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Stream: theory: category theory

Topic: nominal substitution monoids

Tom Hirschowitz (May 05 2021 at 14:13):

This is a question for nominal specialists out there. From skimming a few papers, it seems that the nominal literature on generating syntax with variable binding tends to characterise syntax as the initial algebra for the relevant endofunctor. On the other hand, the Fiore-Plotkin-Turi approach characterises it as the free monoid algebra, which readily equips it with substitution. It seems plausible that one could proceed similarly in the nominal world, by defining a tensor product looking like $(X\otimes Y)(n) = \int^{p \in \mathrm{Inj}} X(p) \times Y(n)^p$. This is probably only skew monoidal, but the Fiore-Plotkin-Turi characterisation still applies in that case (@Ambroise and I proved it last year). Has anyone seen anything like this, or obstruction to it?

Nathanael Arkor (May 05 2021 at 14:29):

John Power discusses this, and gives relevant several theorems (albeit without proofs), in Abstract Syntax: Substitution and Binders (see Theorems 4.6 and 4.7 in particular). As far as I know, there has been no follow up.

Tom Hirschowitz (May 05 2021 at 15:34):

Aaaah, this looks great, thank you!

Tom Hirschowitz (May 12 2021 at 14:23):

Follow up: I got the tensor product wrong. It should be

$$(X\otimes Y)(n) = \int^{p \in \mathrm{Inj}} X(p) \times Y^{\boxtimes p}(n),$$

where $\boxtimes$ denotes Day convolution, so $Y^{\boxtimes p}(n) = \int^{n_1,\ldots,n_p} Y(n_1) \times \ldots \times Y(n_p) \times \mathrm{Inj}(\sum_i n_i,n)$.

Is it obvious to the expert that it lifts to nominal sets, i.e., that $X\otimes Y$ preserves pullbacks if $X$ and $Y$ do?

Nathanael Arkor (May 12 2021 at 14:40):

that $X\otimes Y$ preserves pullbacks if $X$ and $Y$ do?

I had also wondered previously whether there was an elementary argument – I would like to know if you find one!

Tom Hirschowitz (May 12 2021 at 14:47):

And conversely! :grinning_face_with_smiling_eyes:

fosco (May 13 2021 at 11:09):

the serendipity of this finding is striking; I am myself looking for a "unified" way to express the following phenomenon:

let $\mathbb S$ be a suitably nice PROP, then there is a substitution monoidal structure defined on presheaves on $\mathbb S$, that exploits iterated Day convolution defined as

$\displaystyle \int^n Fn \times G^{*n}(-)$

where $G^{*n}$ is $n$-fold Day convolution. All boils down to the property that $(G\otimes H)^{*n}\cong G^{*n}\otimes H$, if the assigment $(n,G)\mapsto G^{*n}$ is a functor of $n$ (so, $\mathbb S$ must be something like "natural numbers and bijections")

I am in the process of formalising this, but I'm somewhat clumsy; is there a definition for such a thing already?

fosco (May 13 2021 at 11:10):

Also, what is a nominal set?

Nathanael Arkor (May 13 2021 at 12:12):

@fosco: I'd take a look at the work of Power and Tanaka on A unified category-theoretic formulation of typed binding signatures. They investigate general notions of substitution monoidal structure that subsumes the operadic versions.

fosco (May 13 2021 at 12:17):

how can you know all these papers :grinning:

Tom Hirschowitz (May 14 2021 at 06:45):

The Cambridge school...

Tom Hirschowitz (May 14 2021 at 06:54):

A nominal set is just an object of the Schanuel topos: a pullback-preserving functor $\mathrm{Inj} \to \mathrm{Set}$, where $\mathrm{Inj}$ denotes the category of finite ordinals and injections between them. Computer scientists are more familiar with an equivalent presentation: it is a set $X$, equipped with an action of the group of permutations of $\mathbb{N}$ with finite support, such that all elements of $X$ have finite support (in a more general sense). This alternative viewpoint is explained efficiently in @Daniela Petrisan's thesis.

fosco (May 14 2021 at 07:59):

Allow me to rephrase the situation in a language I'm familiar with: my motivation comes from operads.

Let $A$ be a subcategory of $\sf Set$, say endowed with a functor $J : A \to \sf Set$; left Kan extension and precomposition induce an adjunction

$[A,{\sf Set}] \leftrightarrows [{\sf Set},{\sf Set}]$

Examples of interesting $A$'s:

1. Finite sets and all functions: $J$ is the tautological full embedding
2. Finite sets and bijections: $J$ is the tautolgical nonfull functor
3. Finite ordinals and monotone bijections: $J$ forgets the order, and then it's like in 2.
4. Finite sets and injections: your example up there
5. Finite sets and surjections: 4 $^\text{op}$
6. Finite sets and bijections "over" a fixed set $C$ of colours: $J : ({\sf Fin}/C)_{bij} \to {\sf Fin}_{bij} \to {\sf Set}$

The adjunction above now induces an equivalence of categories between $\mathcal U(A) \subseteq \widehat{A}$ and $\mathcal F({\sf Set})\subseteq [{\sf Set},{\sf Set}]$.

If $\mathcal F({\sf Set})\subseteq [{\sf Set},{\sf Set}]$ is closed under monoidal product, one can then transport the composition-of-functors monoidal structure on it to the category $\mathcal F({\sf Set})$ to a (highly nonsymmetric) monoidal structure $\lhd$ on $\mathcal U(A)$; this is how you define the "substitution" monoidal structure in most of those examples,

$F\lhd G := \displaystyle \int^n Fn\times G^{*n}(-)$

and operads as substituiton monoids (1. above yields cartesian operads or clones; 2. yields symmetric operads; 6. coloured operads (=multicategories)).

More interestingly, and slightly more abstractly, the substitution monoidal structure can be presented as a skew monoidal structure in disguise, because the substitution monoidal structure $F\lhd G$ is nothing but $\text{Lan}_J F\circ G$ (one easily sees this opening the coend that defines the Lan). Since in examples 1, 2, 3, 6 the functor $J$ is sufficiently good, it turns out that the skew structure is in fact not skew (more precisely, $J$ is dense, so $J\lhd F = \text{Lan}_JJ\circ F \cong F$, fully faithful, so $F\lhd J = \text{Lan}_J F \circ J\cong F$, and $\text{Lan}_J$ "preserves itself" in a certain way that makes the associator invertible).

Now, you're saying something like "in example 4 [and 5] above the skew monoidal structure stays skew, but we can still reason operadically". Is this correct?

Joachim Kock (May 14 2021 at 09:45):

'Reasoning operadically about skew-monoidal structures' sounds like the notion of operadic category of Batanin and Markl. Steve Lack found a characterisation of operadic categories in terms of skew-monoidal collections. It seems (although I don't know for sure) that the operadic categories that come from operads are those for which the skew-monoidal structure is actually monoidal.

Presheaves on $\mathbf{Inj}$ are the restriction species of Schmitt, which intuitively are the species with enough extra structure to have an incidence coalgebra. Schmitt also considered (co)presheaves on $\mathbf{Surj}$, and more importantly (co)presheaves on partial surjections. These are the hereditary species (and have incidence bialgebras). My former student Louis Carlier showed that hereditary species induce operadic categories. (Restriction species induce operadic categories too, but this is not as interesting because they will be operadic categories with only unary operations.)

I don't know so much about nominal sets, but it seems there are some interesting connections to sort out!

Nathanael Arkor (May 14 2021 at 10:40):

Since in examples 1, 2, 3, 6 the functor $J$ is sufficiently good

Isn't example 2 not good? It's faithful, but not full. (6 is also not full.)

Tom Hirschowitz (May 14 2021 at 10:46):

I haven't checked whether or not the first tensor product above is skew (my guess is yes). It is the "relative monads" one, in the sense that monoids for it are monads relative to the embedding $\mathrm{Inj} \hookrightarrow \mathrm{Set}$. But because relative monads do not depend on morphisms in the domain category, I think they are exactly the same as monads relative to $\mathrm{FinSet} \to \mathrm{Set}$, which are exactly Fiore-Plotkin-Turi monoids, aka finitary monads on sets.

On the other hand, John Power claims that the second tensor product, the one based on Day convolution, is not skew. Monoids for it are a kind of affine "operads", in the sense of being able to discard some arguments. But perhaps they only deserve to be called operads when restricted to pullback-preserving functors.

Regarding your final question, I'm not sure what you mean by reasoning operadically, could you try to narrow it down?

My initial claim (hope?) was more that, in some of these settings, letting $I$ denote the (skew) monoidal unit, one could show that for any nice endofunctor $\Sigma$, the initial $\Sigma$-monoid is the initial $(I+\Sigma)$-algebra. A $\Sigma$-monoid is an object with both monoid and $\Sigma$-algebra structure, subject to a coherence condition.

Nathanael Arkor (May 14 2021 at 10:56):

Tom Hirschowitz said:

My initial claim (hope?) was more that, in some of these settings, letting $I$ denote the (skew) monoidal unit, one could show that for any nice endofunctor $\Sigma$, the initial $\Sigma$-monoid is the initial $(I+\Sigma)$-algebra. A $\Sigma$-monoid is an object with both monoid and $\Sigma$-algebra structure, subject to a coherence condition.

How would this relate to the main theorem of Power and Tanaka? You'd like to generalise from monoidal to skew monoidal structure, or something stronger than this?

fosco (May 14 2021 at 11:03):

Nathanael Arkor said:

Since in examples 1, 2, 3, 6 the functor $J$ is sufficiently good

Isn't example 2 not good? It's faithful, but not full. (6 is also not full.)

yet you obtain a monoidal, non skew, structure in 2 and 6! That's why I've been sloppy defining what being "good" means.

Nathanael Arkor (May 14 2021 at 11:09):

Hmm. There's something I don't understand going on here, then. What I'd like to be able to do is define a symmetric operad as a $\mathbb P \to \mathrm{Set}$-relative monad, given by your J in (2). But I have two conflicting confusions: one is that this functor isn't full, so "should" just define a skew monoidal structure rather than a monoidal structure. But also that relative monads shouldn't depend on the morphisms in the domain anyway, so these relative monads should be the same as relative monads for $\mathrm{FinSet} \to \mathrm{Set}$. There's some flaw in my understanding here, but I'm not sure where it lies.

Tom Hirschowitz (May 14 2021 at 12:39):

I think it's exactly what I tried (but obviously failed) to explain above.
In summary:

• There is a tensor product, say $\boxtimes$, on $\mathbb{P} \to \mathrm{Set}$ whose monoids are symmetric operads.
• There is a different tensor product, say $\otimes$, whose monoids are monads relative to the embedding $\mathbb{P} \to \mathrm{Set}$, or equivalently finitary monads on sets.

The reason they're different is that the latter takes a hom in sets, so you get $\int^p X(p) \times Y(n)^p$: variables are "shared" amongst all arguments from $Y$.

Does this help?

Tom Hirschowitz (May 14 2021 at 12:41):

Oh, and yes, replacing the $Y(n)^p$ above by $p$-fold Day convolution seems to miraculously unskew the tensor product.

Nathanael Arkor (May 14 2021 at 12:42):

That does help, thanks :)

Nathanael Arkor (May 14 2021 at 12:42):

Is there some nice relationship between these two structures? Are they duoidal?

Nathanael Arkor (May 14 2021 at 12:44):

And just to check: these tensor products coincide for $\mathbb F \to \mathrm{Set}$, right? That would explain why I was conflating them.

Nathanael Arkor (May 14 2021 at 12:45):

These categories also have products and coproducts, so they're (skew) monoidal in quite a number of ways.

Nathanael Arkor (May 14 2021 at 12:47):

I feel there are pieces of this story scattered across the literature. It would be nice to have a single coherent explanation of these phenomena (ideally from the bicategorical perspective).

Tom Hirschowitz (May 14 2021 at 12:55):

Regarding your earlier question:

Nathanael Arkor said:

How would this relate to the main theorem of Power and Tanaka? You'd like to generalise from monoidal to skew monoidal structure, or something stronger than this?

I don't know, I'm quite confused now! I know that the FPT theorem generalises to the skew monoidal case (with some care about the notion of force). But what was the original question already? I was wondering whether there was an FPT theorem for nominal sets.

It now seems that there may be two:

• one for the Power tensor product $\boxtimes$, whose $\Sigma$-monoids will be symmetric operads with $\Sigma$-algebra structure, and

• one for $\otimes$, whose $\Sigma$-monoids will be finitary monads with $\Sigma$-algebra structure.

The former is a proper FPT theorem, for pointed strong $\Sigma$, while the latter is a generalised one, for what @Ambroise and I called structurally strong $\Sigma$.

However, for the latter to hold, it remains to show that $\otimes$ restricts to nominal sets. Monoids for this lifted tensor product should be cartesian operads, right?

Assuming this works, there should be a forgetful functor from nominal $\Sigma$-$\otimes$-monoids to
nominal $\Sigma$-$\boxtimes$-monoids, lying over the one from cartesian operads to plain operads, and importantly preserving the initial object since the FPT characterisation seems to work in both cases. How plausible is this?

Tom Hirschowitz (May 14 2021 at 13:01):

Nathanael Arkor said:

And just to check: these tensor products coincide for $\mathbb F \to \mathrm{Set}$, right? That would explain why I was conflating them.

Oh! Yes, this seems right, nice point. This looks like a strong hint that we're missing part of the picture here, don't you think?

Nathanael Arkor (May 14 2021 at 13:03):

I think the first tensor product $\boxtimes$ gives a notion of operad, while the second $\otimes$ gives a notion of relative monad. When we consider the FPT setting we get cartesian operads and finitary monads, respectively. It turns out that these coincide, but this is most likely an artefact of the cartesian structure. My guess would be that, in general, the initial algebras don't necessarily coincide.

Nathanael Arkor (May 14 2021 at 13:04):

These two tensor products provide a "bridge" in some sense between the world of operads and (relative) monads. And the FPT setting sits right in the middle.

Nathanael Arkor (May 14 2021 at 13:06):

I would hope there was a functor from one category of monoids to the other, but I'm not sure what structure I would expect it to preserve. I think I had never noticed before that there were really two constructions occurring that were different, as opposed to simply different perspectives, because I only really ever thought about the cartesian setting.

Nathanael Arkor (May 14 2021 at 13:08):

I think I would expect these constructions to coincide more generally in the "well-behaved" relative monad setting of ACU, which includes the FPT setting.

Tom Hirschowitz (May 14 2021 at 13:08):

Yes, I agree.

Tom Hirschowitz (May 14 2021 at 13:09):

But it would be cool to have a bridge between Power-Tanaka style tensors from distributive laws and ACU-style tensors from functors.

Nathanael Arkor (May 14 2021 at 13:12):

The Power–Tanaka-style tensors should be given by a Kleisli composition following the work of FGHW. I think the ACU-style tensors may be given by a similar composition structure in a skew bicategory instead (using the technology of Lack and Street on skew warpings).

Nathanael Arkor (May 14 2021 at 13:13):

So perhaps one has a skew bicategory and a bicategory related by an identity-on-objects pseudofunctor, or something like this…

Tom Hirschowitz (May 14 2021 at 13:25):

Haha you're so good on acronyms!!! For any normal person trying to read the above: ACU = Altenkirch-Chapman-Uustalu, FGHW = Fiore-Gambino-Hyland-Winskel.

Nathanael Arkor (May 14 2021 at 13:26):

In a little more detail: given a distributive law forming a composite (relative) pseudomonad $\mathbb P\mathbb S$, we can consider the unit $\eta_A : A \to \mathbb P\mathbb S A$. In the case where $\mathbb P$ is the presheaf construction (or similar), we can uncurry to get a functor $\mathbb SA° \times A \to \mathbb P1$, to which we apply the ACU construction. I'd hope these are "coherent" in $A$ in the sense that they can be glued together appropriately to form a skew bicategory, rather than just a skew monoidal category. Now we recover relative monads by considering monoids in the skew monoidal hom-categories. The operads are given by monoids in the monoidal hom-categories Kleisli bicategory for $\mathbb P\mathbb S$. And, assuming this coherent glueing can be carried out, we should have a pseudofunctor from one bicategory to the other.

Nathanael Arkor (May 14 2021 at 13:28):

Tom Hirschowitz said:

Haha you're so good on acronyms!!! For any normal person trying to read the above: ACU = Altenkirch-Chapman-Uustalu, FGHW = Fiore-Gambino-Hyland-Winskel.

Sorry, I forget there may be other people reading along :flushed: The relevant papers are Monads need not be endofunctors and Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures.

Tom Hirschowitz (May 14 2021 at 13:32):

Sorry that's not enough for me to get it. Are functors $\mathbb SA° \to \mathbb P1$ supposed to be the objects of your skew bicategory?
I'll have a look at skew warpings, I barely know of their existence --- although I coincidentally was in Sydney when Steve et al were looking for a french translation of "warping"!

Tom Hirschowitz (May 14 2021 at 13:32):

(That was for the mandatory french abstract in Cahiers.)

Nathanael Arkor (May 14 2021 at 13:34):

The functors $\mathbb SA° \to \mathbb P1$ are intended to be 1-cells of the skew bicategory. The objects ought to be small categories. The intention is to recover the category $[\mathbb SA, \mathrm{Set}]$ as a hom-category of the skew bicategory, but with different monoidal structure than the Kleisli bicategory.

Nathanael Arkor (May 14 2021 at 13:35):

I realised I started using $\mathbb P$ for the presheaf construction, whereas I previously used it for the free monoidal category, whoops. I've used $\mathbb S$ in the most recent message to avoid the conflict in notation.

Tom Hirschowitz (May 14 2021 at 13:36):

But then what are the source and target? (Sorry for being thick...)

Nathanael Arkor (May 14 2021 at 13:36):

The paper of Lack–Street I'm thinking of is Monads and warpings, in particular section 7.

Nathanael Arkor (May 14 2021 at 13:38):

Tom Hirschowitz said:

But then what are the source and target? (Sorry for being thick...)

You're not being thick: I was being vague because I wasn't sure of the details myself :sweat_smile:

Nathanael Arkor (May 14 2021 at 13:38):

The source and target should be $1$ and $A$, I think.

Nathanael Arkor (May 14 2021 at 13:40):

That won't quite match the Kleisli bicategory, though.

Tom Hirschowitz (May 14 2021 at 13:41):

RIght, it looks more like a 2-sided Kleisli, but for that you'd need $\mathbb{S}$ to be a comonad, right?

Nathanael Arkor (May 14 2021 at 13:41):

Ideally, 1-cells from $X$ to $Y$ should be functors $\mathbb SY° \times X \to \mathbb P1$, to match the Kleisli construction. But you're right that what I described looks like a two-sided Kleisli construction.

Tom Hirschowitz (May 14 2021 at 13:43):

Ok, I think I'll give it a rest for now and get back to, you know, those other things I should be doing. Hope we'll return to this at some point!

Nathanael Arkor (May 14 2021 at 13:45):

Yes, I think it calls for some more thought.

Nathanael Arkor (May 14 2021 at 14:04):

I've clarified some of my thoughts a little. Let $\mathbb P\mathbb S$ be a composite pseudomonad for which $\mathbb P$ acts "like the presheaf construction". We can form the Kleisli bicategory for $\mathbb P\mathbb S$, for which Kleisli 1-cells $X$ to $Y$ are 1-cells $X \to \mathbb P\mathbb SY$, hence 1-cells $\mathbb S(Y°) \to \mathbb P(X°)$. The hom-category $[\mathbb S(X°), \mathbb P(X°)]$ thus has monoidal structure, given by Kleisli composition. Now consider the unit of the composite pseudomonad, $\eta_A : A \to \mathbb P\mathbb SA$. This is equivalently a 1-cell $\mathbb S(A°) \to \mathbb P(A°)$ by the same reasoning as before. Since $\mathbb P(A°)$ is cocomplete, the requisite left extensions exist, and we can construct a skew monoidal structure on $[\mathbb S(A°), \mathbb P(A°)]$. If we substitute $X = A$ in the Kleisli hom-category, then we have a skew monoidal and a monoidal structure on $[\mathbb S(A°), \mathbb P(A°)]$. Taking $\mathbb P$ to be the presheaf construction, and $\mathbb S$ to be the free symmetric monoidal category construction, this recovers operads and relative monads. So we can talk about relative monads and operads for any such pseudodistributive law that recovers the familiar examples.

Nathanael Arkor (May 14 2021 at 14:06):

The monoidal structure in the first case comes from Kleisli composition; the obstruction to extending the skew monoidal structure in the second case to a skew bicategorical composition is that I used the unit of the pseudomonad, which is homogenous (i.e. we get a 1-cell involving two occurrences of $A$, rather than an $A$ and a $B$). So one would need to figure out how to extend this appropriately to the heterogenous case.

Nathanael Arkor (May 14 2021 at 14:10):

However, I think the same reasoning as earlier shows that any 1-cell $f : A \nrightarrow B$ in the Kleisli bicategory induces a skew monoidal structure on $[\mathbb S(B°), \mathbb P(A°)]$ whose unit is given by $f$. The classical setting is given by taking $1_A : A \nrightarrow A$.

fosco (May 14 2021 at 14:25):

Nathanael Arkor said:

Is there some nice relationship between these two structures? Are they duoidal?

Substitution and convolution form a normal duoidal structure, yes; it's Proposition 36 of Commutativity.

Nathanael Arkor (May 14 2021 at 14:27):

fosco said:

Nathanael Arkor said:

Is there some nice relationship between these two structures? Are they duoidal?

Substitution and convolution form a normal duoidal structure, yes; it's Proposition 36 of Commutativity.

Which one of these tensor products is convolution? Isn't convolution another monoidal structure entirely?

Nathanael Arkor (May 14 2021 at 14:27):

Substitution and convolution are both monoidal; but the $\otimes$ tensor product (i.e. the relative monad composition) is skew monoidal.

Nathanael Arkor (May 14 2021 at 14:28):

So I think there are three tensor products (along with product and coproduct).

fosco (May 14 2021 at 15:39):

oh, so you are denoting $\otimes$ the relmonad composition

Ouff.

Nathanael Arkor (May 14 2021 at 15:41):

(Which, as a side note, could do with a proper name. Perhaps something like the "extension tensor product".)

Tom Hirschowitz (May 14 2021 at 16:03):

Neat! Do you really mean $(\mathbb{P}X)^\circ$, or should it be $\mathbb{P}(X^\circ)$?

Nathanael Arkor (May 14 2021 at 16:10):

I do mean $(\mathbb PX)^\circ$, because $X \to [Y°, \mathrm{Set}]$ is equivalent to $Y° \to [X, \mathrm{Set}]$, hence to $Y° \to (\mathbb PX)°$. However, it doesn't make a big difference, because we should always be able to essentially ignore these opposites by considering morphisms $X° \nrightarrow Y°$ instead, so that everything cancels out.
I mixed up my variances.

Tom Hirschowitz (May 14 2021 at 16:14):

I think I understand each step except the conclusion. And your use of fonts is also mysterious...

Tom Hirschowitz (May 14 2021 at 16:15):

Oh wait, you do have in mind $\mathbb{P}(X) = [X^\circ,\mathrm{Set}]$, right?

Nathanael Arkor (May 14 2021 at 16:16):

Yes. I had some typos, and I may still have some remaining, so if something looks odd, it's probably a mistake!

Tom Hirschowitz (May 14 2021 at 16:16):

Haha, that's cheating, now my message looks crazy!

Nathanael Arkor (May 14 2021 at 16:17):

(I should really use the "Preview message" button more often.)

Tom Hirschowitz (May 14 2021 at 16:17):

So you maintain that $[X,\mathrm{Set}] = (\mathbb{P}X)^\circ$?

fosco (May 14 2021 at 16:18):

I lost track of this conversation; probably because I don't know FPT.

Nathanael Arkor (May 14 2021 at 16:20):

Tom Hirschowitz said:

So you maintain that $[X,\mathrm{Set}] = (\mathbb{P}X)^\circ$?

Whoops, no. Let me go back and edit my messages to confuse future readers again.

Tom Hirschowitz (May 14 2021 at 16:21):

FPT says: given a nice endofunctor $\Sigma$ on a nice monoidal category $\mathbf{C}$, $\Sigma$-monoids are monadic over $\mathbf{C}$ and the initial one has as carrier the free $(I+\Sigma)$-algebra.

[Reminder: $\Sigma$-monoids $=$ $\Sigma$-algebras with compatible monoid structure.]

Nathanael Arkor (May 14 2021 at 16:21):

Tom Hirschowitz said:

Neat! Do you really mean $(\mathbb{P}X)^\circ$, or should it be $\mathbb{P}(X^\circ)$?

So I agree all the $°$'s should be inside the parentheses, sorry.

Tom Hirschowitz (May 14 2021 at 16:22):

I have to stop now. But next step is to try and use your setting to relate both categories of monoids, right?

Nathanael Arkor (May 14 2021 at 16:30):

I'd also like to understand how the Power–Tanaka framework fits into this perspective better.

Nathanael Arkor (May 14 2021 at 16:33):

$\mathrm{Kl}(\mathbb P\mathbb S)(X, X)$ is monoidal, so that a strong monad $T$ on $\mathrm{Kl}(\mathbb P\mathbb S)(X, X)$ induces a skew-monoidal structure on $T\text{-}\mathrm{Alg}$ (by unpublished work by Saville–Fiore). The monoids in this category are $T$-monoids in the sense of FPT.

Nathanael Arkor (May 14 2021 at 16:36):

Then, by a general theorem of Power–Tanaka, the free $T$-monoid should be given by $TI$, which by Saville–Fiore is given by equipping an initial algebra with monoidal structure.

Nathanael Arkor (May 14 2021 at 16:39):

So I think the bicategorical structure is no longer relevant for the initial $T$-monoid constructions: it suffices to have the monoidal structure, and the theory essentially follows that of Power–Tanaka.