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1. • CommentRowNumber1.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 16th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexCreated a new page [[generalized graph]] based on the definition given in Hackney, Robertson and Yau's recent book, which appears to be influenced at least in part by the paper of Kock cited on the page. As far as I can tell none of the other things on the nlab (e.g. [[quiver]] or [[graph]] or their associated sub-entries about pseudographs and so forth) deal with the case of the "exceptional cell." If the notion I describe on this page is already somewhere on the nlab, I'd be happy to know that and get rid of the page I made.

   Created a new page generalized graph based on the definition given in Hackney, Robertson and Yau’s recent book, which appears to be influenced at least in part by the paper of Kock cited on the page. As far as I can tell none of the other things on the nlab (e.g. quiver or graph or their associated sub-entries about pseudographs and so forth) deal with the case of the “exceptional cell.” If the notion I describe on this page is already somewhere on the nlab, I’d be happy to know that and get rid of the page I made.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 16th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexI'm not really following. Where is the output and input data in the definition? Which part is vertices and which part is edges?

   I’m not really following. Where is the output and input data in the definition? Which part is vertices and which part is edges?

3. • CommentRowNumber3.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 16th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexAh you know what. I need to beef up this definition a bit, since the "directedness" doesn't show up in this generality. But the vertices are given by the partition indices, and the flags being a given part of the partition are the ones that "attach" to that vertex.

   Ah you know what. I need to beef up this definition a bit, since the “directedness” doesn’t show up in this generality.

   But the vertices are given by the partition indices, and the flags being a given part of the partition are the ones that “attach” to that vertex.

4. • CommentRowNumber4.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 16th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexShould have worked on it quite a bit more before posting a link to it here. Doing that now.

   Should have worked on it quite a bit more before posting a link to it here. Doing that now.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 17th 2016
   • (edited Jul 17th 2016)
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   Author: Todd_Trimble
   Format: MarkdownItexWhat role does the infinite set $\mathcal{S}$ play? Also, I wouldn't call those "properties" of a generalized graph; those are [structures](https://ncatlab.org/nlab/show/stuff,+structure,+property). Are all the cells supposed to be nonempty, including the exceptional cell? (Which would be usual when speaking of a partition, but I'm just checking to make sure.) I feel as though I'd have something more to say later about this page.

   What role does the infinite set $\mathcal{S}$ play?

   Also, I wouldn’t call those “properties” of a generalized graph; those are structures.

   Are all the cells supposed to be nonempty, including the exceptional cell? (Which would be usual when speaking of a partition, but I’m just checking to make sure.)

   I feel as though I’d have something more to say later about this page.

6. • CommentRowNumber6.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 17th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexThe cells don't need to be non-empty. In fact there is, for instance, an empty graph. Perhaps I'm saying things incorrectly. I'm trying to not stray too far from Hackney, Robertson and Yau's work. The ultimate point of all this is that one can then define infinity properads, which generalize infinity categories and infinity operads, except that the indexing category, rather than being the [[simplex category]] or the [[tree category]], is a certain category of [[generalized graphs]].

   The cells don’t need to be non-empty. In fact there is, for instance, an empty graph. Perhaps I’m saying things incorrectly. I’m trying to not stray too far from Hackney, Robertson and Yau’s work. The ultimate point of all this is that one can then define infinity properads, which generalize infinity categories and infinity operads, except that the indexing category, rather than being the simplex category or the tree category, is a certain category of generalized graphs.

7. • CommentRowNumber7.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 17th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexOh, and the set $\mathcal{S}$ is just the set of names of half edges. One could easily use any infinite set.

   Oh, and the set $\mathcal{S}$ is just the set of names of half edges. One could easily use any infinite set.

8. • CommentRowNumber8.
   • CommentAuthorJon Beardsley
   • CommentTimeJul 17th 2016
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   Author: Jon Beardsley
   Format: MarkdownItexBut honestly, looking back at it, the set $\mathcal{S}$ really isn't necessary. That was just something I ported over directly from the book, but in retrospect there's no reason for it.

   But honestly, looking back at it, the set $\mathcal{S}$ really isn’t necessary. That was just something I ported over directly from the book, but in retrospect there’s no reason for it.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 17th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexRight. My only thought here is that maybe this $\mathcal{S}$ is similar to wanting to have a countable stock of names of variables. (In operadic or string-diagrammy pictures, it's sometimes useful to translate between diagrams and and more traditional syntactic terms where external edges are named by free variables and internal edges by bound variables. But taken on their own terms, the string diagrams don't need such labels; the edges can be left to name themselves.) I'm not really grasping yet the reason for the terminology "flags", "cells", etc., although I'm familiar with such terms from other contexts. Maybe the reason will emerge later, but I'd like to put a pin through that. In my impatience, I feel an urge to rewrite the definition in terms more familiar to me (which is something I can do in the privacy of my own home), but I do encourage avoiding term usages like "partition" here that actually clash with usage at e.g. [[partition]]. Part of the data seems just to be a map $s: F \to V + 1$ which maps a flag to the vertex it lies over, except that exceptional flags map to the summand $1$. (Leaving out mention of the involutions.) If I were any good at graphics, I'd try to help out by drawing up some pictures.

   Right. My only thought here is that maybe this $\mathcal{S}$ is similar to wanting to have a countable stock of names of variables. (In operadic or string-diagrammy pictures, it’s sometimes useful to translate between diagrams and and more traditional syntactic terms where external edges are named by free variables and internal edges by bound variables. But taken on their own terms, the string diagrams don’t need such labels; the edges can be left to name themselves.)

   I’m not really grasping yet the reason for the terminology “flags”, “cells”, etc., although I’m familiar with such terms from other contexts. Maybe the reason will emerge later, but I’d like to put a pin through that.

   In my impatience, I feel an urge to rewrite the definition in terms more familiar to me (which is something I can do in the privacy of my own home), but I do encourage avoiding term usages like “partition” here that actually clash with usage at e.g. partition. Part of the data seems just to be a map $s: F \to V + 1$ which maps a flag to the vertex it lies over, except that exceptional flags map to the summand $1$. (Leaving out mention of the involutions.) If I were any good at graphics, I’d try to help out by drawing up some pictures.

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2016
    • (edited Jul 17th 2016)
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    Author: Todd_Trimble
    Format: MarkdownItexOr maybe I do sort of get it? A flag is supposed to conjure perhaps a picture of a node and half-open edge sticking out from it? So that if two flags are paired by the involution, you're supposed to think of sticking them together (identifying the interiors of the half-open edges) to get a full-fledged edge with two endpoints. Or at least so in the unexceptional case.

    Or maybe I do sort of get it? A flag is supposed to conjure perhaps a picture of a node and half-open edge sticking out from it? So that if two flags are paired by the involution, you’re supposed to think of sticking them together (identifying the interiors of the half-open edges) to get a full-fledged edge with two endpoints. Or at least so in the unexceptional case.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 17th 2016
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    Author: Mike Shulman
    Format: MarkdownItexSince $F_\epsilon$ is finite, the condition $\iota F_\epsilon \subseteq F_\epsilon$ is equivalent to $\iota F_\epsilon = F_\epsilon$, right? So now the structure induced on the finite set $F_\epsilon$ is an involution $\iota$, together with a further free involution $\pi$ on the $\iota$-fixed points. That seems to me to be the same as giving one free involution, $\iota'$ say, defined by $\iota'(x) = \iota(x)$ if $\iota(x)\neq x$ and $\iota'(x) = \pi(x)$ if $\iota(x)=x$, together with a subset of $F_\epsilon$ closed under $\iota'$ (the $\iota$-fixed points). It also seems to me that we could get rid of the other $\iota$-fixed-points by adding new flags in $F_\epsilon$ to be their images; the previous $F_\epsilon$ would then simply be the "loops" therein. Then the entire involution $\iota$ would be free, and so we could simply pass to its quotient to obtain a set $E$ of edges, with a function assigning to each edge the 1- or 2-element set consisting of its vertices. In other words, it looks to me as though a "generalized graph" is just 1. An ordinary undirected graph (allowing loops and multiple parallel edges) 1. with a specified vertex $\epsilon$ called "exceptional", and 1. a specified subset of the loops at $\epsilon$ called "exceptional legs". Is this true? If so, is there a reason this way of presenting it is less useful? I find it much easier to visualize. (And in any case, I don't understand the role of the "exceptional legs".)

    Since $F_\epsilon$ is finite, the condition $\iota F_\epsilon \subseteq F_\epsilon$ is equivalent to $\iota F_\epsilon = F_\epsilon$, right? So now the structure induced on the finite set $F_\epsilon$ is an involution $\iota$, together with a further free involution $\pi$ on the $\iota$-fixed points. That seems to me to be the same as giving one free involution, $\iota'$ say, defined by $\iota'(x) = \iota(x)$ if $\iota(x)\neq x$ and $\iota'(x) = \pi(x)$ if $\iota(x)=x$, together with a subset of $F_\epsilon$ closed under $\iota'$ (the $\iota$-fixed points).

    It also seems to me that we could get rid of the other $\iota$-fixed-points by adding new flags in $F_\epsilon$ to be their images; the previous $F_\epsilon$ would then simply be the “loops” therein. Then the entire involution $\iota$ would be free, and so we could simply pass to its quotient to obtain a set $E$ of edges, with a function assigning to each edge the 1- or 2-element set consisting of its vertices.

    In other words, it looks to me as though a “generalized graph” is just

    1. An ordinary undirected graph (allowing loops and multiple parallel edges)
    2. with a specified vertex $\epsilon$ called “exceptional”, and
    3. a specified subset of the loops at $\epsilon$ called “exceptional legs”.

    Is this true? If so, is there a reason this way of presenting it is less useful? I find it much easier to visualize. (And in any case, I don’t understand the role of the “exceptional legs”.)

12. • CommentRowNumber12.
    • CommentAuthorJon Beardsley
    • CommentTimeJul 17th 2016
    • (edited Jul 17th 2016)
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    Author: Jon Beardsley
    Format: MarkdownItexYeah, so there are definitely other ways to define this type of stuff. I'm not personally attached to this way of doing it. Basically I want to write up these basic things so I can extend the dictionary of segal spaces giving infinity categories and segal dendroidal spaces giving infinity operads to segal graphical spaces giving infinity properads.

    Yeah, so there are definitely other ways to define this type of stuff. I’m not personally attached to this way of doing it. Basically I want to write up these basic things so I can extend the dictionary of segal spaces giving infinity categories and segal dendroidal spaces giving infinity operads to segal graphical spaces giving infinity properads.

13. • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2016
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    Author: Todd_Trimble
    Format: MarkdownItexDo we have by the way some discussion (somewhere in the nLab) of Joachim Kock's nice work on trees and polynomial endofunctors? That could help illuminate some of what is going on here...

    Do we have by the way some discussion (somewhere in the nLab) of Joachim Kock’s nice work on trees and polynomial endofunctors? That could help illuminate some of what is going on here…

14. • CommentRowNumber14.
    • CommentAuthorJon Beardsley
    • CommentTimeJul 17th 2016
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    Author: Jon Beardsley
    Format: MarkdownItexIn response to Mike's comments also, I am way out of my depth in being able to respond to these questions. I trust your analysis of the redundancies in the definition. I can give a reason for the necessity of exceptional edges, but am less clear on exceptional legs myself.

    In response to Mike’s comments also, I am way out of my depth in being able to respond to these questions. I trust your analysis of the redundancies in the definition. I can give a reason for the necessity of exceptional edges, but am less clear on exceptional legs myself.

15. • CommentRowNumber15.
    • CommentAuthorJon Beardsley
    • CommentTimeJul 17th 2016
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    Author: Jon Beardsley
    Format: MarkdownItexUltimately, one is going to want to have a ``graphical category" (which is frustratingly denoted $\Gamma$ in HRY) in which the morphisms are basically graph substitution. That way functors $\Gamma\to Set$ or $\Gamma\to sSet$ will define properads or simplicial properads. One definitely needs an exceptional edge here when one gets around to defining things like inner face maps and substitutions products and so forth. An ``exceptional leg" is basically just a half edge with no vertices (I think of it as either being the tail or the head of an arrow). I guess the main idea is that this is useful in gluing generalized graphs, since you need to glue them along things, so if you want to do this as a pushout, you need be able to pick out legs.

    Ultimately, one is going to want to have a “graphical category” (which is frustratingly denoted $\Gamma$ in HRY) in which the morphisms are basically graph substitution. That way functors $\Gamma\to Set$ or $\Gamma\to sSet$ will define properads or simplicial properads. One definitely needs an exceptional edge here when one gets around to defining things like inner face maps and substitutions products and so forth.

    An “exceptional leg” is basically just a half edge with no vertices (I think of it as either being the tail or the head of an arrow). I guess the main idea is that this is useful in gluing generalized graphs, since you need to glue them along things, so if you want to do this as a pushout, you need be able to pick out legs.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2016
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    Author: Urs
    Format: MarkdownItexCould you say again how this is unifying (if it is?) simplicial sets with [[dendroidal sets]] etc.? Maybe it would be good to have a remark on this in the entry.

    Could you say again how this is unifying (if it is?) simplicial sets with dendroidal sets etc.? Maybe it would be good to have a remark on this in the entry.

17. • CommentRowNumber17.
    • CommentAuthorJon Beardsley
    • CommentTimeJul 19th 2016
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    Author: Jon Beardsley
    Format: MarkdownItexOh sure. The idea is that there is a series of inclusions $\Delta\hookrightarrow \Omega\hookrightarrow \Gamma$, where $\Gamma$ is basically the category of [[wheeled graph|wheel-free graphs]] with morphisms generated by certain face and degeneracy maps (which restrict to the usual ones on $\Delta$ and $\Omega$). Then there are three categories: [[simplicial sets]], [[dendroidal sets]] and [[graphical sets]] with the relevant adjunctions between them. The analogy goes much further. For instance, you can define inner horns in graphical sets that restrict to inner horns in simplicial sets (and the inner horns of dendroidal sets defined by Cisinski and Moerdijk). By enriching everything, one can construct simplicial model category structures which present: $\infty$-categories, $\infty$-operads, and $\infty$-properads. The point of properads of course is that they are less general than PROPs (hence easier to work with) but that pretty much every interesting example of a PROP is in fact a properad (e.g. bialgebras, bialgebras with (co)modules, Lie bialgebras).

    Oh sure. The idea is that there is a series of inclusions $\Delta\hookrightarrow \Omega\hookrightarrow \Gamma$, where $\Gamma$ is basically the category of wheel-free graphs with morphisms generated by certain face and degeneracy maps (which restrict to the usual ones on $\Delta$ and $\Omega$). Then there are three categories: simplicial sets, dendroidal sets and graphical sets with the relevant adjunctions between them. The analogy goes much further. For instance, you can define inner horns in graphical sets that restrict to inner horns in simplicial sets (and the inner horns of dendroidal sets defined by Cisinski and Moerdijk). By enriching everything, one can construct simplicial model category structures which present: $\infty$-categories, $\infty$-operads, and $\infty$-properads. The point of properads of course is that they are less general than PROPs (hence easier to work with) but that pretty much every interesting example of a PROP is in fact a properad (e.g. bialgebras, bialgebras with (co)modules, Lie bialgebras).

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2016
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    Author: Urs
    Format: MarkdownItexGreat, thanks. Maybe you could find the time to add this paragraph at a suitable place on the $n$Lab and cross-link with _[[model structure on dendroidal sets]]_? That would be useful

    Great, thanks. Maybe you could find the time to add this paragraph at a suitable place on the $n$Lab and cross-link with model structure on dendroidal sets? That would be useful

19. • CommentRowNumber19.
    • CommentAuthorJon Beardsley
    • CommentTimeJul 19th 2016
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    Author: Jon Beardsley
    Format: MarkdownItexYes definitely. I'm still trying to figure out how exactly to structure all of this. I think the next step is to create a page for the graphical category, and that's where the above paragraph would probably go. There are some slight intimations of that kind of structure on the [[wheeled graph]] page, under the part where I define all the different types of wheeled graphs, but I'll me more explicit on this other page. The only issue with putting together the graphical category page is that I have to explain the morphisms in it, which basically come from graph substitution (but only certain kinds of graph substitution), which makes it a bit unwieldy to write out.

    Yes definitely. I’m still trying to figure out how exactly to structure all of this. I think the next step is to create a page for the graphical category, and that’s where the above paragraph would probably go. There are some slight intimations of that kind of structure on the wheeled graph page, under the part where I define all the different types of wheeled graphs, but I’ll me more explicit on this other page. The only issue with putting together the graphical category page is that I have to explain the morphisms in it, which basically come from graph substitution (but only certain kinds of graph substitution), which makes it a bit unwieldy to write out.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJul 20th 2016
    • (edited Jul 20th 2016)
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    Author: Mike Shulman
    Format: MarkdownItex> pretty much every interesting example of a PROP is in fact a properad (e.g. bialgebras, bialgebras with (co)modules, Lie bialgebras). I assume you mean something like "most small props that present 'theories' can in fact also be presented by a properad". I don't think a prop can "be" a properad since it has more operations, except in the sense you don't mean that every prop has an underlying properad. Also I think there are lots of (colored) props that are not in any sense properads, e.g. the underlying (colored) prop of a symmetric monoidal category. (None of which detracts from your real point, of course, I'm just being pedantic (-: except that I do care about props of the latter sort myself.)

    pretty much every interesting example of a PROP is in fact a properad (e.g. bialgebras, bialgebras with (co)modules, Lie bialgebras).

    I assume you mean something like “most small props that present ’theories’ can in fact also be presented by a properad”. I don’t think a prop can “be” a properad since it has more operations, except in the sense you don’t mean that every prop has an underlying properad. Also I think there are lots of (colored) props that are not in any sense properads, e.g. the underlying (colored) prop of a symmetric monoidal category.

    (None of which detracts from your real point, of course, I’m just being pedantic (-: except that I do care about props of the latter sort myself.)