Why are commutative diagrams called that way? I don't see what's commutative about them. Here is the commutative diagram for the Fundamental Theorem of Homomorphisms, in case a particular diagram is needed for the answer.

Commutative diagram

(I've only encountered commutative diagrams in the context of abstract algebra, I am not familiar with category theory or uses of this diagram outside of abstract algebra).


1 Answer 1


This terminology originates from diagrams of the form $$\require{AMScd} \begin{CD} A @>>> B\\ @V{}VV @V{}VV \\ C @>{}>> D \end{CD}$$ where commutativity says that going from $A$ to $B$ and then $B$ to $D$ is the same as going from $A$ to $C$ and then $C$ to $D$. That is, if you start from an element of $A$, then using the maps in the diagram to go right and then down is the same as going down and then right. Or, crudely, "going right commutes with going down" in the diagram.

The term "commutative diagram" was then generalized from diagrams of this shape to more general diagrams such as the triangular diagram you showed, where there aren't two things that are "commuting" anymore so it doesn't make as much sense.


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