I already said that I think commutative diagram is better as a redirect to diagram -- does anyone want to speak up in favor of separate pages?

I don't understand your point about poset/preorder. If a diagram commutes whenever any two parallel paths through it are equal, then a diagram with two anti-parallel morphisms commutes when those two maps are isomorphisms, which doesn't necessarily imply that the two objects are equal. So that's a commutative diagram which factors through a preorder but not a poset.

I think one thing that came out of the discussion on diagrams is that the word really is used in at least two different ways, one more restricted than the other. The page could certainly be improved pedagogically, but I think it's helpful to be made aware of this difference.

]]>Only now realize what has happened at diagram meanwhile.

I won't have more energy for this, but three brief comments:

I find it not so elegant and not so pedagogical to stat "Definition 1" and "Definition 2" when the second is nothing but a special case of the first.

I see you changed my "poset" to "preorder". In situations where the difference matters, one typically has in the text extra discussion of how the anti-parallel morphisms are to be read as far as commutativity goes. If not the poset interpretation is meant, then what is usually meant is a shorthand for two different commuting diagrams, which are poset images after all.

I had originally made commutative diagram redirect to diagram. I now see that it has been split off as a separate entry, but I don't see that the new separate entry is an improvement over just the redirect.

OK, I have put in the elementary definition.

]]>Well, it would be good to fill in the elementary definition of commutative diagram. For one thing, that would help Harry with his problems with circularity; the definition of functors in terms of commutative diagrams *is* incomplete so far.

I did a bit of editing at diagram; I think it is looking good. Do we really need commutative diagram to be a separate page from diagram?

]]>Thanks, Todd. I am not sure if the discussion in revision 24 is really what I was thinking of (since I believe that there was discussion on other pages as well, such as what is now graph theory > history, as well as on the Forum and perhaps at latest changes). However, there are some relevant comments there, especially if you search in the page for ‘digraph’.

See also my new discussion thread on the Forum which I made about this issue.

]]>Oh, I didn't know what you meant by #24! I'll check it out right now.

]]>Let's let Toby decide what he was talking about. Most of the discussion I was talking about was removed, but may be found at version #24, which is linked to in the graph article.

]]>@Todd: I think he was talking about the entry quiver, since that was a much bolder (and I think mistaken) change of definitions. The discussion there is also longer.

]]>Toby #24: There was a ton of discussion surrounding the entry graph, which may be what you were thinking.

]]>At quiver, John decided to take it upon himself to change the definition of a quiver from a multidigraph to its free category. I think we should change the current page over at quiver to free category on a quiver (or on a multidigraph or on a Diagram Scheme), and also change the page called directed graph to quiver (or multidigraph or Diagram Scheme), since a quiver is a generalization of a directed graph.

I see no legitimate reason to change DiGraph->Quiver->Free Category on a Quiver. We should try to use standard terminology unless there is a really compelling argument against it (and in this case, there is not).

]]>Sorry, the discussion at directed graph isn't really relevant. I know that there was some discussion about this stuff somewhere; I'll try to find it.

]]>A quiver

isa multidigraph.

Right. Does it still say different somewhere? (I wouldn't be surprised.)

The finiteness thing that I noted was at the very beginning of the page.

You can't draw it on a page, which is what I was writing about, if it's not finite. But the extension to infinite quivers is pretty natural.

]]>That is the wrong definition of a quiver. A quiver *is* a multidigraph. The free category of a multidigraph/quiver/diagram scheme is just called a free category. I don't understand why we should change this definition from Representation Theory, but we can't change the obviously much worse name of a prestack from Algebraic Geometry. We should be consistent!

The finiteness thing that I noted was at the very beginning of the page. However, the diagram for infinite composition (with or without the cone point) is not finite but very obviously standard.

]]>Harry, read the discussion at directed graph.

Also, finiteness is not part of the definition of a quiver, so the comment about finite quivers seems a bit misleading.

??? Did you mean to say that finiteness *is* part of the definition of a quiver (which I would *not* say, but some people do say that). Because nothing that I wrote should imply that finiteness *is* part of the definition.

Quivers are directed graphs that allow loops and multiple edges (so they're not really directed graphs), so you can't really equate those two terms (a quiver is a directed multigraph). A directed graph is a much more strict condition.

Also, finiteness is not part of the definition of a quiver, so the comment about finite quivers seems a bit misleading.

Another (albeit much more minor) problem is that the article switches back and forth from "directed graph" and "quiver".

There are three correct terms for the kind of graphs we want to consider: multidigraph (directed multigraph?), quiver, or "Diagram Scheme", which is a term that I've seen and like somewhat (although this usage is restricted to category theory-related stuff). Right now there appears to be a problem where these three identical concepts are given different names on different parts of the nLab.

Question: Which one should the nLab use in general (with exceptions where applicable, obviously)?

]]>I've redone diagram in accordance with the conversation on this humongous thread which I have read but don't want to reply to.

]]>I wrote something at commutative diagram.

]]>Commutative diagrams are an informal notation for arguments that could be handled completely formally with a minimal amount of effort. I do not think that there is a sufficiently expressive diagram calculus for modern mathematics to include all of the information in the diagram.

Simplicial sets and quasicategories give us a combinatorial way of formalizing diagram arguments, but it is often impossible to draw the pictures at that point.

]]>Identity morphism can cause problems for commutative diagrams.

I am with those who say that if you draw a loop (but *not* looking like that, since ‘’ should only be written once) and claim that it commutes, then you are claiming that . In particular, an operation on objects and morphisms that preserves all commutative diagrams must preserve identities (and thus be a functor).

However, not everything involving an identity morphism can actually be drawn in this way. For example, given , there's no way to indicate that the composition in *one* direction is the identity without also catching the *other* direction. If you tell me that the diagram that I've just drawn commutes, then I'll take it to mean that and are inverses (although I'll feel more certain if you draw the arrows curved on opposite sides). If you only want one composite to be the identity, then you can either put an identity arrow in explicitly (making a triangle) or adopt a convention that (say) the curved arrow is a section and the straight arrow is the retraction.

The reason that I'd feel a little uncomfortable taking that diagram literally (to mean that and are inverses) is that there's another convention about parallel arrows with no gap between them. For example, if you have , then to say that the diagram that I've just drawn commutes should mean that . But in fact, that often appears as a small part of a larger commutative diagram where you *don't* mean that , but simply mean that the diagram commutes with either version. (If you really want to say that , then again it's better if the arrows are curved on opposite sides.)

So in a straightforward mathematical theory of commutative diagrams, then you certainly want a loop to be an identity, and to be inverses in , and and to be equal in . But in practice, people might adopt conventions that cancel those conclusions.

]]>In any case, you will all be pleased to learn that I have returned to Mac Lane (Categories for the Working Mathematician) while still keeping Awodey close at hand.

Incidentally, I have a concrete reason for the graph theory supplement, though. Bob Coecke has developed a semi-graphical description of dagger-categories and I'm trying to figure out the "method behind the madness," as it were.

As for social circles, despite having degrees in engineering, physics, and mathematics, I live in Maine which means my social circle tends to consist of things such as moose, bear, coyotes, and trout. :) ]]>

Scenes from "A Beautiful Mind" come to mind :)

]]>On the train I must, when scribbling in my little black book all the arcane symbols of our field, look a little odd too. Understandable when I was a student, and scruffy looking, but now in business attire... Some of my best work was done on the train. :)

]]>Commute.. Isn't that when you take the train to work?

Yes. Yes it is :)

By the way, on my commute this morning, the image of myself in my pin-striped suit hunched over Goldblatt on a train must have been interesting to observe :)

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