This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence classes, where this operation is referred to as a "section".
With this term, given an equivalence relation $\sim$ on a set $X$, I can define a "canonical section" $s_c$ as an injective map $s_c \colon X/\sim \to X$ and then add that the image by $s_c$ of an equivalence class is called its "canonical representative". This notation comes in very handy for formal derivations, where I can use $s_c(C)$ rather than natural-language descriptions like "and then we take the canonical representative of C,...".
But this use of the term "section" doesn't seem to be common at all in the textbooks about algebra, so I am not sure if I should use it in my current research. I think it comes from the use of "section" and "retraction" in category theory, but would it be understood by people with a background in other fields of mathematics? Is there any well-known textbook dealing with equivalence relations where the word "section" is used in this way?