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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 28th 2013
   • (edited Jul 28th 2013)
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   Author: Todd_Trimble
   Format: MarkdownItexI'm having some trouble deciding how seriously I want to take the discussion title, so I thought I'd ask others here for their opinions. Ordinarily, generalized multicategories are discussed in the context of a [[cartesian monad]] $T$ on a finitely complete category $C$. One may pass to a bicategory of $T$-spans (ordinary spans in $C$ of the form $T X \leftarrow R \to Y$), composed in Kleisli fashion. A $T$-multicategory is then a monad in the bicategory of $T$-spans. One could play a very similar game with *weakly* cartesian monads $T$ on a *regular* category $C$ (meaning that $T$ preserves weak pullbacks, and the naturality squares for unit and multiplication are weak pullbacks). One may pass to a locally posetal bicategory of $T$-relations (ordinary jointly monic spans in $C$ of the form $T X \leftarrow R \to Y$), composed again in Kleisli fashion. The weak cartesianness is just what is needed to make this work. Then a relational $T$-module (or whatever one wants to call it) is a monad in the 2-category of $T$-relations. That's all very well and good. And in the case of $T$ = ultrafilter monad on $C = Set$, this all works out, except for one annoying detail: the unit transformation is not weakly cartesian. (Everything else is fine: $T$ preserves weak pullbacks, and the naturality squares for the multiplication are weak pullbacks.) So as far as I can tell, there *is no* 2-category of $T$-relations. Is there some way to circumvent this detail to make a nice parallel development with usual $T$-multicategories? What do people think?

   I’m having some trouble deciding how seriously I want to take the discussion title, so I thought I’d ask others here for their opinions.

   Ordinarily, generalized multicategories are discussed in the context of a cartesian monad $T$ on a finitely complete category $C$. One may pass to a bicategory of $T$-spans (ordinary spans in $C$ of the form $T X \leftarrow R \to Y$), composed in Kleisli fashion. A $T$-multicategory is then a monad in the bicategory of $T$-spans.

   One could play a very similar game with weakly cartesian monads $T$ on a regular category $C$ (meaning that $T$ preserves weak pullbacks, and the naturality squares for unit and multiplication are weak pullbacks). One may pass to a locally posetal bicategory of $T$-relations (ordinary jointly monic spans in $C$ of the form $T X \leftarrow R \to Y$), composed again in Kleisli fashion. The weak cartesianness is just what is needed to make this work. Then a relational $T$-module (or whatever one wants to call it) is a monad in the 2-category of $T$-relations.

   That’s all very well and good. And in the case of $T$ = ultrafilter monad on $C = Set$, this all works out, except for one annoying detail: the unit transformation is not weakly cartesian. (Everything else is fine: $T$ preserves weak pullbacks, and the naturality squares for the multiplication are weak pullbacks.) So as far as I can tell, there is no 2-category of $T$-relations.

   Is there some way to circumvent this detail to make a nice parallel development with usual $T$-multicategories? What do people think?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 28th 2013
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   Author: Mike Shulman
   Format: MarkdownItexHave a look at my paper with Geoff Cruttwell.

   Have a look at my paper with Geoff Cruttwell.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 28th 2013
   • (edited Jul 28th 2013)
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   Author: Todd_Trimble
   Format: MarkdownItexI hope I didn't cause offense with my question, Mike (and sorry if I did)! I hadn't realized that this observation is actually due to you two. I actually thought I was raising a good question. Is there a particular place in the paper where you address this concern? Example A.7 looks pertinent. **Edit:** Well, you can hold off on replying if you'd like -- I'm looking through your paper with Cruttwell now. It looks as though you're introducing a still more general notion of multicategory than the one adumbrated in my opening post, and also that one of the crucial inputs for one set of instances of this notion is the notion of taut monad, and the ultrafilter monad is an example of that. This looks like a very nice paper.

   I hope I didn’t cause offense with my question, Mike (and sorry if I did)! I hadn’t realized that this observation is actually due to you two.

   I actually thought I was raising a good question. Is there a particular place in the paper where you address this concern? Example A.7 looks pertinent.

   Edit: Well, you can hold off on replying if you’d like – I’m looking through your paper with Cruttwell now. It looks as though you’re introducing a still more general notion of multicategory than the one adumbrated in my opening post, and also that one of the crucial inputs for one set of instances of this notion is the notion of taut monad, and the ultrafilter monad is an example of that.

   This looks like a very nice paper.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJul 28th 2013
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   Author: Mike Shulman
   Format: MarkdownItexNo, no offense taken. I was just in a hurry this morning, sorry. That's right, we're proposing a more general notion of multicategory. The notion of taut monad, and the observation that relational $\beta$-modules are an example of generalized multicategories relative to a taut monad, is not due to us either; there's a whole group of people (Clementino, Hofmann, Tholen, Seal) who've developed a theory of "generalized multicategories" of that sort, which we discuss more generally in B.6. What we were trying to do is find a general framework that includes both the CHTS sort of generalized multicategories and also the "Leinster-Burroni" sort that you described, as well as others.

   No, no offense taken. I was just in a hurry this morning, sorry.

   That’s right, we’re proposing a more general notion of multicategory. The notion of taut monad, and the observation that relational $\beta$-modules are an example of generalized multicategories relative to a taut monad, is not due to us either; there’s a whole group of people (Clementino, Hofmann, Tholen, Seal) who’ve developed a theory of “generalized multicategories” of that sort, which we discuss more generally in B.6. What we were trying to do is find a general framework that includes both the CHTS sort of generalized multicategories and also the “Leinster-Burroni” sort that you described, as well as others.