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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 2nd 2018
   • (edited Apr 2nd 2018)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI've decided to bite the bullet, intending to clean up the various messes which have been left for a few years at [[cycle category]] and [[cyclic order]]. On Mike's advice in a discussion box in the [remarks section](https://ncatlab.org/nlab/show/cyclic+order#remarks) at [[cyclic order]], I had a look at the [Berger-Moerdijk paper](https://arxiv.org/pdf/0809.3341.pdf) on generalized Reedy categories, and after a bit of work their description of Connes's category $\Lambda$ became quite clear to me, but they stop short of giving an exposition in the style of "Australian category theory" -- not their style, perhaps. I've been having a think about all this, and would like to run a few things past you good category theorists. One observation is that the material on what they call a "crossed $R$-group" ($R$ a small category), and its total category, fits pretty neatly into the language of distributive laws. Recall that a category can be regarded as a monad in a bicategory of spans. This means that if I have two categories = monads $M: C \nrightarrow C$ and $N: C \nrightarrow C$ in $Span$, and a distributive law between them, say $$\theta: N \circ M \nrightarrow M \circ N,$$ then I can form a new category = monad $M \circ N$ in the bicategory of spans. There are many, many examples of this sort of construction. In the context of their paper, their notion of $R$-group is such an example of a distributive law (whose total category is exactly the monad composite). One of the categories is $R$ regarded as a span $$R_0 \leftarrow R_1 \to R_0$$ with its monad structure. Another is a sum of groups $G_r$ indexed by $r \in R_0$, which forms a groupoid G, regarded as a span $$R_0 \leftarrow \sum_{r \in R_0} G_r \to R_0$$ again with a monad structure. What they call a *crossed $R$-group* consists precisely (I believe) of two such monads equipped with a distributive law $\theta: G \circ R \to R \circ G$ between them, where I write composition in Leibnizian order so that $G \circ R$ refers to a pullback of a diagram $$\array{ & & R_1 & & & & \sum_r G_r & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & \\ R_0 & & & & R_0 & & & & R_0 }$$ More precisely, $\theta(\alpha: r \to s, g \in G_s)$ would be, in their notation, $(\alpha^\ast(g), g_\ast(\alpha))$ (lying in the fiber $G_r \times \hom(r, s)$ of the span composite $R \circ G$). Then they describe the Connes cycle category $\Lambda$ as a total category which in our notation would be denoted $\Delta \circ Z$, where the groupoid $Z$ is the sum of all finite cyclic groups $Z_n$ and $\Delta: \mathbb{N} \nrightarrow \mathbb{N}$ is the simplex category. They describe the relevant distributive law, modulo the translations above, in terms of a restriction of another distributive law of the form $\theta: \Delta \circ \mathbb{P} \to \mathbb{P} \circ \Delta$, where $\mathbb{P}$ is the groupoid of finite permutations. It's at this point where a more conceptual description of what is truly going on seems possible. The idea is that there is a forgetful functor from symmetric monoidal categories to monoidal categories, and another from monoidal categories with a monoid object to monoidal categories. Both of these are (2-)monadic, I believe, and can be described in terms of clubs. As we know, distributive laws have to do with liftings of monads through monadic functors. In the present case, it seems that the distributive law $\theta: \Delta \mathbb{P} \to \mathbb{P} \Delta$ has to do with a doctrinal lifting of a club $\Delta \wr_{\mathbb{N}} -: MonCat \to MonCat$ (attached to the doctrine "monoidal categories with a monoid") through the forgetful functor $U: SymMonCat \to MonCat$. It seems to me that there ought to be very conceptual grounds on which to expect the existence of that particular lifting. In any case, I want to think of the category $\mathbb{P} \circ \Delta$ as just a "symmetrification" of the walking monoid $\Delta$, i.e., we're just applying a change-of-doctrine map to the theory of monoids, from monoidal categories to symmetric monoidal categories. Similarly, I guess there ought to be a doctrine intermediate between monoidal categories and symmetric monoidal categories, called "cyclic monoidal categories". I can't find mention of what I mean through a quick Google search, but whatever they are, I am assuming they can be described as algebras of a club governed by the groupoid $Z$ (just as symmetric monoidal categories are algebras of a club $\mathbb{P}$). In other words, I am proposing that doctrinally, the Connes cyclic category is the "cyclification" of the walking monoid. Do people discuss such things, somewhere, somehow?

   I’ve decided to bite the bullet, intending to clean up the various messes which have been left for a few years at cycle category and cyclic order. On Mike’s advice in a discussion box in the remarks section at cyclic order, I had a look at the Berger-Moerdijk paper on generalized Reedy categories, and after a bit of work their description of Connes’s category $\Lambda$ became quite clear to me, but they stop short of giving an exposition in the style of “Australian category theory” – not their style, perhaps. I’ve been having a think about all this, and would like to run a few things past you good category theorists.

   One observation is that the material on what they call a “crossed $R$-group” ($R$ a small category), and its total category, fits pretty neatly into the language of distributive laws. Recall that a category can be regarded as a monad in a bicategory of spans. This means that if I have two categories = monads $M: C \nrightarrow C$ and $N: C \nrightarrow C$ in $Span$, and a distributive law between them, say

   $$\theta: N \circ M \nrightarrow M \circ N,$$

   then I can form a new category = monad $M \circ N$ in the bicategory of spans.

   There are many, many examples of this sort of construction. In the context of their paper, their notion of $R$-group is such an example of a distributive law (whose total category is exactly the monad composite). One of the categories is $R$ regarded as a span

   $$R_0 \leftarrow R_1 \to R_0$$

   with its monad structure. Another is a sum of groups $G_r$ indexed by $r \in R_0$, which forms a groupoid G, regarded as a span

   $$R_0 \leftarrow \sum_{r \in R_0} G_r \to R_0$$

   again with a monad structure. What they call a crossed $R$-group consists precisely (I believe) of two such monads equipped with a distributive law $\theta: G \circ R \to R \circ G$ between them, where I write composition in Leibnizian order so that $G \circ R$ refers to a pullback of a diagram

   $$\array{ & & R_1 & & & & \sum_r G_r & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & \\ R_0 & & & & R_0 & & & & R_0 }$$

   More precisely, $\theta(\alpha: r \to s, g \in G_s)$ would be, in their notation, $(\alpha^\ast(g), g_\ast(\alpha))$ (lying in the fiber $G_r \times \hom(r, s)$ of the span composite $R \circ G$).

   Then they describe the Connes cycle category $\Lambda$ as a total category which in our notation would be denoted $\Delta \circ Z$, where the groupoid $Z$ is the sum of all finite cyclic groups $Z_n$ and $\Delta: \mathbb{N} \nrightarrow \mathbb{N}$ is the simplex category. They describe the relevant distributive law, modulo the translations above, in terms of a restriction of another distributive law of the form $\theta: \Delta \circ \mathbb{P} \to \mathbb{P} \circ \Delta$, where $\mathbb{P}$ is the groupoid of finite permutations.

   It’s at this point where a more conceptual description of what is truly going on seems possible. The idea is that there is a forgetful functor from symmetric monoidal categories to monoidal categories, and another from monoidal categories with a monoid object to monoidal categories. Both of these are (2-)monadic, I believe, and can be described in terms of clubs.

   As we know, distributive laws have to do with liftings of monads through monadic functors. In the present case, it seems that the distributive law $\theta: \Delta \mathbb{P} \to \mathbb{P} \Delta$ has to do with a doctrinal lifting of a club $\Delta \wr_{\mathbb{N}} -: MonCat \to MonCat$ (attached to the doctrine “monoidal categories with a monoid”) through the forgetful functor $U: SymMonCat \to MonCat$.

   It seems to me that there ought to be very conceptual grounds on which to expect the existence of that particular lifting. In any case, I want to think of the category $\mathbb{P} \circ \Delta$ as just a “symmetrification” of the walking monoid $\Delta$, i.e., we’re just applying a change-of-doctrine map to the theory of monoids, from monoidal categories to symmetric monoidal categories.

   Similarly, I guess there ought to be a doctrine intermediate between monoidal categories and symmetric monoidal categories, called “cyclic monoidal categories”. I can’t find mention of what I mean through a quick Google search, but whatever they are, I am assuming they can be described as algebras of a club governed by the groupoid $Z$ (just as symmetric monoidal categories are algebras of a club $\mathbb{P}$).

   In other words, I am proposing that doctrinally, the Connes cyclic category is the “cyclification” of the walking monoid.

   Do people discuss such things, somewhere, somehow?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 2nd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe description of "crossed groups" as distributive laws in $Span$ is really nice! I don't quite follow what's going on with the rest of it though; how do you get from a distributive law in $Span$ to a distributive law of clubs/monads? To answer your question, this doesn't ring any bells for me. Although the notion of "cyclic monoidal category" might be related to the notion of "shadow" defined by [Kate Ponto](https://arxiv.org/abs/0910.1306). A shadow on a monoidal category $V$ is a functor $\langle\langle - \rangle\rangle : V \to T$, for some other category $T$, together with cyclicity isomorphisms $\langle\langle x \otimes y \rangle\rangle \cong \langle\langle y \otimes x \rangle\rangle$ satisfying appropriate axioms. Maybe a "cyclic monoidal category" is related to one equipped with a shadow where $\langle\langle-\rangle\rangle$ is the identity functor, or something?

   The description of “crossed groups” as distributive laws in $Span$ is really nice! I don’t quite follow what’s going on with the rest of it though; how do you get from a distributive law in $Span$ to a distributive law of clubs/monads?

   To answer your question, this doesn’t ring any bells for me. Although the notion of “cyclic monoidal category” might be related to the notion of “shadow” defined by Kate Ponto. A shadow on a monoidal category $V$ is a functor $\langle\langle - \rangle\rangle : V \to T$, for some other category $T$, together with cyclicity isomorphisms $\langle\langle x \otimes y \rangle\rangle \cong \langle\langle y \otimes x \rangle\rangle$ satisfying appropriate axioms. Maybe a “cyclic monoidal category” is related to one equipped with a shadow where $\langle\langle-\rangle\rangle$ is the identity functor, or something?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 2nd 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSorry if I wasn't being clear. The basic hope was that if you have an algebraic theory $T$ (like the theory of monoids) that is expressible in the doctrine of monoidal categories, let's say just a one-sorted theory to keep things simple for now, and you want to switch that theory over to a richer doctrine such as the doctrine of symmetric monoidal categories, then it should be given by some simple construction like $T \circ \mathbb{P}$. That is: the one-sorted theory $T$ is expressed by a monoidal category called a "pro" (prop minus the p), with underlying set of objects = arities $\mathbb{N}$. As a category, this category = span monad has the form $\mathbb{N} \leftarrow A \to \mathbb{N}$. Then you have the free symmetric monoidal category $\mathbb{P}$ generated by a single sort, which as a span also has the form $\mathbb{N} \leftarrow \mathbb{P} \to \mathbb{N}$, and you want to compose them in some way, in effect adjoining symmetries in a sensible way. We can put the club stuff on hold, if this seems to be a clearer description of what I was aiming for.

   Sorry if I wasn’t being clear. The basic hope was that if you have an algebraic theory $T$ (like the theory of monoids) that is expressible in the doctrine of monoidal categories, let’s say just a one-sorted theory to keep things simple for now, and you want to switch that theory over to a richer doctrine such as the doctrine of symmetric monoidal categories, then it should be given by some simple construction like $T \circ \mathbb{P}$. That is: the one-sorted theory $T$ is expressed by a monoidal category called a “pro” (prop minus the p), with underlying set of objects = arities $\mathbb{N}$. As a category, this category = span monad has the form $\mathbb{N} \leftarrow A \to \mathbb{N}$. Then you have the free symmetric monoidal category $\mathbb{P}$ generated by a single sort, which as a span also has the form $\mathbb{N} \leftarrow \mathbb{P} \to \mathbb{N}$, and you want to compose them in some way, in effect adjoining symmetries in a sensible way.

   We can put the club stuff on hold, if this seems to be a clearer description of what I was aiming for.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 2nd 2018
   • (edited Apr 2nd 2018)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAnd just to add an additional, hopefully clarifying comment: I meant it when I said (monoidal) *algebraic* theories $T$, i.e., I mean for the generating operations of $T$ to have multiple inputs but just one output, in order for such a simple-minded construction like $T \circ \mathbb{P}$ to do its job properly. The underlying point here is that the relevant distributive law $\theta: \mathbb{P} \circ T \to T \circ \mathbb{P}$ is to be guided purely by naturality considerations. A simple example should get the point across: if we have $T$-operations $f: s_1 \otimes \ldots \otimes s_m \to s$ and $g: t_1 \otimes \ldots \otimes t_n \to t$, and we want to think of the theory $T$ qua the doctrine of symmetric monoidal categories, then naturality of the symmetry would force the equation $$s_1 \otimes \ldots \otimes s_n \otimes t_1 \otimes \ldots \otimes t_n \stackrel{f \otimes g}{\to} s \otimes t \stackrel{\sigma_{s t}}{\to} t \otimes s\;\;\; = \;\;\; s_1 \otimes \ldots \otimes s_n \otimes t_1 \otimes \ldots \otimes t_n \stackrel{\sigma_{\vec{s} \vec{t}}}{\to} t_1 \otimes \ldots \otimes t_n \otimes s_1 \otimes \ldots \otimes s_n \stackrel{g \otimes f}{\to} t \otimes s$$ which means the distributive law $\theta$ should take the element $(f \otimes g, \sigma_{s t})$ of the span $\mathbb{P} \circ T$ to the element $(\sigma_{\vec{s},\vec{t}}, g \otimes f)$ of $T \circ \mathbb{P}$. That's what's really going on in the description of $\Lambda$ (as found in Berger-Moerdijk). But if you think about it a little, such a simple description of change-of-doctrine applied to a theory, from monoidal categories to symmetric monoidal categories or cyclic monoidal categories, as being just $T \circ \mathbb{P}$ or $T \circ Z$, wouldn't be so simple if we were considering more general pros $T$. Here's another way of putting approximately the same thought: if you have a purely *coalgebraic* pro $T$, then you'd better use $\mathbb{P} \circ T$ instead to switch doctrines (and again let naturality guide the form of the relevant distributive law $T \circ \mathbb{P} \to \mathbb{P} \circ T$). (By the way -- thanks Mike not just for your response but for pointing me to the paper by you and Kate; I'll have a look.)

   And just to add an additional, hopefully clarifying comment: I meant it when I said (monoidal) algebraic theories $T$, i.e., I mean for the generating operations of $T$ to have multiple inputs but just one output, in order for such a simple-minded construction like $T \circ \mathbb{P}$ to do its job properly.

   The underlying point here is that the relevant distributive law $\theta: \mathbb{P} \circ T \to T \circ \mathbb{P}$ is to be guided purely by naturality considerations. A simple example should get the point across: if we have $T$-operations $f: s_1 \otimes \ldots \otimes s_m \to s$ and $g: t_1 \otimes \ldots \otimes t_n \to t$, and we want to think of the theory $T$ qua the doctrine of symmetric monoidal categories, then naturality of the symmetry would force the equation

   $$s_1 \otimes \ldots \otimes s_n \otimes t_1 \otimes \ldots \otimes t_n \stackrel{f \otimes g}{\to} s \otimes t \stackrel{\sigma_{s t}}{\to} t \otimes s\;\;\; = \;\;\; s_1 \otimes \ldots \otimes s_n \otimes t_1 \otimes \ldots \otimes t_n \stackrel{\sigma_{\vec{s} \vec{t}}}{\to} t_1 \otimes \ldots \otimes t_n \otimes s_1 \otimes \ldots \otimes s_n \stackrel{g \otimes f}{\to} t \otimes s$$

   which means the distributive law $\theta$ should take the element $(f \otimes g, \sigma_{s t})$ of the span $\mathbb{P} \circ T$ to the element $(\sigma_{\vec{s},\vec{t}}, g \otimes f)$ of $T \circ \mathbb{P}$.

   That’s what’s really going on in the description of $\Lambda$ (as found in Berger-Moerdijk). But if you think about it a little, such a simple description of change-of-doctrine applied to a theory, from monoidal categories to symmetric monoidal categories or cyclic monoidal categories, as being just $T \circ \mathbb{P}$ or $T \circ Z$, wouldn’t be so simple if we were considering more general pros $T$. Here’s another way of putting approximately the same thought: if you have a purely coalgebraic pro $T$, then you’d better use $\mathbb{P} \circ T$ instead to switch doctrines (and again let naturality guide the form of the relevant distributive law $T \circ \mathbb{P} \to \mathbb{P} \circ T$).

   (By the way – thanks Mike not just for your response but for pointing me to the paper by you and Kate; I’ll have a look.)

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeApr 3rd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOkay, I get a bit of the idea I think. Normally when I think of lifting a theory $T$ from one doctrine $D_1$ to a richer doctrine $D_2$, I think of doing it "universally" so that a $\hat{T}$-model in a $D_2$-category is the same as a $T$-model in its underlying $D_1$-category. That sort of lifting doesn't require any extra structure on $T$, it just exists universally. Are you saying we can instead *choose* some extra structure on $T$ (namely, a distributive law) to lift it to $D_2$ in a different (hence not universal) way? This also reminds me of section 6.7 (page 169) of [HOHC](https://arxiv.org/abs/math/0305049) on "change of shape" morphisms between generalized multicategories, which are also constructed as certain composites in bicategories of spans.

   Okay, I get a bit of the idea I think. Normally when I think of lifting a theory $T$ from one doctrine $D_1$ to a richer doctrine $D_2$, I think of doing it “universally” so that a $\hat{T}$-model in a $D_2$-category is the same as a $T$-model in its underlying $D_1$-category. That sort of lifting doesn’t require any extra structure on $T$, it just exists universally. Are you saying we can instead choose some extra structure on $T$ (namely, a distributive law) to lift it to $D_2$ in a different (hence not universal) way?

   This also reminds me of section 6.7 (page 169) of HOHC on “change of shape” morphisms between generalized multicategories, which are also constructed as certain composites in bicategories of spans.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 3rd 2018
   • (edited Apr 3rd 2018)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNo, I didn't think I was saying that. I thought what I was doing just now was giving a simple direct construction of what you call $\hat{T}$ in terms of $T$, at least when $T$ is an algebraic theory in $D_1$ = doctrine of monoidal categories, and $D_2$ is something like the doctrine of symmetric monoidal categories, or braided monoidal categories, or cartesian categories... or indeed cyclic monoidal categories (modeled after your "shadows"). The thread started off being about the cyclic category $\Lambda$, and the description I'm putting forth, which I've never seen before, is that it is the "theory of monoids within the doctrine of cyclic monoidal categories", i.e., it classifies monoids within this doctrine: if $C$ is a cyclic monoidal category, then the category of cyclic monoidal functors $\Lambda \to C$ is equivalent to the category of monoids in $C$. Parallel to the way that $\Delta$ classifies monoids within the doctrine of monoidal categories. Moreover, I am suggesting in #3 and #4 that this abstract conceptual description leads directly to its explicit concrete description, as $\Delta \circ Z$. If someone asks you how to write down the distributive law $Z \circ \Delta \to \Delta \circ Z$ used in this description, you can answer that it's just the obvious law you need in order to ensure that the cyclic monoidal structure constraints $M \otimes N \to N \otimes M$ (which come from $Z$, the groupoid of finite cyclic permutations) do in fact behave as natural transformations when viewed in $\Delta \circ Z$, which is what I was trying to explain in #4 via an example. Thanks for your feedback, and please let me know if I'm not being clear.

   No, I didn’t think I was saying that. I thought what I was doing just now was giving a simple direct construction of what you call $\hat{T}$ in terms of $T$, at least when $T$ is an algebraic theory in $D_1$ = doctrine of monoidal categories, and $D_2$ is something like the doctrine of symmetric monoidal categories, or braided monoidal categories, or cartesian categories… or indeed cyclic monoidal categories (modeled after your “shadows”).

   The thread started off being about the cyclic category $\Lambda$, and the description I’m putting forth, which I’ve never seen before, is that it is the “theory of monoids within the doctrine of cyclic monoidal categories”, i.e., it classifies monoids within this doctrine: if $C$ is a cyclic monoidal category, then the category of cyclic monoidal functors $\Lambda \to C$ is equivalent to the category of monoids in $C$. Parallel to the way that $\Delta$ classifies monoids within the doctrine of monoidal categories. Moreover, I am suggesting in #3 and #4 that this abstract conceptual description leads directly to its explicit concrete description, as $\Delta \circ Z$. If someone asks you how to write down the distributive law $Z \circ \Delta \to \Delta \circ Z$ used in this description, you can answer that it’s just the obvious law you need in order to ensure that the cyclic monoidal structure constraints $M \otimes N \to N \otimes M$ (which come from $Z$, the groupoid of finite cyclic permutations) do in fact behave as natural transformations when viewed in $\Delta \circ Z$, which is what I was trying to explain in #4 via an example.

   Thanks for your feedback, and please let me know if I’m not being clear.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeApr 3rd 2018
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   Author: Mike Shulman
   Format: MarkdownItexOh, are you suggesting that for *any* $D_1$-theory $T$ there is a "canonically defined" distributive law that we can use to construct $\hat{T}$?

   Oh, are you suggesting that for any $D_1$-theory $T$ there is a “canonically defined” distributive law that we can use to construct $\hat{T}$?

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 3rd 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexYes, that is what I'm hoping to be able to prove, restricting for now to $D_1$ = doctrine of monoidal categories, and $D_2$ a doctrine of somewhat more enriched monoidal type (symmetric, braided, cyclic, etc.), since I'm not quite sure what shape a more general theorem would have. And also with the proviso that a $D_1$-theory means generated from operations each having a single output sort, and that equations declared in the theory are between terms of the same type $s_1 \otimes \ldots \otimes s_n \to t$, again with single output sort $t$. (The last is probably already understood, but I'm playing it safe here; it would ruin the hoped-for theorem if we allowed $D_1$-equations to occur between terms with output $T = t_1 \otimes \ldots \otimes t_m$ that don't come about by tensoring together $m$ equations of the type I just described.) Not to belabor the point, but in the case where $D_2$ is symmetric monoidal categories, the idea is to adjoin symmetry isomorphisms to the $D_1$-theory, and be able to rewrite a map $\phi \circ (f_1 \otimes \ldots \otimes f_m)$ where $\phi$ is a symmetry isomorphism $t_1 \otimes \ldots \otimes t_m \to t_{\sigma(1)} \otimes \ldots t_{\sigma(m)}$ as $(f_{\sigma(1)} \otimes \ldots \otimes f_{\sigma(m)}) \circ \phi'$ where $\phi'$ is another symmetry isomorphism. Seems simple enough, right? String-diagrammatically, push symmetry isomorphisms to the top, by a kind of "combing" procedure. That dictates the form of the distributive law: $\theta(\phi, f_1\otimes \ldots \otimes f_m) = (f_{\sigma(1)} \otimes \ldots \otimes f_{\sigma(m)}, \phi')$.

   Yes, that is what I’m hoping to be able to prove, restricting for now to $D_1$ = doctrine of monoidal categories, and $D_2$ a doctrine of somewhat more enriched monoidal type (symmetric, braided, cyclic, etc.), since I’m not quite sure what shape a more general theorem would have. And also with the proviso that a $D_1$-theory means generated from operations each having a single output sort, and that equations declared in the theory are between terms of the same type $s_1 \otimes \ldots \otimes s_n \to t$, again with single output sort $t$. (The last is probably already understood, but I’m playing it safe here; it would ruin the hoped-for theorem if we allowed $D_1$-equations to occur between terms with output $T = t_1 \otimes \ldots \otimes t_m$ that don’t come about by tensoring together $m$ equations of the type I just described.)

   Not to belabor the point, but in the case where $D_2$ is symmetric monoidal categories, the idea is to adjoin symmetry isomorphisms to the $D_1$-theory, and be able to rewrite a map $\phi \circ (f_1 \otimes \ldots \otimes f_m)$ where $\phi$ is a symmetry isomorphism $t_1 \otimes \ldots \otimes t_m \to t_{\sigma(1)} \otimes \ldots t_{\sigma(m)}$ as $(f_{\sigma(1)} \otimes \ldots \otimes f_{\sigma(m)}) \circ \phi'$ where $\phi'$ is another symmetry isomorphism. Seems simple enough, right? String-diagrammatically, push symmetry isomorphisms to the top, by a kind of “combing” procedure. That dictates the form of the distributive law: $\theta(\phi, f_1\otimes \ldots \otimes f_m) = (f_{\sigma(1)} \otimes \ldots \otimes f_{\sigma(m)}, \phi')$.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 3rd 2018
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   Author: Todd_Trimble
   Format: MarkdownItex(I wonder if the language of (covariant) clubs would be useful in finding a nice general shape for the hoped-for theorem.)

   (I wonder if the language of (covariant) clubs would be useful in finding a nice general shape for the hoped-for theorem.)

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeApr 3rd 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI often use the word "theory" to imply the unary-output restriction (e.g. that's what $D$-multicategories and the relevant kind of type theory represent), so I'm totally on board with you there. Are you sure that you want to do this construction in $Span$ rather than $D_1$-$Span$? The latter being the bicategory in which monads coincide with $D_1$-multicategories, i.e. exactly unary-output $D_1$-theories, so it seems a more natural place to work to me. Just spitballing a bit, let $D_1$ be the free-monoid monad on $Set$, which extends to $Span$, and let $D_1' = Mod(Span(D_1))$ be the induced monad on $Prof = Mod(Span(Set))$. Let $D_2$ be another monad on $Prof$ with a monad morphism $D_1' \to D_2$ such that each functor $D_1'(X) \to D_2(X)$ is bijective on objects. Then an ordinary multicategory $X$, i.e. a (co-unary) $D_1$-theory, is a span $X_0 \leftarrow X_1 \to D_1 X_0$ with a monad structure in $D_1$-$Span$. But we can also apply $D_2$ to the discrete category $X_0^d$ on $X_0$ regarded as an object of $Prof$, to obtain a category that we can represent by a span $(D_2 X_0^d)_0 \leftarrow (D_2 X_0^d)_1 \to (D_2 X_0^d)_0$. But since $D_1'\to D_2$ is bijective on objects, $(D_2 X_0^d)_0 = D_1 X_0$, so $D_2 X_0^d$ is also a span $D_1 X_0 \leftarrow (D_2 X_0^d)_1 \to D_1 X_0$. So maybe this span has a distributive law over the span $X_0 \leftarrow X_1 \to D_1 X_0$ such that the composite is the free $D_2$-multicategory generated by $X$. The latter has another explicit description as the composite of the profunctor $X_1 : X_0 ⇸ D_1' X_0$ with (the representable profunctor determined by) the functor $D_1' X_0 \to D_2 X_0$; maybe we could show that this coincides with the above distributive-law composite.

    I often use the word “theory” to imply the unary-output restriction (e.g. that’s what $D$-multicategories and the relevant kind of type theory represent), so I’m totally on board with you there. Are you sure that you want to do this construction in $Span$ rather than $D_1$-$Span$? The latter being the bicategory in which monads coincide with $D_1$-multicategories, i.e. exactly unary-output $D_1$-theories, so it seems a more natural place to work to me.

    Just spitballing a bit, let $D_1$ be the free-monoid monad on $Set$, which extends to $Span$, and let $D_1' = Mod(Span(D_1))$ be the induced monad on $Prof = Mod(Span(Set))$. Let $D_2$ be another monad on $Prof$ with a monad morphism $D_1' \to D_2$ such that each functor $D_1'(X) \to D_2(X)$ is bijective on objects. Then an ordinary multicategory $X$, i.e. a (co-unary) $D_1$-theory, is a span $X_0 \leftarrow X_1 \to D_1 X_0$ with a monad structure in $D_1$-$Span$. But we can also apply $D_2$ to the discrete category $X_0^d$ on $X_0$ regarded as an object of $Prof$, to obtain a category that we can represent by a span $(D_2 X_0^d)_0 \leftarrow (D_2 X_0^d)_1 \to (D_2 X_0^d)_0$. But since $D_1'\to D_2$ is bijective on objects, $(D_2 X_0^d)_0 = D_1 X_0$, so $D_2 X_0^d$ is also a span $D_1 X_0 \leftarrow (D_2 X_0^d)_1 \to D_1 X_0$. So maybe this span has a distributive law over the span $X_0 \leftarrow X_1 \to D_1 X_0$ such that the composite is the free $D_2$-multicategory generated by $X$. The latter has another explicit description as the composite of the profunctor $X_1 : X_0 ⇸ D_1' X_0$ with (the representable profunctor determined by) the functor $D_1' X_0 \to D_2 X_0$; maybe we could show that this coincides with the above distributive-law composite.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeApr 3rd 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex(I'm using the language of [UFGM](https://arxiv.org/abs/0907.2460), which unsurprisingly I am most comfortable with, but probably the language of clubs would also suffice.)

    (I’m using the language of UFGM, which unsurprisingly I am most comfortable with, but probably the language of clubs would also suffice.)

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex> so it seems a more natural place to work to me. Yes, that makes total sense. At the time I wrote #1, I was in the midst of understanding $\Lambda$ just as a *category*, but the conversation has moved on to a point where your suggestion makes more sense -- thanks. I'll need more time to study your second paragraph.

    so it seems a more natural place to work to me.

    Yes, that makes total sense. At the time I wrote #1, I was in the midst of understanding $\Lambda$ just as a category, but the conversation has moved on to a point where your suggestion makes more sense – thanks.

    I’ll need more time to study your second paragraph.