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• CommentRowNumber1.
• CommentAuthorHarry Gindi
• CommentTimeFeb 12th 2011
• (edited Feb 12th 2011)

Interestingly enough, I was reading the introduction to the Como, Italy proceedings, and I noticed that Lawvere also calls the category of presheaves of sets on the category $\bullet \rightrightarrows \bullet$quivers”, which seems like the correct term based on the standard definitions. Is there any reason why the lab has reverted to the old, (traditionally) incorrect, terminology of “directed graphs” since we last discussed this a year or so ago? In particular, the current nLab page quiver refers to the free category on a quiver. I’m particularly bothered by the way that the current page asserts something that is demonstrably false (namely that these presheaves are only called quivers when considering their representations) (in this case demonstrated by Lawvere’s essay). Directed graphs are quivers satisfying the property that the source and target map are jointly injective.

At the very least, I think that it’s definitely necessary to remove the outright falsehoods on the page quiver, but what I would really prefer to do is change directed graph to quiver, quiver merged into free category, and make a new page directed graph that gives the more traditional definition.

If the last part is unacceptable, then maybe we could replace directed graph by diagram scheme, which is also a standard term.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeFeb 12th 2011

It is actually pretty common in category theory circles to use “directed graph” for what you want to call a quiver; the nLab is not breaking new ground there. Yes, it’s different from what many graph-theorists mean, but it’s a fact. The current page directed graph points this out and refers to graph, which I think is sufficient; there aren’t many graph-theorists working on the nLab at the moment. And this is the first I’ve heard of anyone using “quiver” for a directed (multi)graph when not considering the free category it generates. But I’d be perfectly happy to use “quiver” to mean the same as directed (multi)graph (as opposed to the free category on such).

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

I don’t think that anything’s been reverted; it’s just that the ideas resolved in that discussion were never implemented.

• CommentRowNumber4.
• CommentAuthorHarry Gindi
• CommentTimeFeb 13th 2011

@Mike: You’re forgetting something: There are two left adjoints that are applied at the same time: First, we give a map of a quiver into the underlying quiver of the category of complex vector spaces. By free-forgetful-nonsense, this gives us a functor from the free category on the quiver into the category of complex vector spaces. Then finally, by a generalization of the monoid algebra, we get the Path algebra/quiver algebra. The reason you don’t see representation theorists using them without a free category structure is because representation theory is all about mapping the object in question (quiver, group, etc) into some category of vector spaces or modules or wherever. However, a map from a quiver into a category always extends to a category.

@Toby: I thought I remembered some substantial changes made. I could be wrong!

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

Incidentally to all: The previous discussion was here (where some even earlier discussion is also cited).

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

I have moved DiGraph to Quiv and changed some appearances of “digraph” in the Lab to “quiver”. In particular this allowed me to remove some discussion at codiscrete groupoid.

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

I’ve moved the definition from directed graph to quiver. The latter then had several places where the same terminological points were made, and I consolidated them, but only partially.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeFeb 13th 2011

The previous discussion quickly got diverted into other directions, so I think I didn’t notice that some people had concluded we should avoid using “directed graph” when we mean to allow parallel edges and loops and use “quiver” instead. I think that’s silly; “directed graph” in this sense is used in lots of places in category theory and I don’t see any reason to change it. I would rather that quiver focus more on the representation-theoretic aspect, since that’s where that terminology seems to be used, with directed graph for the more common category-theoretic uses, and a remark to see graph for the graph-theoretic use. Since, as I said, there are not many graph-theorists working on the nLab.

• CommentRowNumber9.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

I don’t mind using “directed graph” to mean a quiver (directed multipseudograph, whatever). But it is ambiguous, whereas “quiver” is not. So it would be handy if the links, at least, to go to quiver (via [[quiver|directed graph]], if appropriate), although I often went further and wrote something like “directed graph (or quiver)” in the pages that I found.

And I thought that we established that “quiver” is not restricted to representation-theoretic usage. So I don’t see why we should let quiver focus on the representation theory; if we want a page that focuses on that, surely we would put it at representation of a quiver or quiver representation? (These currently redirect to representation.)

I added something about identifying quivers with their free categories to quiver. I’m not sure what to do with the discussions at directed graph and quiver.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeFeb 14th 2011

Ok, sure, we can have it all on one page. But are there any links on the nLab to directed graph which are really about the graph-theorists’ version? If not, why not just make directed graph redirect to quiver?

• CommentRowNumber11.
• CommentAuthorTobyBartels
• CommentTimeFeb 14th 2011

are there any links on the nLab to directed graph which are really about the graph-theorists’ version?

Yes. However, none of the links seem to be about simple directed graphs in general; instead, they point out that something is a directed graph, and of course that something is always also a quiver.

• CommentRowNumber12.
• CommentAuthorHarry Gindi
• CommentTimeFeb 14th 2011
• (edited Feb 15th 2011)

For what it’s worth, I think that the term quiver for a presheaf on $\bullet \rightrightarrows\bullet$ is an extremely apt comparison, what with it being the weakest type of structure in which we can encode objects who have a “front end” and a “back end”.

Or more topos-theoretically, they’re “arrow-shaped”sets, or sets whose elements are arrow-shaped.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeFeb 15th 2011

Okay, I concede – “quiver” it is. We should make sure that all the links go to the right place, and maybe get rid of the leftover discussion at directed graph.

I would also like to merge omega-graph and directed n-graph into globular set, and implement the suggestion of Marc at the latter that the default notion should be the non-reflexive one. If no one objects, maybe I’ll have some time to do that in a bit.

• CommentRowNumber14.
• CommentAuthorHarry Gindi
• CommentTimeFeb 15th 2011

You’ll also have to change the definition of the globe category, I think.

• CommentRowNumber15.
• CommentAuthorTobyBartels
• CommentTimeFeb 15th 2011

No opinion on the stuff about directed n-graph, but I think that I’ve already fixed all of the current links. While there are still several links to directed graph, they occur in contexts where that term is appropriate and there is (if applicable, which is usually the case) also a link to quiver. Many of these could be further edited to say (and link to) only “quiver”, but I don’t think that this is necessary.

• CommentRowNumber16.
• CommentAuthorTobyBartels
• CommentTimeFeb 15th 2011

I’ve removed the old discussion and noted this at a new forum post, where they may more easily be found.

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

Since nobody objected, I implemented my suggestion at #13. I also added some remarks about n-globular sets.

• CommentRowNumber18.
• CommentAuthorzskoda
• CommentTimeMar 12th 2011
• (edited Mar 12th 2011)

In mirror symmetry and some parts of representation theory people also talk about quivers with relations, and sometimes just shortly say quiver for a quiver with relations, i.e. a pair $(Q,R)$ of a quiver $Q$ and a set of relations $R$ on the path category of the quiver.

• CommentRowNumber19.
• CommentAuthorHarry Gindi
• CommentTimeMar 12th 2011
• (edited Mar 12th 2011)

@Zoran: That agrees with the idea of a presentation of a group, where you give a free generating set and a set of relations on the free group generated by the set.

• CommentRowNumber20.
• CommentAuthorzskoda
• CommentTimeMar 13th 2011

What do you want to say, “agree” ? We are discussing terminology on what we call a quiver of some kind ? Is it that some group theorists call group presentations “quivers with relations” ?

• CommentRowNumber21.
• CommentAuthorTobyBartels
• CommentTimeMar 13th 2011

Harry’s drawing an analogy, in which group theorists would say “sets with relations” instead. (However, they don’t actually say that; perhaps they should.)

• CommentRowNumber22.
• CommentAuthorHarry Gindi
• CommentTimeMar 13th 2011

I meant something like “a presentation for a group consists of a set and a set of relations on its free group”, so a presentation for a category (or other algebraic quiver-based structure) should be given by a quiver and a family of relations on its free category.

• CommentRowNumber23.
• CommentAuthorzskoda
• CommentTimeMar 14th 2011

It is well known what is a presentation of a category. We were talking here terminology quivers vs. various kinds of graphs, and how extensive is the notion of a quiver. There were opinions at the beginning that a quiver is a free category more or less, though I object – terminology distinguished quiver from a path category on a quiver – but people also label the cases with relations as sort of quiver.

• CommentRowNumber24.
• CommentAuthorHarry Gindi
• CommentTimeMar 14th 2011
• (edited Mar 14th 2011)

Sure, but I mean, you would just say something like “a presentation for a category (or other quiver-based structure, although I can’t think of one offhand) is given by the data of a quiver with relations (on its associated path category)”.

Anyway, it’s really of no consequence, since on whichever lab page, we could say something like

A presentation for a category, sometimes called a quiver with relations in representation theory, comprises the data of a quiver and a family of relations on its path category.

• CommentRowNumber25.
• CommentAuthorzskoda
• CommentTimeMar 15th 2011

There are some cases of big categories (not even essentially small) which allow for presentations, while the quivers are always with small set of objects and small set of arrows in between.

• CommentRowNumber26.
• CommentAuthorMike Shulman
• CommentTimeMar 15th 2011

I think you can’t mean what it sounded to me like you meant. If I start with a small quiver, then the free category it generates is small, and any quotient of that free category will also be small.

• CommentRowNumber27.
• CommentAuthorzskoda
• CommentTimeMar 15th 2011

I said that presentation of categories is somewhat more general than presentation of quivers. Harry defines a presentation of a category by means of quivers, what, in standard conventions, assumes smallness, hence does not mean the general case of presentation of categories.

• CommentRowNumber28.
• CommentAuthorMike Shulman
• CommentTimeMar 15th 2011

Okay. But a large quiver can present a large category just as well as a small quiver can present a small category, right?

• CommentRowNumber29.
• CommentAuthorzskoda
• CommentTimeMar 15th 2011

I do not think that “large quiver” is a used term, unless we play Bourbaki. The quiver literature rarely goes beyond th finite case, and I know of no usages of the terminology with uncountable infinity of vertices, or a fortiori a proper class. I mean the term quiver is a specific term in one area of applications and hence they have a special terminology. Large graph is however widely used term in category theory.

• CommentRowNumber30.
• CommentAuthorMike Shulman
• CommentTimeMar 15th 2011

Well, even if no one’s ever said “large quiver” before, it has an obvious meaning.

• CommentRowNumber31.
• CommentAuthorzskoda
• CommentTimeMar 15th 2011
• (edited Mar 15th 2011)

Right, of course, but the point of quiver entry is in my understanding to record that some subfield of math has an alternative terminology with its conventions and uses. Most classical sources in category like literature just say “graph”, unadorned. With having large graphs and small graphs: the latter being internal graphs in Set. Entries like quiver need attention just because of alternative communities using this terminology. Main uses being from representation theory and mathematical physics. I would never say quiver and never need to say quiver if there were not those people. Of course, I do not mind talking large quivers in discussions among us, but I do not think it is a good pedagogical and normative effort to loose completely the spirit and inclination that the quiver terminology is pretty much tied to specific parts of mathematics. From logical point of view, there is nothing wrong in doing it, of course.

• CommentRowNumber32.
• CommentAuthorTobyBartels
• CommentTimeMar 15th 2011

The phrase “large quiver”, should we ever have cause to use it, shouldn’t be seen as a new term. This is because “large” is an adjective that can be applied to any noun, as in “The surreal numbers form a large field.”, etc. Since the meaning might be unclear from context, we write “large quiver” (although one of those links isn’t satisfied yet).

• CommentRowNumber33.
• CommentAuthorMike Shulman
• CommentTimeMar 15th 2011

Thanks Toby, that’s what I was trying to say.

• CommentRowNumber34.
• CommentAuthorEric
• CommentTimeJan 26th 2012

Hi,

I missed this discussion, but just looked up directed n-graph and see it now redirects to globular set. I think of a directed n-graph as different than a globular set. Would it be better to redirect to computad or polygraph?

• CommentRowNumber35.
• CommentAuthorMike Shulman
• CommentTimeJan 26th 2012

Personally, I would find it surprising for “directed n-graph” to mean “computad” rather than n-globular set. Actually, why do we need a term like “directed n-graph” at all, when we have both “globular set” and “computad” with established meanings?

• CommentRowNumber36.
• CommentAuthorTodd_Trimble
• CommentTimeJan 26th 2012

Yes, what is the meaning of directed $n$-graph, if not an $n$-globular set?

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeJan 26th 2012

One could argue that also simplicial sets and cubical sets can be thought of as higher directed graphs.

• CommentRowNumber38.
• CommentAuthorTodd_Trimble
• CommentTimeJan 26th 2012

Urs: yes, but Eric said an directed $n$-graph is different from an $n$-globular set – so I wanted to see what exact notion he meant.

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeJan 26th 2012
• (edited Jan 26th 2012)

I don’t know for sure, but unless somebody hijacked Eric’s account, I am sensing a disposition towards the cubical.

• CommentRowNumber40.
• CommentAuthorEric
• CommentTimeJan 27th 2012

Urs can usually express what I mean better than I can :)

At some point, I wrote a page directed n-graph that explained the best I could (which probably wasn’t very good) what I meant by directed $n$-graph. I had a reason recently to start thinking about this again and went to remind myself what I wrote only to find myself redirected to globular set. While trying to figure out what happened, I came across this thread. Any idea how I might be able to access that old material?

One reason I don’t like being redirected to globular set may be related to the fact I still think of directed graph as a diagram scheme (like MacLane does), i.e. a functor from the category with two objects and two parallel non-identity morphisms to $\mathbf{Set}$, whereas the nLab seems to have chosen to make directed graph and quiver synonymous.

So, in my opinion, directed n-graph should be something like a higher diagram scheme and, as such, maybe it should not redirect anywhere and should be a separate page.

Maybe we (I would if I could) could create a separate page diagram scheme which covered what might be called higher diagram schemes, redirect directed n-graph there, and add a reference to it from directed graph.

While I’m on the subject…

I was wondering if instead of a set $E$ of “edges” and a set $V$ of “vertices” and two functions $s,t: E\to V$, if it would make sense to just define a set $D$ of directed cells and two functions $s,t: D\to D$?

Unless there is some problem I haven’t thought of, I’d be tempted to call this a directed $n$-graph or higher diagram scheme.

The idea is that a directed $n$-cell would have source and target $(n-1)$-cells, which would themselves have source and target $(n-2)$-cells etc.

This probably expresses what I mean the best I possibly can.

Urs,

What I’m ultimately trying to do is define a discrete calculus on a discrete bundle (where there is likely a name clash here as what I mean by “discrete bundle” is probably different than a standard term, if one exists). Motivated by bundle, I thought one way to do that is to define a category of discrete calculi and bundle is just a morphism $E\to B$.

A directed graph generates a discrete calculus (and a discrete calculus often implies a directed graph). I was hoping to formalize this idea.

This is coming about at the moment as a result of a challenge by John here.

• CommentRowNumber41.
• CommentAuthorMike Shulman
• CommentTimeJan 27th 2012

@Eric: The old version is here. I thought we concluded on the discussion at that page that what you really wanted was a computad. I didn’t realize that meant that you wanted “directed n-graph” to mean computad! Why not just say “computad” if that’s what you want to say?

I still think of directed graph as … a functor from the category with two objects and two parallel non-identity morphisms to Set

Exactly how does that differ from a quiver?

• CommentRowNumber42.
• CommentAuthorEric
• CommentTimeJan 27th 2012

@Mike 41: Thanks for the link. The old discussion on that page was also helpful to jog my memory. However, the definition I think of is the first one here. That definition survived until I added a bit about identity assigning maps in revision 10. It wasn’t until revision 15, where identity morphisms were included in the definition (by me - I don’t recall why).

Exactly how does that differ from a quiver?

At the risk of repeating previous discussions, a quiver is a directed graph, but not all directed graphs are quivers. For example, consider the sets

$E = \{i\to j, j\to k\}$

and

$V = \{i,j,k\}.$

The source map is given by

$s(i\to j) = i \quad\text{and}\quad s(j\to k) = j$

and the target map is given by

$t(i\to j) = j\quad\text{and}\quad t(j\to k) = k.$

The collection $(E,V,s,t)$ is a perfectly fine directed graph according to revisions 1-12 (and MacLane), but it is not a quiver. However, from this directed graph we can generate a quiver by filling in identity edges and composite edges resulting in the following edge set:

$E' = \{i\to i, j\to j, i\to j,j\to k, i\to k\}.$

The collection $(E',V,s,t)$ is a quiver.

To avoid clashes, we could create a page for diagram scheme and on directed graph mention “Some people refer to diagram schemes as directed graphs.”

But this gets me back to my question in 40 above…

I’m wondering if we can glob $E$ and $V$ into a single set

$D = \{i,j,k,i\to j,j\to k\}$

and define source and target maps $s,t:D\to D$ with $s(i) = s(j) = s(k) = \emptyset$ or something.

• CommentRowNumber43.
• CommentAuthorTodd_Trimble
• CommentTimeJan 27th 2012
• (edited Jan 27th 2012)

Eric, it sounds as if you’re identifying a quiver with the free category it generates. The page on quiver is unequivocal in saying that it’s the same as what category theorists (but not graph theorists) call a directed graph. That page has a section on “identifying a quiver with its free category”, but the point was that this can get you into trouble if you’re not careful, and it’s not really kosher.

Here’s the deal; it has to do with mathematical subcultures. Whenever you hear big-shots talking about quivers, it’s almost always in the context of so-called quiver representations, which are, in nLab parlance, functors from the free category on the quiver to some category (very often some linear category like the category of vector spaces). So if you find yourself listening in on some such conversation about representations of quivers, it pays to translate by saying, “Oh, they’re really discussing representations of the free category generated by the quiver”. But as a category theorist, you don’t allow yourself to get mixed up about this.

We did once have a very long discussion about this; I’m too lazy to look it up right now, but if memory serves, someone (Harry Gindi perhaps) didn’t like the cultural ambiguity of “directed graph” (which means one thing to category theorists, and a different thing to graph theorists), and was proposing we use quiver from now on, as that doesn’t have the same problem. Well, it sort of does have the same problem, because some people act as if they mean the free category generated by a quiver. But we needn’t get mixed up about this.

Edit: Oh, heck, the long discussion is right here! (See comment 1.)

• CommentRowNumber44.
• CommentAuthorEric
• CommentTimeJan 27th 2012
• (edited Jan 27th 2012)

Thanks Todd.

When I’m in front of a computer, I’ll read this thread from beginning to end, but yeah, I’m not sure how calling directed graphs quivers helps because I’ve always thought quiver meant “free category generated from a directed graph.”

I like Harry’s suggestion to use diagram scheme from 1, but maybe that was shot down in subsequent posts (?). Sorry for resurrecting this yet again.

• CommentRowNumber45.
• CommentAuthorEric
• CommentTimeJan 27th 2012
• (edited Jan 27th 2012)

Ok. I’ve slightly modified directed graph to make it a little more clear that quiver is the term we’re using for what someone studying category theory would usually think of when they hear directed graph. It is unfortunate the change was made given that the nLab should cater to category theorists, but c’est la vie.

PS: By the way, quiver is an awesome page. The only thing that would make it better would be if “quiver” were replaced with “directed graph”.

• CommentRowNumber46.
• CommentAuthorTodd_Trimble
• CommentTimeJan 27th 2012

Eric #45: I agree with you that “directed graph” ought to be fine. (I don’t like “digraph”, because that sounds like something a graph theorist might say.) I myself wasn’t completely happy with the idea to have directed graph be only for the meaning graph theorists give it, and I think “quiver” has its own problems, but no solution is perfect.

• CommentRowNumber47.
• CommentAuthorEric
• CommentTimeJan 27th 2012

@Mike #41: Sorry. I replied to your question thinking I knew what a quiver was, i.e. a free category generated by a directed graph, but now see that what is described at quiver is precisely what I meant by directed graph. Sorry.

• CommentRowNumber48.
• CommentAuthorMike Shulman
• CommentTimeJan 27th 2012

@Eric: That’s ok. It’s not your fault you’re confused; the page quiver used to say that a quiver is the free category on a directed graph. I think this was an innovation due to John Baez who originally created the page quiver. Eventually the rest of us decided to conform to the usage of “quiver” in the rest of the world; I think that was part of this very discussion.

Also, as you can see from the earlier parts of this discussion, I also wanted to keep “directed graph” with the category-theorists’ meaning. But I don’t object so much to using “quiver” instead, since it is (now) completely unambiguous.

• CommentRowNumber49.
• CommentAuthorEric
• CommentTimeJan 28th 2012

Now that I’m less confused about terminology, this gets back to the reason for me to resurrect this thread in the first place.

I’m interested in what I thought should be call directed $n$-graphs, which, to be consistent with the nLab, should now be $n$-quivers or $n$-diagram schemes. What would be a good definition for $n$-quiver? I would like to record Urs’ comment somewhere:

## Idea

A directed $n$-graph, or $n$-digraph or $n$-quiver, is a higher dimensional generalization of a quiver (a category theorist's digraph) with $r$-dimensional edges spanning $(r-1)$-dimensional vertices.

A directed $n$-graph is like an n-category with units and composition forgotten. Indeed, an $n$-category is a directed $n$-graph with extra structure. To formalize this idea, we say there is a forgetful functor

$U : n\Cat \to n\DiGraph$

where $n\DiGraph$ is the category of directed $n$-graphs and $n\Cat$ is the category of small $n$-categories. Moreover, this forgetful functor has a left adjoint

$F : n\DiGraph \to n\Cat$

sending each directed $n$-graph to the free $n$-category on that $n$-graph. A free $n$-category on an $n$-graph is called an $n$-quiver.

Maybe we should resurrect that page and rename it to n-quiver. For example, we could change the above to:

## Idea

An $n$-quiver or $n$-diagram scheme is a higher dimensional generalization of a quiver with $r$-dimensional edges spanning $(r-1)$-dimensional vertices.

An $n$-quiver is like an n-category with units and composition forgotten. Indeed, an $n$-category is an $n$-quiver with extra structure. To formalize this idea, we say there is a forgetful functor

$U : n\Cat \to n\Quiver$

where $n\Quiver$ is the category of $n$-quivers and $n\Cat$ is the category of small $n$-categories. Moreover, this forgetful functor has a left adjoint

$F : n\Quiver \to n\Cat$

sending each directed $n$-quiver to the free $n$-category on that $n$-quiver.

Ack. But I see n-quiver currently redirects to globular set. Is that what we want? A globular set is definitely not what I would think of as an n-diagram scheme. Am I just not thinking about it properly?

I’m a little stuck because the ($n$-)quivers I work with, by definition, do not have intermediate edges, i.e. if the quiver contains $i\to j$ and $j\to k$, it explicitly does not contain a directed edge $i\to k$, which it would seem to me precludes globular sets.

• CommentRowNumber50.
• CommentAuthorTodd_Trimble
• CommentTimeJan 28th 2012

Eric: the Ideas in #49 don’t help me see what pictures are in your head. Notice that there are a number of “competing” definitions of $n$-category. Some are globular sets with extra structure, some are based on simplicial sets, others on other types of shape like opetopes. Unless you tell what notion of $n$-category you have in mind, it’s hard to guess what the underlying $n$-quivers you have in mind are supposed to look like.

(IMO, it’s better to use the idea section to describe the geometric shapes one is picturing – the stuff about forgetful functors and their left adjoints and whatnot is putting a cart before a horse, because $n$-quivers are presumably simpler than whatever idea of $n$-category one has in mind – it’s probably not a good idea to describe some something in terms of a more complicated something else.)

But anyway, you seem to have a wrong impression of globular set. What intermediate edges?

The cells of a globular set look like “globes” or $n$-disks. That is, each $n$-cell has a domain $(n-1)$-cell (like the earth has a northern hemisphere) and a codomain $n$-cell. There are natural-looking axioms that say the domain of the domain of an $n$-cell equals the domain of the codomain, and the codomain of the domain equals the codomain of the codomain – if one knows what 2-cells in an ordinary 2-category look like, this will seem like an obvious axiom.

I don’t see how you’re getting intermediate edges out of this.

• CommentRowNumber51.
• CommentAuthorEric
• CommentTimeJan 28th 2012
• (edited Jan 28th 2012)

Hi Todd,

The shape I have in mind at the moment is fairly simple. I call it an $n$-diamond, but it is basically a special kind of directed $n$-cube. An example of a 3-diamond is illustrated below:

Globbing $n$-diamonds together gives a kind of directed space with time flowing along the major diagonal.

Recently, I’ve been working on a series of articles applying the stuff from our paper to practical problems following John’s lead:

In any quiver algebra associated with a quiver $(G_1,G_0,s,t:G_1\to G_0)$, there is a special element which is the simple sum of all directed edges

$\mathbf{G} = \sum_{G_1} (i\to j).$

If there are no intermediate edges in the quiver, you can define a (graded) differential on the quiver algebra

$d f = [\mathbf{G},f],$

where $[ , ]$ is a graded commutator and the grading is given by the length of the path in the quiver algebra. Applying this twice, we get

$d^2 f = [\mathbf{G}^2,f].$

This generates a differential graded algebra by imposing the relation $\mathbf{G}^2 = 0$ in the quiver algebra. Imposing this relation can be interpreted geometrically as promoting paths of length 2 to 2-dimensional cells, i.e. imposing this relation means the sum of paths of length 2 whose source and target vertices coincide must vanish. For a square bounded by the paths $(i\to j\to k)$ and $(i\to j'\to k)$ in the example above, we have

$(i\to j\to k) = -(i\to j'\to k).$

In other words, the relation $\mathbf{G}^2 = 0$ induces an orientation on higher dimensional cells in the $n$-diamond.

John’s latest article

inspired me to try to formulate classical mechanics on an $n$-diamond. To do this, John nudged me in the following direction:

Hi! Yes, I’ve been following your blog posts, but pretty distracted by other things.

It would be good if you could discretize the concept of “cotangent bundle” and the “tautological 1-form” on this bundle. Here’s a nice coordinate-free formula for the tautological 1-form.

For any manifold $X$ there’s a map

$p: T^* X \to X$

sending any cotangent vector $\mu$ at the point $x \in X$ to the point $x$. Differentiating this we get a map sending tangent vectors on $T^* X$ to tangent vectors on $X$:

$d p: T(T^* X) \to T X$

The tautological 1-form $\alpha$ on $T^* X$ eats any tangent vector $v$ at the point $\mu \in T^* X$ and gives a number as follows:

$\alpha(v) = \mu(d p(v))$

It’s mind-blowingly tautological, eh? In case your brain starts to melt, it may help to keep in mind.

$x \in X$ $\mu \in T^*_x X \subseteq T^* X$ $v \in T_\mu (T^* X) \subseteq T (T^* X)$

I only recently noticed the more appealing but equivalent formulation I gave in the blog article, but it was already in Wikipedia.

It’s a good exercise to see why the formula I just gave is equivalent to the tautological property of $\alpha$ that I gave in my blog article, and also to the explicit coordinate-ridden formula I gave for $\alpha$. For anyone who gets stuck, Wikipedia gives some hints.

I’m giving this challenge a try and fell back to bundle for ideas. From there, it seems I should define a category of DGAs on n-diamonds. I suppose I could do that directly in a “nuts and bolts” sort of way, but was trying to be clever (rarely a good idea for me!).

I hope it is clear enough that given a quiver $G$, we can generate a discrete DGA $\Omega(G)$ as outlined above, i.e. we have a map $G\to \Omega(G)$. It would be cool if given any DGA, there was an underlying quiver so there is a forgetful functor $U:\DGA\to \Quiver$. If there is no such thing, it would be cool if there was an underlying $n\Quiver$ for any DGA and a forgetful functor $U:DGA\to n\Quiver$. If so, then we can talk about the “free DGA on a(n $n)\Quiver$” and what I described above would seem much less mysterious (I think).

So I am more interested in $n$-quivers as they may (or may not) pertain to DGAs than $n$-categories, but I think it is all related and I’m still just basically thinking out loud…

Edit: After digging around a bit, I found there is absolutely a straightforward way (and I think I knew this at some point) to obtain a quiver from a DGA if $\Omega^0$ has a unit and if that unit can be written as a sum of primitive idempotents

$1 = \sum_i \mathbf{e}^i.$

The vertices of the quiver are the primitive idempotents and the directed edges are given by $\mathbf{e}^i \Omega^1 \mathbf{e}^j$ for all pairs of primitive idempotents. So starting with a DGA $\Omega$, you can construct a quiver $G$, and from that quiver, you can construct a DGA $\tilde\Omega(G)$. I wonder how well $\tilde\Omega(G)$ approximates $\Omega$?

• CommentRowNumber52.
• CommentAuthorTodd_Trimble
• CommentTimeJan 28th 2012
• (edited Jan 28th 2012)

Eric, getting back to #49 again, it seems that you said something stronger than the way I was interpreting you in #50. You said

I’m a little stuck because the (n-)quivers I work with, by definition, do not have intermediate edges, i.e. if the quiver contains $i \to j$ and $j \to k$, it explicitly does not contain a directed edge $i \to k$

Thus, the notion of 1-quiver you work with is not the same as the notion of directed graph in the category theorist’s sense, since directed graphs can have such triangles (not that they must have, which is what I thought you were claiming at first). I definitely wasn’t expecting a triangle-forbidding restriction. Is that what you want?

By that convention, you could not have an underlying 1-quiver of any category except the empty category. For if you take $i = j = k$ in a category and both $i \to j$ and $j \to k$ to be the identity arrow, you still have the identity arrow $i \to k$, which you said you want to forbid.

Even if you include the hypothesis that $i$, $j$, $k$ are distinct vertices, this still rules out being able to take the underlying 1-quiver of any but pretty trivial categories.

• CommentRowNumber53.
• CommentAuthorEric
• CommentTimeJan 28th 2012

Hi Todd,

A definition of $n$-quiver that allows composite $r$-edges is fine as long as it doesn’t require composite $r$-edges (and by saying this, I don’t mean to imply any current definition does require this, but see my concern below). The category theory definition of 1-quiver does not require composite triangles, so I have no complaints and I can simply think of my 1-quivers as “special”.

In my example above, I would like to think of a square as a directed 2-edge. This 2-edge is not bounded by two 1-edges, as it would be if it were a globular set. Rather, my 2-edge is bounded by four distinct 1-edges $(i\to j)$, $(j\to k)$, $(i\to j')$, and $(j'\to k)$ with no composites. I may have made my statement too strong because I was trying to argue against having $n$-quiver redirect to globular set because I think any definition of $n$-quiver should be general enough to accommodate my “diamonds” (at least). If we insist on having $n$-quiver be synonymous with globular set, then my 2-quiver would have to contain composite edges $(i\to j)\circ (j\to k)$ and $(i\to j')\circ(j'\to k)$ and the square would be a directed 2-edge $(i\to j')\circ(j'\to k)\Rightarrow (i\to j)\circ (j\to k)$. Alternatively, I could simply be forbidden from associating my diamonds with $n$-quivers, but that would be an unnecessary shame I think because diamonds are useful toys with many applications and make perfectly sensible higher diagram schemes.

As far as the “underlying 1-quiver of any category”, I’m not too concerned about that because looking at any such underlying quiver is not something I would do. Instead, I may start with a 1-quiver $G$ with no composite triangles and then generate the free path category $P G$. The path category $P G$ will contain paths of length 2 from $i$ to $k$, but no path of length 1 from $i$ to $k$. The length matters to me because the length of a path gets promoted to the dimension of an edge in my $n$-quiver when generating the DGA.

One thing I may like to do given a finite category would be to determine the smallest quiver that can reproduce the underlying quiver of that finite category by filling in identities and composites, but that is just a curiosity.

• CommentRowNumber54.
• CommentAuthorTodd_Trimble
• CommentTimeJan 28th 2012

Okay, I see better what you are after. Thanks.

Alternatively, I could simply be forbidden from associating my diamonds with $n$-quivers, but that would be an unnecessary shame I think because diamonds are useful toys with many applications and make perfectly sensible higher diagram schemes.

Of course they are perfectly sensible, and useful. Simplicial sets are also sensible and useful, in different ways. As are globular sets. They can all be taken as reasonable notions of diagram scheme; each is good for some (but not all) purposes.

I don’t know who introduced this term $n$-quiver to begin with (it’s not a term I’m familiar with from anywhere except this discussion), and I don’t know who decided it should mean the same thing as ($n$-)globular set. I don’t think we need an extra name for globular set; that seems a little silly to me! But at the same time, I don’t see any compelling reason that “$n$-quiver” should be commandeered to mean one of these cubical or diamond-like structures you fancy.

It sounds like what you might really be after is some combinatorial notion of diagram scheme which is flexible enough to handle all these cases: globular, simplicial, cubical, etc. There are a number of such notions out there: parity complex (Street), pasting scheme (Michael Johnson), directed complex (Steiner?), and there may be others I’m forgetting. Each of these can be used to generate $n$-categories. Is this the sort of thing you’re looking for?

But I don’t think we need to rename any of these $n$-quiver either, since they already have perfectly good names.

• CommentRowNumber55.
• CommentAuthorMike Shulman
• CommentTimeJan 28th 2012

if we insist on having n-quiver be synonymous with globular set, then my 2-quiver would have to contain composite edges

No, a globular set does not require any composite edges.

I agree with Todd that we don’t need new names for any existing shapes. Your picture in #51 suggests to me that maybe what you want is either a cubical set or a multisimplicial set.

• CommentRowNumber56.
• CommentAuthorEric
• CommentTimeJan 29th 2012

Thanks Todd.

Simplicial sets are also sensible and useful, in different ways. As are globular sets. They can all be taken as reasonable notions of diagram scheme; each is good for some (but not all) purposes.

I totally agree of course :)

I don’t think we need an extra name for globular set; that seems a little silly to me! But at the same time, I don’t see any compelling reason that “$n$-quiver” should be commandeered to mean one of these cubical or diamond-like structures you fancy.

Yeah. I agree. Just like I’m happy my quivers are special cases of quiver, I would be happy if my diamonds were a special cases of $n$-quiver, or whatever we end up calling it.

It sounds like what you might really be after is some combinatorial notion of diagram scheme which is flexible enough to handle all these cases: globular, simplicial, cubical, etc. …[snip]… Is this the sort of thing you’re looking for?

I suppose so. Yes :)

But I don’t think we need to rename any of these $n$-quiver either, since they already have perfectly good names.

Sure. There is no need to introduce new names for existing stuff. I’m not crazy about “quiver” or “$n$-quiver” either. I’m tempted to replace quiver with diagram scheme (as Harry suggested in comment 1 above) since that term does not seem to be as overloaded as either directed graph or quiver. I’m not sure that a definition of higher diagram scheme (aka $n$-quiver) needs to be able to generate $n$-categories, but should only describe diagrams of higher categories in the same way diagram schemes describe diagrams of categories. Because of this, higher diagram schemes may even be more general than those alternatives you listed.

I proposed one possible definition of higher diagram scheme above, i.e. a set $D$ with two functions $s,t:D\to D$ where sources and targets of vertices are the empty set. This has a slick definition (I think) as a functor from a category with one object $\bullet$ and two non-identity endomorphisms $s,t:\bullet\to\bullet$ to Set. I think this encompasses the usual definition of diagram scheme when we define $D = E\cup V$ and $s(v) = t(v) = \emptyset$ for all $v\in V$.

• CommentRowNumber57.
• CommentAuthorTobyBartels
• CommentTimeJan 29th 2012

I support making n-quiver and directed n-graph redirect to a new page (possibly called one of those or something like ‘pasting diagram $n$-scheme’ whatever) about general shapes of higher categories. We ought to have a page on that topic and, given the multiplicity of shapes, ‘$n$-quiver’ can’t really mean anything more specific.

• CommentRowNumber58.
• CommentAuthorTobyBartels
• CommentTimeJan 29th 2012

Note that the page directed graph is really a disambiguation page; it favours neither the graph theorists’ nor the category theorists’ terminology, directing readers to simple directed graph (which redirects to graph now) and quiver (respectively) instead.

• CommentRowNumber59.
• CommentAuthorTodd_Trimble
• CommentTimeJan 29th 2012
• (edited Jan 29th 2012)

We already have a page pasting diagram which contains the basic idea, and this could be expanded (and maybe renamed to, e.g., notions of pasting diagram). However, I can see such a page getting pretty big, unless it were to serve basically as a hub for pages on individual notions like parity complex, pasting scheme, etc. For now, I’ll see about doing a little editing of the idea section at pasting diagram.

Edit: The idea section at pasting diagram has now been expanded upon.

• CommentRowNumber60.
• CommentAuthorMike Shulman
• CommentTimeFeb 1st 2012

Another very general notion of “shape” is specified by a direct category.

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeFeb 2nd 2012

There is also test category as a notion of “shape”. I have added links back and forth these entries. But no discussion yet about their relation.