Meaning of commutative diagram

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?

Solution 1:

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 arrow 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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