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.
]]>Another very general notion of “shape” is specified by a direct category.
]]>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.
]]>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.
]]>I support making n-quiver and directed n-graph redirect to a new page (possibly called one of those or something like ‘pasting diagram -scheme’ whatever) about general shapes of higher categories. We ought to have a page on that topic and, given the multiplicity of shapes, ‘-quiver’ can’t really mean anything more specific.
]]>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 “-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 -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 -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 “-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 -quiver) needs to be able to generate -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 with two functions 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 and two non-identity endomorphisms to Set. I think this encompasses the usual definition of diagram scheme when we define and for all .
]]>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.
]]>Okay, I see better what you are after. Thanks.
Alternatively, I could simply be forbidden from associating my diamonds with -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 -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 (-)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 “-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 -categories. Is this the sort of thing you’re looking for?
But I don’t think we need to rename any of these -quiver either, since they already have perfectly good names.
]]>Hi Todd,
A definition of -quiver that allows composite -edges is fine as long as it doesn’t require composite -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 , , , and with no composites. I may have made my statement too strong because I was trying to argue against having -quiver redirect to globular set because I think any definition of -quiver should be general enough to accommodate my “diamonds” (at least). If we insist on having -quiver be synonymous with globular set, then my 2-quiver would have to contain composite edges and and the square would be a directed 2-edge . Alternatively, I could simply be forbidden from associating my diamonds with -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 with no composite triangles and then generate the free path category . The path category will contain paths of length 2 from to , but no path of length 1 from to . The length matters to me because the length of a path gets promoted to the dimension of an edge in my -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.
]]>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 and , it explicitly does not contain a directed edge
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 in a category and both and to be the identity arrow, you still have the identity arrow , which you said you want to forbid.
Even if you include the hypothesis that , , are distinct vertices, this still rules out being able to take the underlying 1-quiver of any but pretty trivial categories.
]]>Hi Todd,
The shape I have in mind at the moment is fairly simple. I call it an -diamond, but it is basically a special kind of directed -cube. An example of a 3-diamond is illustrated below:
Globbing -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 , there is a special element which is the simple sum of all directed edges
If there are no intermediate edges in the quiver, you can define a (graded) differential on the quiver algebra
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
This generates a differential graded algebra by imposing the relation 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 and in the example above, we have
In other words, the relation induces an orientation on higher dimensional cells in the -diamond.
John’s latest article
inspired me to try to formulate classical mechanics on an -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 there’s a map
sending any cotangent vector at the point to the point . Differentiating this we get a map sending tangent vectors on to tangent vectors on :
The tautological 1-form on eats any tangent vector at the point and gives a number as follows:
It’s mind-blowingly tautological, eh? In case your brain starts to melt, it may help to keep in mind.
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 that I gave in my blog article, and also to the explicit coordinate-ridden formula I gave for . 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 , we can generate a discrete DGA as outlined above, i.e. we have a map . It would be cool if given any DGA, there was an underlying quiver so there is a forgetful functor . If there is no such thing, it would be cool if there was an underlying for any DGA and a forgetful functor . If so, then we can talk about the “free DGA on a(n ” and what I described above would seem much less mysterious (I think).
So I am more interested in -quivers as they may (or may not) pertain to DGAs than -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 has a unit and if that unit can be written as a sum of primitive idempotents
The vertices of the quiver are the primitive idempotents and the directed edges are given by for all pairs of primitive idempotents. So starting with a DGA , you can construct a quiver , and from that quiver, you can construct a DGA . I wonder how well approximates ?
]]>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 -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 -category you have in mind, it’s hard to guess what the underlying -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 -quivers are presumably simpler than whatever idea of -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 -disks. That is, each -cell has a domain -cell (like the earth has a northern hemisphere) and a codomain -cell. There are natural-looking axioms that say the domain of the domain of an -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.
]]>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 -graphs, which, to be consistent with the nLab, should now be -quivers or -diagram schemes. What would be a good definition for -quiver? I would like to record Urs’ comment somewhere:
Idea
A directed -graph, or -digraph or -quiver, is a higher dimensional generalization of a quiver (a category theorist's digraph) with -dimensional edges spanning -dimensional vertices.
A directed -graph is like an n-category with units and composition forgotten. Indeed, an -category is a directed -graph with extra structure. To formalize this idea, we say there is a forgetful functor
where is the category of directed -graphs and is the category of small -categories. Moreover, this forgetful functor has a left adjoint
sending each directed -graph to the free -category on that -graph. A free -category on an -graph is called an -quiver.
Maybe we should resurrect that page and rename it to n-quiver. For example, we could change the above to:
Idea
An -quiver or -diagram scheme is a higher dimensional generalization of a quiver with -dimensional edges spanning -dimensional vertices.
An -quiver is like an n-category with units and composition forgotten. Indeed, an -category is an -quiver with extra structure. To formalize this idea, we say there is a forgetful functor
where is the category of -quivers and is the category of small -categories. Moreover, this forgetful functor has a left adjoint
sending each directed -quiver to the free -category on that -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 (-)quivers I work with, by definition, do not have intermediate edges, i.e. if the quiver contains and , it explicitly does not contain a directed edge , which it would seem to me precludes globular sets.
]]>@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.
]]>@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.
]]>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.
]]>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”.
]]>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.
]]>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.)
]]>@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
and
The source map is given by
and the target map is given by
The collection 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:
The collection 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 and into a single set
and define source and target maps with or something.
]]>@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?
]]>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 -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 , 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 of “edges” and a set of “vertices” and two functions , if it would make sense to just define a set of directed cells and two functions ?
Unless there is some problem I haven’t thought of, I’d be tempted to call this a directed -graph or higher diagram scheme.
The idea is that a directed -cell would have source and target -cells, which would themselves have source and target -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 .
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.
]]>I don’t know for sure, but unless somebody hijacked Eric’s account, I am sensing a disposition towards the cubical.
]]>Urs: yes, but Eric said an directed -graph is different from an -globular set – so I wanted to see what exact notion he meant.
]]>