Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like
"For every [bla bla] there exists
a [yadda yadda] such that
the following diagram commutes."
and perhaps it left you wondering what it all meant. I was certainly mystified when I first came across a commutative diagram. And it didn't take long to realize that the phrase "the following diagram commutes" was (and is) quite ubiquitous. It appears in theorems, propositions, lemmas and corollaries almost everywhere!
So what's the big deal with diagrams? And what does commute mean anyway?? It turns out the answer is quite simple.
Do you know what a composition of two functions is?
Then you know what a commutative diagram is!
A commutative diagram is simply
the picture behind function composition.
But be careful.
Not every diagram is a commutative diagram.
"...there exists a map $g$ such that $z$ factors through $g$"The word "factors" means just what you think it means. The diagram commutes if and only if $z=g\circ f$, and that notation suggests that we may think of $g$ as a factor of $z$, analogous to how $2$ is a factor of $6$ since $6=2\cdot 3$.
And as you can imagine, there are more complicated diagrams than triangular ones. For instance, suppose we have two more maps $i:A\to D$ and $j:D\to C$ such that $h$ is equal to not only $g\circ f$ but also $j\circ i$. Then we can express the equality $g\circ f=h=j\circ i$ by a square:
Next time we'll see that diagrams not only help us keep track of compositions of maps, but are themselves the image of some map.
How's that for a brain teaser? A map of WHAT? you may be wondering. And what in the world are the domain and codomain? It turns out that the answer is fairly simple---and pretty cool!---using the language of category theory.
Until next time!