25 2020 at 19:45):

And so then in the properad situation, we replace the trees $T_n$ with the "corollas" $C_{n,m}$, each of which looks like a single vertex with n inputs and m outputs.

Jonathan Beardsley (Mar 25 2020 at 19:46):

And these have the obvious "edge contraction" and "vertex insertion" maps.

philip hackney (Mar 25 2020 at 19:46):

I know what a properad is

Jonathan Beardsley (Mar 25 2020 at 19:47):

But, again, the functor out of our category of graphs has to take $C_{n,m}$ to the "set of operations with n inputs and m outputs" and then we have all kinds of "composition" things to do.

Jonathan Beardsley (Mar 25 2020 at 19:47):

Well yeah @philip hackney is a FAR better resource for this stuff than I am.

Jonathan Beardsley (Mar 25 2020 at 19:47):

He literally wrote a book on properads.

Jonathan Beardsley (Mar 25 2020 at 19:48):

And all I'm really doing is regurgitating all the reading I did three years ago in preparation for going to Australia to talk to him and Marcy Robertson.

philip hackney (Mar 25 2020 at 19:49):

haha I didn't jump in just to make jb feel self-conscious

Jonathan Beardsley (Mar 25 2020 at 19:49):

It's okay. Also I have no idea if anyone is even reading anymore.

Jonathan Beardsley (Mar 25 2020 at 19:49):

And my "work time" is almost up now anyway. Gotta go be a dad. I was "supposed" to be grading a quiz. Whoops.

philip hackney (Mar 25 2020 at 19:50):

I'll try to pop back in later on today to see if @Joe Moeller came back

Ben Steffan (Mar 25 2020 at 19:50):

I'm certainly still reading :)

philip hackney (Mar 25 2020 at 19:51):

Oh! OKay well I'll come back a bit later today when I have more time

Nathanael Arkor (Mar 25 2020 at 19:51):

I'm just butting into the conversation here, but can a properad be described as a special kind of dioperad, or are they not related like that?

Jonathan Beardsley (Mar 25 2020 at 19:51):

@Nathanael Arkor dioperads are a special kind of properad

Nathanael Arkor (Mar 25 2020 at 19:52):

oh, is there a specific name for a coloured properad, analogous to a polycateory for a dioperad?

Jonathan Beardsley (Mar 25 2020 at 19:53):

where, if I recall correctly, you can only "compose by attaching two edges at a time"

Ben Steffan (Mar 25 2020 at 19:53):

@philip hackney Ah, please don't inconvenience yourself for my sake. I'm happy sucking up whatever knowledge falls off between you, Jonathan and Joe.

Jonathan Beardsley (Mar 25 2020 at 19:53):

in a properad, you can attach any collection of outputs to any collection of inputs

Jonathan Beardsley (Mar 25 2020 at 19:53):

also, properads are generally always "colored"

Jonathan Beardsley (Mar 25 2020 at 19:54):

@philip hackney is there a special name for a properad with one color?

Jonathan Beardsley (Mar 25 2020 at 19:54):

so you've got "operad = 1-colored multicategory" and "dioperad = 1-colored polycategory" and then "properad = properad"

Jonathan Beardsley (Mar 25 2020 at 19:55):

I think.

Nathanael Arkor (Mar 25 2020 at 19:55):

where, if I recall correctly, you can only "compose by attaching two edges at a time"

oh, is this analogous to the situation where you can present operads in either a "partial" or "full" style?

Jonathan Beardsley (Mar 25 2020 at 19:56):

I'm not sure.

Jonathan Beardsley (Mar 25 2020 at 19:56):

I think there are properads that are NOT dioperads.

Jonathan Beardsley (Mar 25 2020 at 19:57):

Like, basically you can attach one edge at a time in a dioperad, but I sort of think there are properads with composition operations that cannot be built up in this way

Nathanael Arkor (Mar 25 2020 at 19:57):

"properad = properad"

ah, it's sad if the naming convention isn't "-ad" = one-coloured

Nathanael Arkor (Mar 25 2020 at 19:57):

sort of think there are properads with composition operations that cannot be built up in this way

oh, I'd be very interested to see an example

Nathanael Arkor (Mar 25 2020 at 19:57):

I was under the impression that the partial and full styles of operads were equivalent, at least

Jonathan Beardsley (Mar 25 2020 at 19:57):

Yeah, I'm not 100% sure off the top of my head. Again, @philip hackney really would be the guy to ask here.

Nathanael Arkor (Mar 25 2020 at 19:58):

(but that may not be the analogous situation anyway)

Joe Moeller (Mar 25 2020 at 20:21):

The Segal condition for $\Delta^{op}$ is what I normally call "$\Delta^{op}$ is the Lawvere theory for monoids."

Joe Moeller (Mar 25 2020 at 20:24):

I finished my salad, so now I'm back.

Joe Moeller (Mar 25 2020 at 20:28):

So I'm sorta confused what the difference between a properad and a prop is.

Joe Moeller (Mar 25 2020 at 20:37):

I think this distinction is especially important for me, because I actually construct my operads from symmetric monoidal categories, which happen to be props/colored props.

Nathanael Arkor (Mar 25 2020 at 20:37):

I've found conflicting information about the relationship: on the nLab it says that PROPs are more general than properads, but in the Handbook of Algebra, it says that every PROP is a properad

John Baez (Mar 25 2020 at 20:40):

You gotta be careful about this "more general" business, because for example every Lawvere theory gives a prop, and every prop gives a Lawvere theory, but it's just an adjunction.

Nathanael Arkor (Mar 25 2020 at 20:40):

what's the nonrepresentable version of a PROP?

John Baez (Mar 25 2020 at 20:41):

I don't know. Maybe a "strict symmetric polycategory"???

John Baez (Mar 25 2020 at 20:42):

I don't know if people have even defined symmetric polycategories yet, but they will someday if not today.

Nathanael Arkor (Mar 25 2020 at 20:42):

polycategories only allow composition along single objects at a time, whereas we need something that allows multiple composition

Nathanael Arkor (Mar 25 2020 at 20:42):

Richard Garner defines symmetric polycategories in https://arxiv.org/abs/math/0606735

John Baez (Mar 25 2020 at 20:44):

"polycategories only allow composition along single objects at a time, whereas we need something that allows multiple composition" - oh, okay.

John Baez (Mar 25 2020 at 20:44):

How about a "properad", then? I'm trying to grok the elegant definition here:

https://ncatlab.org/nlab/show/properad

Nathanael Arkor (Mar 25 2020 at 20:45):

yes, we've been struggling to decipher what it is :big_smile:

Joe Moeller (Mar 25 2020 at 20:45):

Phil will straighten us out as soon as he returns. I wonder how far we can get before then though.

Joachim Kock (Mar 25 2020 at 20:51):

There is an alternative description of properads as algebras for a monad on the category of presheaves on elementary graphs. It's a Segal-condition-style nerve theorem just like the ones quoted above for operads.

John Baez (Mar 25 2020 at 20:53):

Can you just tell me what a properad is in one sentence as if I were a five-year-old? :slight_smile:

John Baez (Mar 25 2020 at 20:54):

It's a bunch of things with inputs and outputs, where you can glue together all (? some?) of the outputs of one thing with the inputs of the next.... something like that.

Joachim Kock (Mar 25 2020 at 20:55):

It's an algebraic rule that allow you to contract acyclic connected graph configurations of operations to a single operation. Operations are many-in/many-out. Sorry, that was two sentences. Argh now we up to four sentences :-(

Nathanael Arkor (Mar 25 2020 at 20:57):

how does this differ from how you would describe a PROP in the same language?

John Baez (Mar 25 2020 at 20:57):

Okay, I get it now. So if I have an operation A with 2 inputs and 2 outputs, and an operation B with 2 inputs and 2 outputs, I cannot, in a properad, compose them by attaching both outputs of A to inputs of B. Right, @Joachim Kock?

Joachim Kock (Mar 25 2020 at 20:57):

In a prop, you are also allowed to contract nonconnected (acyclic, directed) graphs.

John Baez (Mar 25 2020 at 20:58):

....

Jonathan Beardsley (Mar 25 2020 at 20:58):

Also.... Wheeled properads?

Joachim Kock (Mar 25 2020 at 20:58):

How can I reply to a specific message?

Nathanael Arkor (Mar 25 2020 at 20:58):

@Joachim Kock: there's a drop-down menu if you hover over the right side of a message — and there's an option "Quote and reply"

John Baez (Mar 25 2020 at 20:58):

The best you can do is click on it, at right, and quote it.

John Baez (Mar 25 2020 at 20:59):

"Quote and reply".

John Baez (Mar 25 2020 at 20:59):

I think in some ways we are still behind the technology of "usenet" discussions back in 1989, where we could have arbitrary discussion trees.

Joachim Kock (Mar 25 2020 at 21:01):

John Baez said:

Okay, I get it now. So if I have an operation A with 2 inputs and 2 outputs, and an operation B with 2 inputs and 2 outputs, I cannot, in a properad, compose them by attaching both outputs of A to inputs of B. Right, Joachim Kock?

Yes, you are allowed to connect those pairs of wires, because even if it's slightly loopy, it's not a directed loop. By acyclic, I meant 'no directed cycles'.

John Baez (Mar 25 2020 at 21:02):

Oh, wow. Hmm.

Joe Moeller (Mar 25 2020 at 21:02):

Oh, and props do let you do that. Right?

Joe Moeller (Mar 25 2020 at 21:02):

Just as you would with any symmetric monoidal category.

John Baez (Mar 25 2020 at 21:02):

So Joachim's "yes" means "you're wrong, Baez, you can do that".

Joe Moeller (Mar 25 2020 at 21:03):

oh whoops. So what's the difference then?

Joachim Kock (Mar 25 2020 at 21:03):

I feel young with all this technology.

John Baez (Mar 25 2020 at 21:03):

I think someone needs to make a chart of all $2^{20}$ different variations on these ideas.

Joachim Kock (Mar 25 2020 at 21:04):

In a prop, you can take for example the disjoint union of seven (1,1) operations and contract the whole thing to a (7,7)-corolla.

Nathanael Arkor (Mar 25 2020 at 21:04):

ahh

Joe Moeller (Mar 25 2020 at 21:05):

And this is just "tensoring the morphisms", right?

Nathanael Arkor (Mar 25 2020 at 21:05):

so in a properad, you just have the notion of composition; but in a prop, you also have a notion of concatenation (given by functoriality of the tensor product)

Joachim Kock (Mar 25 2020 at 21:05):

There is a nice survey by Markl.

Joachim Kock (Mar 25 2020 at 21:05):

Joe Moeller said:

And this is just "tensoring the morphisms", right?

Right.

John Baez (Mar 25 2020 at 21:06):

A prop is just a strict symmetric monoidal category with $\mathbb{N}$ as objects, so there's no mystery about them.

Joe Moeller (Mar 25 2020 at 21:06):

Oh, this is making sense now. Right, this is precisely what you don't have in operads too. Should've known. :upside_down:

Nathanael Arkor (Mar 25 2020 at 21:06):

so is a (coloured) dioperad the same as a (coloured) properad?

Nathanael Arkor (Mar 25 2020 at 21:06):

or is one the "partial composition" version of the other?

Nathanael Arkor (Mar 25 2020 at 21:06):

or something else entirely?

John Baez (Mar 25 2020 at 21:07):

Dioperads are definitely not the same as properads!

Nathanael Arkor (Mar 25 2020 at 21:08):

I want to establish how polycategories are different from coloured properads

Nathanael Arkor (Mar 25 2020 at 21:08):

because they seem very similar to me at the moment

Joachim Kock (Mar 25 2020 at 21:09):

(Actually I what I described should be called a graphical prop. It is not precisely the same thing as a prop in the symmetric-monoidal-category sense of Mac Lane. The difference was spotted only recently by Michael Batanin: the difference is that in a Mac Lane prop, the (0,0) operations form a
commutative monoid, by the Eckmann-Hilton argument. In a graphical prop it can be any monoid.)

Joachim Kock (Mar 25 2020 at 21:10):

In a dioperad you are only allowed to contract contractible graphs. That is, John's example is not allowed. I think this is the same thing as polycategory. (For me everything is always coloured.)

John Baez (Mar 25 2020 at 21:11):

Nathanael said that polycategories only let you attach one output to one input; Joachim just said that colored properads let you attach multiple outputs to multiple inputs. So, they can't be the same.

Nathanael Arkor (Mar 25 2020 at 21:12):

okay, so if this is the case, then dioperads are to properads what operads with partial composition are to operads with multiple composition

Nathanael Arkor (Mar 25 2020 at 21:12):

I was under the impression that operads with partial and operads with multiple composition were equivalent

John Baez (Mar 25 2020 at 21:13):

Yes, they are.

Nathanael Arkor (Mar 25 2020 at 21:13):

and so I would have imagined dioperads and properads to be similarly equivalent?

John Baez (Mar 25 2020 at 21:13):

Joachim explained how they're not, in response to my question.

Joachim Kock (Mar 25 2020 at 21:13):

Joachim Kock said:

contract contractible graphs

OK, that sounded funny. By contractible graph I meant simply-connected geometric realisation.

Joachim Kock (Mar 25 2020 at 21:14):

So now I have also been able to reply to myself. Now I can continue the conversation even when all you other go to bed.

Nathanael Arkor (Mar 25 2020 at 21:15):

Joachim explained how they're not, in response to my question.

ah, of course!

Nathanael Arkor (Mar 25 2020 at 21:16):

everything feels like it's fitting into place now — thank you for explaining, everyone!

Nathanael Arkor (Mar 25 2020 at 21:18):

to me, coloured properads seem more deserving of the name "polycategory" than coloured dioperads

Joachim Kock (Mar 25 2020 at 21:23):

Wheeled properads? You just allow yourself directed cycles (but still connected). Sounds easy, but that's really a can of worms :-( because of the nodeless loop. Philip and his friends wrote a whole book about it...

Joachim Kock (Mar 25 2020 at 21:24):

Sorry, that was not a proper attribution (I just mentioned Philip because he was around in this chat). I meant: Hackney-Robertson-Yau.

philip hackney (Mar 25 2020 at 23:14):

Yeah I'm the only one of us around in the chat. :-)

philip hackney (Mar 25 2020 at 23:18):

But since someone else who has written about properads appeared in the chat while I was gone, I'm not sure there's anything I should add.

Jonathan Beardsley (Mar 26 2020 at 00:46):

@Nathanael Arkor did you feel like you got a good understanding of why there where properads that could not be dioperads?

Jonathan Beardsley (Mar 26 2020 at 00:47):

I wish someone would make a like... hierarchy

Nathanael Arkor (Mar 26 2020 at 00:49):

yes, @John Baez gave an example of a properad that was not a polycategory here: https://categorytheory.zulipchat.com/#narrow/stream/229199-basic-questions/topic/Properads/near/191809947

Nathanael Arkor (Mar 26 2020 at 00:50):

I think the hierarchy goes like this: coloured PROPs > coloured properads > polycategories > multicompositional multicategories ~ partial-compositional multicategories > categories

Nathanael Arkor (Mar 26 2020 at 00:50):

where > means "generalises" and ~ means "is equivalent to"

Nathanael Arkor (Mar 26 2020 at 00:53):

(and then wheeled properads sound even more general than properads)

Jonathan Beardsley (Mar 26 2020 at 00:56):

Right I'm actually wondering if wheeled properads are PROPS or not

Jonathan Beardsley (Mar 26 2020 at 00:57):

And then you've got modular (pr)operads and cyclic (pr)operads haha

Jonathan Beardsley (Mar 26 2020 at 00:57):

And then you can do everything but with ∞ in front of it

Nathanael Arkor (Mar 26 2020 at 00:57):

it's easy to see how categorifying everything will affect the hierarchy

Nathanael Arkor (Mar 26 2020 at 00:58):

I would be surprised if wheeled properads could be expressed as PROPs

Nathanael Arkor (Mar 26 2020 at 00:58):

I would imagine they're two different generalisations

Jonathan Beardsley (Mar 26 2020 at 00:58):

Yeah I'm getting that feeling as well

Nathanael Arkor (Mar 26 2020 at 00:58):

but I don't yet have an intuition for wheeled properads

Jonathan Beardsley (Mar 26 2020 at 00:59):

Maybe it's worth pointing out some examples of properads: bialgebras, Lie bialgebras are two that I can think of off the top my head.

Jonathan Beardsley (Mar 26 2020 at 00:59):

Maybe there's some notion of, like, a Poisson bialgebra or something

Nathanael Arkor (Mar 26 2020 at 00:59):

cyclic operads also seem like a different generalisation: you no longer have a strict divide between input and output

Nathanael Arkor (Mar 26 2020 at 01:00):

so they don't seem like operads or dioperads

Jonathan Beardsley (Mar 26 2020 at 01:00):

Oh yeah cyclic and modular are going in a different direction completely

Nathanael Arkor (Mar 26 2020 at 01:00):

(personally, I think many of these names are very misleading)

Jonathan Beardsley (Mar 26 2020 at 01:00):

I STILL don't really know wtf is going on with modular operads

Nathanael Arkor (Mar 26 2020 at 01:01):

I hadn't come across modular operads before

Jonathan Beardsley (Mar 26 2020 at 01:01):

I've tried to read the Getzler paper introducing them a couple of times, never seriously, but I've never been able to grok it

Jonathan Beardsley (Mar 26 2020 at 01:02):

Oh and of course coöperads are an example of properads

Nathanael Arkor (Mar 26 2020 at 01:04):

I like your use of the diaeresis :grinning_face_with_smiling_eyes:

Nathanael Arkor (Mar 26 2020 at 01:04):

it would be really nice if all these examples could be captured in some general framework, like Cruttwell–Shulman's approach to multicategories

Jonathan Beardsley (Mar 26 2020 at 01:05):

maybe. i suspect that would be a rather thankless task.

Jonathan Beardsley (Mar 26 2020 at 01:06):

there's also the possibility of messing around with the group acting on the inputs. so you've got non-symmetric operads and symmetric operads, but you can also have _braided_ operads, where the symmetries of the inputs are coming from the braid group

Nathanael Arkor (Mar 26 2020 at 01:11):

there's certainly an interesting question of what the right generalisation of input and output is

Nathanael Arkor (Mar 26 2020 at 01:12):

https://arxiv.org/pdf/0907.2460.pdf captures braided operads already

Nathanael Arkor (Mar 26 2020 at 01:13):

(and really anything that is "some structured input" and a single output)

Nathanael Arkor (Mar 26 2020 at 01:20):

a good person to ask about this could be Mike Shulman, who appears to be in the Zulip server

Jonathan Beardsley (Mar 26 2020 at 01:25):

whoa mike's here, cool! also... whoa there are a LOT of people here!

Jonathan Beardsley (Mar 26 2020 at 01:26):

It's a little weird stepping outside of the homotopy theory bubble and all of the sudden seeing all these people whose names I don't recognize.

John Baez (Mar 26 2020 at 01:48):

I STILL don't really know wtf is going on with modular operads

For those you have to be somewhat comfortable with Riemann surfaces (or maybe algebraic curves). Are you? I forget.

Jonathan Beardsley (Mar 26 2020 at 01:50):

Eh, somewhat. I've spent a little time with the Grothendieck-T*****r group.

Jonathan Beardsley (Mar 26 2020 at 01:50):

I just think about spheres and tori and stuff.

Jonathan Beardsley (Mar 26 2020 at 01:50):

And that seems to get me close enough.

Jonathan Beardsley (Mar 26 2020 at 01:52):

But that does jar my memory a little bit I guess. You've got marked Riemann surfaces and you can like, glue them together along markings, or delete a marking, or something?

John Baez (Mar 26 2020 at 02:08):

Yes. I'm not an expert on modular operads by any means, but looking over the Getzler-Kapranov paper again I see this:

The concept of modular operad is described purely combinatorially: it's a cyclic operad with extra structure. But this extra structure reflects things you can do with Riemann surface with marked points. You can think of a Riemann surface with n+1 labelled marked points as an n-ary operation. You can compose these as usual for operations in an operad, or even a cyclic operad. But you can also take two marked points and glue your Riemann surface to itself by identifying these points. This idea is incorporated into the definition of modular operad: for example, I guess you get ways to take an n-ary operation and produce an (n-2)-ary operation.

Joe Moeller (Mar 26 2020 at 02:13):

Are the marked points supposed to be like a divisor?

John Baez (Mar 26 2020 at 02:14):

Yes.

John Baez (Mar 26 2020 at 02:15):

The "gluing" business is a bit subtle.

Jonathan Beardsley (Mar 26 2020 at 02:17):

right so "cyclic" here means that you can permute the inputs

Jonathan Beardsley (Mar 26 2020 at 02:17):

(or the "labels" on the "input" markings)

Jonathan Beardsley (Mar 26 2020 at 02:18):

oh no sorry

Jonathan Beardsley (Mar 26 2020 at 02:18):

cyclic means you can rotate an input to an output

Jonathan Beardsley (Mar 26 2020 at 02:18):

gah it's too late i need to go to bed

John Baez (Mar 26 2020 at 02:19):

No prob. Yes, cyclic means you can rotate an input to be an output.

John Baez (Mar 26 2020 at 02:20):

There's no fundamental distinction between the n inputs and the 1 output of an operation in a cyclic operad.

Joe Moeller (Mar 26 2020 at 02:26):

On top of all the structures we've been discussing, I want to understand how they relate to hypergraph categories, because they're giving me the same vibes.

Jonathan Beardsley (Mar 26 2020 at 02:27):

I also want to know about the graph complex stuff that comes up w/r/t GT and all that stuff.

Joe Moeller (Mar 26 2020 at 02:29):

What's GT?

John Baez (Mar 26 2020 at 02:30):

Grothendieck-Teichmuller.

John Baez (Mar 26 2020 at 02:30):

There's a book about operads and that stuff by Benoit Fresse.

Jonathan Beardsley (Mar 26 2020 at 02:31):

Yeah, an enormous book.

Jonathan Beardsley (Mar 26 2020 at 02:33):

@Joe Moeller the Grothendieck-T* group (I don't like writing his name b/c he's an actual Nazi) is related to Riemann surfaces and all that jazz, but it also happens to be the homotopy automorphisms of the little 2-discs operad (this isn't SUPER surprising once you learn what GT is), but there's a lot of stuff going on there

John Baez (Mar 26 2020 at 02:41):

I would be happy to talk about something like this GT stuff. I'd have to read stuff to get up to speed and remember what I once knew.

philip hackney (Mar 26 2020 at 03:04):

Jonathan Beardsley said:

I wish someone would make a like... hierarchy

Here's such a thing from the preface of "A Foundation for PROPs, Algebras, and Modules" (but only dealing with "directed" contexts)
Screen-Shot-2020-03-25-at-8.01.56-PM.png

Joe Moeller (Mar 26 2020 at 03:19):

So this idea of having distinguished points in a surface reminds me of modular tensor categories. Is it the same modular as modular operads?

John Baez (Mar 26 2020 at 03:20):

Yes. "Modular" just means "like stuff you can do with Riemann surfaces".

John Baez (Mar 26 2020 at 03:21):

That is, in this context that's what it means. The "modular group" and "modular curves" use the word in a more specific way... but they too are connected to Riemann surfaces.

Joe Moeller (Mar 26 2020 at 03:23):

Great. So what does the operad look like here?

John Baez (Mar 26 2020 at 05:24):

Which operad? As I said, there's not just one "modular operad": a modular operad is a kind of operad, whose properties I sketched.

Joe Moeller (Mar 26 2020 at 05:26):

I meant the one you get for a fixed surface. How do you glue the surface to itself? I'm confused what exactly the operations are.

John Baez (Mar 26 2020 at 05:26):

I didn't say a fixed surface gives an operad.

John Baez (Mar 26 2020 at 05:28):

The concept of modular operad is described purely combinatorially: it's a cyclic operad with extra structure. But this extra structure reflects things you can do with Riemann surfaces with marked points. You can think of a Riemann surface with n+1 labelled marked points as an n-ary operation. You can compose these as usual for operations in an operad, or even a cyclic operad. But you can also take two marked points and glue your Riemann surface to itself by identifying these points. This idea is incorporated into the definition of modular operad: for example, I guess you get ways to take an n-ary operation and produce an (n-2)-ary operation.

John Baez (Mar 26 2020 at 05:29):

I guess it wasn't clear from this that a modular operad is a kind of operad: a cyclic operad with extra structure that's a bit hard for me to explain because I don't understand it very well.

Joe Moeller (Mar 26 2020 at 05:29):

Right, so I read this and I'm confused by it.

Joe Moeller (Mar 26 2020 at 05:30):

I was sorta guessing that a surface gives an example of one of these operads, where the operations are the n+1 points.

John Baez (Mar 26 2020 at 05:30):

No.

John Baez (Mar 26 2020 at 05:30):

In an ordinary operad you can take a bunch of operations, visualize them as trees, and do things with them that you'd do with trees.

Joe Moeller (Mar 26 2020 at 05:31):

So what is a Riemann surface doing here?

John Baez (Mar 26 2020 at 05:31):

I'm saying that in a modular operad, which is a kind of operad, you can take a bunch of operations, visualize them as Riemann surfaces with marked points, and do thigns with them that you'd do with Riemann surfaces with marked points.

John Baez (Mar 26 2020 at 05:31):

So the analogy is "trees are to operads as Riemann surfaces are to modular operads".

John Baez (Mar 26 2020 at 05:32):

You don't get an operad from a specific tree, and you don't get a modular operad from a specific Riemann surface with marked points.

Joe Moeller (Mar 26 2020 at 05:33):

Right, I was misunderstanding what you were saying before.

Joe Moeller (Mar 26 2020 at 05:34):

Great. What's one specific one?

John Baez (Mar 26 2020 at 05:35):

I deleted some stuff that I said that was wrong.

John Baez (Mar 26 2020 at 05:35):

Let me try to give an example of a modular operad.

John Baez (Mar 26 2020 at 05:42):

I'm looking at the paper by Getzler and Kapranov.

John Baez (Mar 26 2020 at 05:43):

This stuff is pretty complicated....

John Baez (Mar 26 2020 at 05:43):

I don't think I can get it right and also be comprehensible - it'll be hard to even get things right!

John Baez (Mar 26 2020 at 05:44):

But the most important idea in this general vicinity is the moduli space of Riemann surfaces with genus g and n marked points.

John Baez (Mar 26 2020 at 05:44):

There are lots of Riemann surfaces with genus g and n marked points (i.e. points labelled 1, 2, ... n).

John Baez (Mar 26 2020 at 05:45):

The set of isomorphism class of these things, roughly, forms a space called M(g,n).

John Baez (Mar 26 2020 at 05:46):

It's incredibly important and people have spend a lot of time trying to understand its topology (it's a topological space) and geometry (it's something like an algebraic variety).

John Baez (Mar 26 2020 at 05:46):

It's a fundamental player in string theory, since you can think of it as the "set of strings of genus g with n inputs/outputs".

John Baez (Mar 26 2020 at 05:47):

Roughly speaking, there's a modular operad whose set of (n-1)-ary operations is the union of all the M(g,n)'s over all g.

John Baez (Mar 26 2020 at 05:47):

This is probably the one to understand.

Joe Moeller (Mar 26 2020 at 05:54):

One operation is some surface with the right number of distinguished points. How does the tree analogy come into play here?

John Baez (Mar 26 2020 at 05:58):

A tree has a certain number of leaves and a root. These are the points where you're allowed to attach one tree to another!

John Baez (Mar 26 2020 at 05:58):

In an ordinary operad you can only attach an output of one operation (root) to an input of another (leaf).

John Baez (Mar 26 2020 at 05:59):

In a cyclic operad, operations should instead be visualized as unrooted trees!

John Baez (Mar 26 2020 at 05:59):

They are just trees in the sense of graph theory, with a bunch of leaves.

John Baez (Mar 26 2020 at 06:00):

In a cyclic operad there's no real distinction between "inputs" and "output" - an operation just has a bunch of "puts". (Nobody calls them that.)

John Baez (Mar 26 2020 at 06:00):

You can attach a "put" of one operation to a "put" of another operation.

John Baez (Mar 26 2020 at 06:01):

A modular operad is more like that... as I mentioned, it's a cyclic operad with extra structure.

Joe Moeller (Mar 26 2020 at 07:01):

So are you supposed to thing of the unrooted tree as embedded in a Riemann surface?

Joe Moeller (Mar 26 2020 at 07:02):

And the leaves are the distinguished points?

Joachim Kock (Mar 26 2020 at 08:46):

Nathanael Arkor said:

it would be really nice if all these examples could be captured in some general framework, like Cruttwell–Shulman's approach to multicategories

Try the operadic categories of Batanin and Markl.

Joachim Kock (Mar 26 2020 at 08:58):

Nathanael Arkor said:

but I don't yet have an intuition for wheeled properads

Did you read this tweet: https://categorytheory.zulipchat.com/#narrow/stream/229199-basic-questions/topic/Properads/near/191812919
I can expand on it if necessary.

Joachim Kock (Mar 26 2020 at 09:06):

Regarding modular operads, the one of marked Riemann surfaces is of course the most famous one, because Riemann surfaces are so famous anyway. But just to grasp the notion, I would stick to graphs, saying that a modular operad is a structure where the operations indexed by corollas (but without distinction between in and out), and where you can contract any (connected) graph of operations to a single operation.

Joachim Kock (Mar 26 2020 at 09:13):

If you want to talk about Riemann surfaces, you have to realise that they are allowed to have singular points. It is cleaner to deal with this in the setting of stable algebraic curves in the sense of Deligne and Mumford: these are algebraic curve allowed to have ordinary double points (but not cusps or worse singularities). (I ignore the actual stability condition, a technical condition to limit the number of automorphisms.)
They also have marked points.

Joachim Kock (Mar 26 2020 at 09:19):

Given a stable curve, consider the dual graph: for each irreducible component, draw a node, and for each double point, draw an edge between the two irreducible components meeting at that point. Since a double point could be that of a component intersecting itself (like in an irreducible nodal plane cubic y^2=x^2(x-1)), the graph can have loops. Finally for each marked point, draw a dangling edge from the node corresponding to the irreducible component. Altogether you get a (non-oriented) graph with open-ended edges allowed. (If you really want to follow the original Getzler-Kapranov definition of modular operad, your graphs should also have a genus decoration at each node, namely the genus of the corresponding irreducible component. But nowadays many users have stripped off this data from the definition, and regard it as further decoration you can add if really needed.)

Joachim Kock (Mar 26 2020 at 09:26):

The overview of all these different kinds of operad-like structures can be made in terms of which graph-like configurations of operations you are allowed to contract.
For a category, you contract linear rooted trees to linear rooted trees with only one node.
For an operad, you contract rooted trees to rooted trees with only one node.
(For a nonsymmetric operad, you contract planar rooted trees to planar rooted trees with only one node.)
For properads, you contract connected directed acyclic graphs to ones with only one node.
For wheeled properads, you also allow directed cycles, and contract to directed acyclic graphs without inner edges.
For (graphical) props, you contract (not-necessarily-connected) directed acyclic graphs to directed acyclic graphs with only one node.
For cyclic operads you contract non-rooted trees to non-rooted trees with only one node.
For modular operads, you contract connected graphs to connected graphs without inner edges.

(For wheeled and modular it is not good enough to say "with only one node". A loop with a single node should not count as an operation.)

Joachim Kock (Mar 26 2020 at 09:28):

In each case, the contracted graph must have the same arity as the graph you contract. (This condition essentially tells you that the structures are algebras for a cartesian monad, but for this to be true you need to upgrade from sets to groupoids.)

Joachim Kock (Mar 26 2020 at 09:31):

In each case there is an operadic category whose objects are the graphs in question, and whose morphisms are contractions of inner edges. I am not an expert in this, so please look at the papers of Batanin and Markl for precise statements and details.

Joachim Kock (Mar 26 2020 at 09:38):

I am talking about the coloured version of these notions. This means that it is important to allow the graph which is just an edge without any nodes. This graph indexes the 'colours'. So 'graph-like configuration of operations' means that each node is decorated by an operation, and each edge by a colour -- in a compatible way.

Joachim Kock (Mar 26 2020 at 09:49):

To actually formalise the intuitions listed above, a main task is to make precise the notions of trees and graphs, and have a good formalism for them. (In particular, they should admit open-ended edges. For trees, these are the leaves and the root.) That's a longer story. I think I should not bend the discussion in that direction, although it is one of my favourite topics.

Nicolas Blanco (Mar 26 2020 at 10:42):

I am a little late to the discussion. Are (coloured) properads the same thing as "compact polycategories" defined by Ross Duncan in his PhD thesis (some slides about those: http://www.cs.ox.ac.uk/people/ross.duncan/ckc-21-jul-2006.pdf)?
These are basically polycategories with multicuts (composition along multiple elements).

Nathanael Arkor (Mar 26 2020 at 12:10):

@Joachim Kock: that's a really helpful explanation, thank you!

Nathanael Arkor (Mar 26 2020 at 12:11):

To actually formalise the intuitions listed above, a main task is to make precise the notions of trees and graphs, and have a good formalism for them.

is this to say that this is not what Batanin–Markl do?

Nathanael Arkor (Mar 26 2020 at 12:12):

that indeed sounds like a fascinating topic — I shall look at operadic categories; thank you for the reference!

Nathanael Arkor (Mar 26 2020 at 12:12):

Joachim Kock said:

Nathanael Arkor said:

but I don't yet have an intuition for wheeled properads

Did you read this tweet: https://categorytheory.zulipchat.com/#narrow/stream/229199-basic-questions/topic/Properads/near/191812919
I can expand on it if necessary.

ah, I did read it — but what I meant was that to feel I properly understood what was going on, I would have to think of some examples, as I don't think I've seen such a structure before

Nathanael Arkor (Mar 26 2020 at 12:16):

@Nicolas Blanco: they do sound like the same thing to me

Nathanael Arkor (Mar 26 2020 at 16:24):

operadic categories seem like a nice abstraction, but it's a bit hard for me to tell exactly what kinds of "generalised multicategory" structures they capture — are all generalised multicategories (in either the Leinster or Cruttwell–Shulman sense) examples of operadic categories?

Joachim Kock (Mar 26 2020 at 17:20):

Nathanael Arkor said:

To actually formalise the intuitions listed above, a main task is to make precise the notions of trees and graphs, and have a good formalism for them.

is this to say that this is not what Batanin–Markl do?

I did not say anything in that direction.

Nathanael Arkor (Mar 26 2020 at 17:25):

the graph-focused view seemed particularly espoused by Markl's Operads and PROPs

Joachim Kock (Mar 26 2020 at 17:31):

But maybe I can clarify this: for each of the examples in the list, it can be tricky to get the combinatorics right. This goes for the situation where you just want to develop your own 'ad hoc' theory for the kind of operad in question, and it also goes for the situation where you want to prove that your trees or graphs form an operadic category. In both cases, care and precision is required.
(The theory of operadic categories is abstract. In principle, it could be developed without ever seeing a tree or a graph.)

Nathanael Arkor (Mar 26 2020 at 17:36):

(The theory of operadic categories is abstract. In principle, it could be developed without ever seeing a tree or a graph.)

the definition doesn't seem to indicate of an graph/tree-based intuition, certainly

Nathanael Arkor (Mar 26 2020 at 17:38):

it also goes for the situation where you want to prove that your trees or graphs form an operadic category

would it be reading too much into your comments to take it that there might be a more "direct" way to go from these kinds of descriptions of operations on graphs to the corresponding kinds of operadic categories (or another, similarly-inclined categorical structure)?

Joachim Kock (Mar 26 2020 at 19:38):

Nathanael Arkor
Yes, that would be reading to much into my comments.

Joachim Kock (Mar 26 2020 at 19:39):

The operadic-category axioms are very powerful. It is normal that it takes some work to check them in a given situation, perhaps a bit like Quillen model structures.
I advertised operadic categories because I recommend them.

Nathanael Arkor (Mar 26 2020 at 19:41):

okay, that makes sense, thank you :)

philip hackney (Mar 27 2020 at 00:06):

Nicolas Blanco said:

I am a little late to the discussion. Are (coloured) properads the same thing as "compact polycategories" defined by Ross Duncan in his PhD thesis (some slides about those: http://www.cs.ox.ac.uk/people/ross.duncan/ckc-21-jul-2006.pdf)?
These are basically polycategories with multicuts (composition along multiple elements).

The pictures in these slides certainly seem promising! I hadn't heard of this thesis before, so can't say for sure. But thanks for the pointer

Chaitanya Leena Subramaniam (Mar 27 2020 at 13:10):

Joachim Kock said:

The operadic-category axioms are very powerful. It is normal that it takes some work to check them in a given situation, perhaps a bit like Quillen model structures.
I advertised operadic categories because I recommend them.

Sorry if this is a digression. I've been wondering if the operadic category definition can be modified to allow for finitely presentable presheaves (on some small category) rather than finite sets to be the "cardinalities". @Nathanael Arkor this is an approach I've been thinking about to get dependently coloured operads/dependent multicategories. In particular, one might hope that globular operads become examples. Note that the (opposite) category of globes is a finitely branching inverse category.

Nathanael Arkor (Mar 27 2020 at 13:39):

I had also been wondering if there were operadic categories whose operads were "dependent multicategories" (in whichever particular sense)

Nathanael Arkor (Mar 27 2020 at 13:40):

but I need to become more familiar with the intuition behind operadic categories first

Nathanael Arkor (Mar 27 2020 at 13:40):

are you sure that globular operads are not already examples covered by operadic categories?

Nathanael Arkor (Mar 27 2020 at 13:41):

(the existing papers on operadic categories don't mention enough examples or non-examples to really give a good idea of what's covered and what isn't)

Chaitanya Leena Subramaniam (Mar 27 2020 at 13:52):

Nathanael Arkor said:

are you sure that globular operads are not already examples covered by operadic categories?

I'm not sure but I would be surprised. The arities of operations of globular operads are not finite sets but rather finite globular sets.

Nathanael Arkor (Mar 27 2020 at 14:06):

it's not clear to me that the cardinality functor should be thought of as an arity functor; it reminds me a little of the length function in a C-system

Nathanael Arkor (Mar 27 2020 at 14:07):

as long as the dependency itself is encoded somewhere, a length function can be a reasonable operation, even when it doesn't capture all the information

Nathanael Arkor (Mar 27 2020 at 14:12):

I don't yet understand why operadic categories are the right notion — they seem to capture many existing notions, but the definition itself isn't suggestive to me of the graph-like intuition, for instance

Nathanael Arkor (Mar 27 2020 at 14:15):

(compared to, say, generalised multicategories, the formulation of which feels very intuitive)

Chaitanya Leena Subramaniam (Mar 27 2020 at 14:16):

Nathanael Arkor said:

it's not clear to me that the cardinality functor should be thought of as an arity functor; it reminds me a little of the length function in a C-system

I believe the moral is that objects in an operadic category are contexts that "look like" finite sets (of "variables" if you like), and that the morphisms (context morphisms) have a "descent-like" property in that they can be chopped up into fibres and glued back together. It's easy to understand what this means for usual coloured operads seen as operadic categories. (Look at the example of the operadic category of "bouquets"). It's precisely this "descent-like" property that one would like to understand for morphisms between dependently-typed contexts.

Chaitanya Leena Subramaniam (Mar 27 2020 at 14:22):

I would say that the cardinality functor describes the "shape" of a context.

Nathanael Arkor (Mar 27 2020 at 14:47):

I see what you mean; I think Lack's reformulation makes this slightly clearer, now that I look at it again — but I still don't have enough of an intuition to see precisely what the obstruction is in thinking about a simple dependent operad — I'll try playing around with some examples to get a better understanding

Nathanael Arkor (Mar 27 2020 at 14:47):

if operadic categories are insufficent, then considering finitely presentable presheaves seems like a promising idea to try

Joachim Kock (Mar 27 2020 at 22:51):

Nathanael Arkor said:

I don't yet understand why operadic categories are the right notion — they seem to capture many existing notions, but the definition itself isn't suggestive to me of the graph-like intuition, for instance

To grasp the intuition, I think the best example to look at is the operadic category Delta (the monoidal delta, including the empty ordinal), which is the operadic category for nonsymmetric operads. This means that a nonsymmetric operad (in a symmetric monoidal category V) is a strictly monoidal lax functor Delta -> Sigma V. Here Sigma V is the one-object 2-category with V as Hom cat. If you spell out the conditions, you get precisely the operad axioms. (This is due to Day and Street, and was one of the motivating examples for the notion of operadic category.)

Nathanael Arkor (Mar 28 2020 at 00:02):

@Joachim Kock: yes, I think it would be helpful for me to explicitly work through that example — one can often only get so far in understanding without going through a construction step-by-step

Nathanael Arkor (Mar 28 2020 at 00:03):

that correspondence (Lemma 1.13 in the original paper on operadic categories) seems quite helpful

Nathanael Arkor (Mar 28 2020 at 00:05):

I think there are quite a few steps of understanding to make here — I've only just taken a look at your paper with Batanin and Weber, and the result that Feynman categories, which can describe many operadic notions that intuitively seem much richer than just operads, are biequivalent to coloured operads seems very surprising at first sight

Nathanael Arkor (Mar 28 2020 at 00:07):

also, the note relating regular patterns and substitutes to generalised species is also very interesting, as I feel slightly more familiar with that work, so I can hopefully draw some more connections to things I understand

Nathanael Arkor (Mar 28 2020 at 00:08):

it seems like a delightful area to start exploring