# Why are commutative diagrams called “commutative”?

Does anyone know the rationale behind the name of "commutative diagrams"? To be precise, what is(are) the reason(s) for calling those diagrams "commutative" and in what sense?

I have previously asked the question here and one can see the discussion that followed.

• My own guess is that it's a back formation from "commutative square", where "horizontal" commutes with "vertical" (i.e., there is an equality relation between the only sensible ways of interpreting "vertical $\circ$ horizontal" and "horizontal $\circ$ vertical"). – Todd Trimble Sep 4 '18 at 18:23
• See also the answers here mathoverflow.net/questions/59456/whence-commutative-diagrams , particularly the one by KConrad. – j.c. Sep 4 '18 at 18:23
• It is kinda similar to Chasles rule with maps viewed as vectors or paths in some abstract space. Writing $f\circ g=g\circ f$ is analogous to $\vec{AC}=\vec{AB}+\vec{BC}=\vec{BC}+\vec{AB}$, since in some sense the considered diagram yields a context in which the composition is "locally abelianized". – Sylvain JULIEN Sep 4 '18 at 18:28
• It's probably important to know the native language of whoever introduced this terminology. For example in French "commuter" can mean to change the position of an electrical switch (a meaning that doesn't seem to exist in English?). One could imagine electricity running through the arrows, to commute them would be to change their path, and the diagram is commutative if you can commute without changing the result. I don't know. If I follow the linked discussion, it was introduced by Hurewicz? So perhaps a Polish speaker can weigh in on what "commute" evokes for them. – Najib Idrissi Sep 4 '18 at 19:41
• @NajibIdrissi Stromwender (Stromwechsler, Kommutator, Inversor, Polwender, Umschalter, Gyrotrop, Pachytrop): a device that serves the purpose of rapidly changing the connection of individual parts of an electrical line. Used by Gauss (1835): “Bei allen drei Apparaten sind Commutatoren (Gyrotrope) mit der Kette verbunden, wodurch man die Richtung des Stroms mit Leichtigkeit umkehren kann” and in English, Maxwell (1867). – Francois Ziegler Sep 5 '18 at 0:55