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What is the meaning of a commutative diagram in mathematics?

For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative diagram?

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It is a diagrammatic expression of morphisms (kinda like functions) depicting different "paths" from some initial object to some final object, such that the end result doesn't depend on which path you take. See Wikipedia. – Willie Wong Jul 5 '12 at 13:54
@Willie: This is an answer, not a comment. – Martin Brandenburg Jul 5 '12 at 15:22
@Martin: I'm afraid that people who truly understood that comment probably already know what a commutative diagram is; to be an answer I would have to explain a lot more. I think there are more qualified people than I for that endeavour. It would, however, help if the original poster can explain a little bit what his/her mathematical background is. This is one of those kind of things that it is possible to give a simple rough overview as well as an in depth technical discussion. – Willie Wong Jul 5 '12 at 18:00

Here is an example of a commutative diagram:

commutative triangle

Here $A$, $B$, and $C$ are mathematical "objects": perhaps sets, groups, or spaces, and $f$, $g$, and $h$ are "arrows", which are some sort of mapping between the objects that preserves their structure.

The prototypical example is that the objects are sets and the arrows are functions, but the idea itself is extremely general and encompasses objects and arrows that are nothing at all like sets and functions.

The diagram above means that $f$ is an from $B$ to $A$, $g$ is an arrow from $A$ to $C$, and $h$ is an arrow from $B$ to $C$. But the most important part of its meaning is that the arrow you get by going from $B$ to $C$ along the top path is the same arrow as the one you get by going along the bottom path. That is, $g$ and $f$ can be composed, and $$g\circ f = h.$$

Diagrams of other shapes are similar: any time there is more than one path between two objects, the diagram asserts that the arrows along the paths can be composed and yield the same result.

Here's another example:

commutative square

This asserts that $$h\circ f = k\circ g.$$

There are a lot of additional twists to the notation: a dotted shaft on an arrow often means that the indicated arrow is unique; a little cross in the corner of a square asserts that the square is a "pullback square", and so on. But the main point is simply to assert an equation between arrows.

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The main exception to this is diagrams that have parallel arrows – see the standard equaliser diagram, for example. – Zhen Lin Jul 5 '12 at 17:20
@ZhenLin Freyd's notation uses a puncture mark between the equalized arrows for exactly that reason. This is a refinement that OP probably does not care about yet. – MJD Jul 5 '12 at 17:23
I think it's worth pointing out that while the diagram part of the name refers literally to the graphical form of the concept in question, the adjective commutative is kind of "loose association" or "metaphor", referring to the fact that when you write down the most basic commutative diagram, that is the square that can be seen above, as a formula then this formula resembles the commutativity axiom for algebraic operations. – Łukasz Maciejewski Jul 5 '12 at 22:00
@ŁukaszMaciejewski I do not see much resemblance with any commutative axiom. The name commutative refers to the fact that you can commute (ie. exchange) two paths with the same starting and ending points, and you get the same result – magma Jul 6 '12 at 0:51
@magma: “$f$ commutes with $g$ under composition” is $f\circ g=g\circ f$ is a special commutative square. The association exists but is very very very loose. – beroal Jul 7 '12 at 17:36

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