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474:(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.
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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
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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
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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
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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
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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
Knowledge when the paper doesn't make
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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".
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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?
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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. -
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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.
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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.
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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. —
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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
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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. -
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1124:\begin{array}{lcl} & X & \overset{f}\rightarrow & Z & \\ &g \downarrow &&\downarrow g'\\ &Y & \underset{f'}\rightarrow& W & \\ \end{array} </math: -->
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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.,
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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.
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sense. The use of the word "commute" in category theory just generalizes the equation above. First, if the objects (and therefore morphisms)
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Basically what I mean to say is, the sentence should be backed up with a specification of in what form this analogy holds.
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but perhaps here is a better place. Apparently it's fairly standard in commutative diagrams to use arrows with a hook (
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This statement is not very helpful without elaboration for those who don't know what role equations play in algebra.
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