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Chaz, could you also produce a diagram with has like two or three squares or triangles? This could serve to illustrate the last point: if all the "small" squares and triangles commute, then the whole diagram is automatically commutative. AxelBoldt 19:29 Oct 26, 2002 (UTC)
- In this diagram, if the subsequence H → G → G* commutes with H → G*; and G → G* → K commutes with G → K; and H → G → K commutes with H → G* → K; then the entire diagram commutes. For example, in group theory, when all the sequences are exact, we say that 0 → H → G → K → 0 and 0 → H → G* → K → 0 are equivalent extensions when this diagram commutes.
If you've got another example in mind, just do it in ASCII and I'll diagram it. In fact, if you bump across any other examples you'd like to see diagrammed this wayt, just post to my talk page, and I'll do it up - just takes me a second. Chas zzz brown 20:30 Oct 29, 2002 (UTC)
I notice from this example that your images have white backgrounds, rather than being transparent. It's not a big deal, since they'll almost always be on pages with white backgrounds, but they probably ought to be transparent instead. — Toby 02:08 Nov 3, 2002 (UTC)
(VPatryshev) It is probably not very smart from to to ask here, but I've been trying to add some entries for toposes, and found that I have no clue how to post images here - you seem to have succeeded; it would be great if you could help me, say, point to a wikipedia how-to page. So far my "ascii art" does not look very convincing. Thank you.
Why are they called commutative diagrams? They say something about function composition, which is, after all, not commutative. What is it that's commuting in the case of a commutative diagram? JeffreyYasskin 20:53, 27 Mar 2005 (UTC)
- The word "commutes" here is not being used in the sense of algebra; instead, it refers to a certain property of a diagram: any two composites of arrows with the same source and target in the diagram are equal. In a 2-category, you can replace this strict equivalence by 2-isomorphisms instead, with a corresponding notion of strict 2-commutativity for 2-arrows. One can replace strict commutativity for 2-arrows by 3-isos, and so on for higher categories. At least this is what I seem to remember from looking at n-categories some time ago. - Gauge 21:52, 12 May 2005 (UTC)
- I don't know the actual history of the use of "commutative" here, but here's a folk etymology I came up with to remind me that the word goes with this concept. Picture a diagram similar to the PQRS diagram on the article page, but where all the objects are identical, the horizontal arrows are both labelled with f, and the vertical arrows are both labelled with g. Then the diagram commutes (in the categorial sense) iff g ○ f = f ○ g — that is, iff f and g commute in the algebraic sense. The use of the word "commute" in category theory just generalizes the equation above. First, if the objects (and therefore morphisms) aren't equal but the two horizontal morphisms are somehow related (they might be components of the same natural transformation, for example), and similarly for the two vertical morphisms, then the commutativity of the diagram says that gQ ○ fP = fR ○ gP. Second, relax the requirement that the morphisms within each pair are related (fP and fR may be unrelated, so call them w and z), and this generalizes to y ○ w = z ○ x as in the body of the article. Finally, relax the requirement that there are only two paths and that each is composed of two morphisms, and you've got the categorial definition, that any paths from A to B are equal. - Owsteele 03:02, 18 November 2006 (UTC)
Request for arrow decorations
I asked over in Talk:Morphism but perhaps here is a better place. Apparently it's fairly standard in commutative diagrams to use arrows with a hook ( or ↪ or ↪) to denote monomorphisms, double-headed arrows ( or ↠ or ↠) to denote epimorphisms, and double arrows ( or ⇒ or ⇒) to denote isomorphisms. So standard, in fact, that algebra papers don't even footnote the usage. It would be nice to mention this usage and use it in the illustrations, for those of us who turn to Wikipedia when the paper doesn't make sense.–Dan Hoeytalk 19:57, 24 May 2007 (UTC)
- Whatever happened to being bold ;-) I've added the changes in a note to the page, feel free to edit to make it flow better (or to add in references) SetaLyas (talk) 09:15, 25 May 2008 (UTC)
- I've never seen for an isomorphism. You mention algebra papers, and while it's true that I'm not an algebraist, I am a category theorist; and that means that I do see a lot of commutative diagrams. I've changed it to what I consider to be standard, which is an arrow with a alongside it: . 126.96.36.199 (talk) 15:32, 26 February 2009 (UTC)
Role in category theory
"Commutative diagrams play the role in category theory that equations play in algebra."
I agree that this statement is somewhat helpful though I full well know the usefulness of equations in algebra: algebra is more or less all about truth-preserving manipulations of equations, such as the multiplicative property of equality. It includes also study of operations which don't perfectly preserve truth-value, such as squaring both sides of an equation; in these cases at least the one equation implies the other. So does this metaphor hold perfectly? Is category theory focused on truth-preserving operations on commutative diagrams?
Shouldn't this be called a "commutativity diagram" instead of a "commutative diagram"? The latter seems, by the normal rules of English, to describe a "diagram that is commutative" (i.e., commutative is an adjective modifying the noun diagram), when what it meant is a "diagram depicting commutativity" (i.e., commutativity is a noun). — Loadmaster (talk) 22:57, 19 December 2013 (UTC)
Diagrams as functors
There is written that a commutative diagram is a diagram of type J, where J is a poset category. Why not even allow prosets (preorders)? I would call a diagram still commutative if there are isomorphisms in and often enough I walk them "backwards". — Preceding unsigned comment added by 188.8.131.52 (talk) 10:04, 9 February 2014 (UTC)
Add definitions for loops and cycles
The comment(s) below were originally left at several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section., and are posted here for posterity. Following
|I'm rating this top-priority, because I think it could grow into a very important article on an idea that has become indispensible throughout many branches of mathematics. Geometry guy 19:45, 10 June 2007 (UTC)|
Last edited at 19:45, 10 June 2007 (UTC). Substituted at 01:54, 5 May 2016 (UTC)