0
$\begingroup$

Sections and Retractions are general concepts in category theory. Given $f:B \rightarrow A$ be a retraction of $g:A \rightarrow B$, $f$ geographically (!) stays on the right side of f; that is, $g \cdot f = 1_{A}$ (with dot notation being arrow composition in the category theory).

Question: Why on earth $f$ is called left inverse? It should have been called right-inverse! Shouldn't it?

ps. After I wrote the question, I noticed I know the answer! (probably because of unfortunate notation of function composition), however, leave the question as it might occur to someone in the future.

  • 1
    $f:A\to B$ is a retraction iff $f\circ g=1_B$ for some $g:B\to A$. In that situation $g$ is a section. Nice mnemonic: "$r\circ s=1$" with $r$ for retraction and $s$ for section. – 2017-01-28
  • 0
    Something with your definitions is wrong here. One of your maps needs to be $B\to A$. So if $f:A\to B$ and $g:B\to A$ with $f\cdot g=1_B$, then we say $f$ is a retraction. So it is indeed "on the left". I like to think that $f$ "retracts" the effect of $g$. – 2017-01-28
  • 0
    Sorry about the domain/codomain mistake. corrected. – 2017-01-28
  • 0
    I usually try to avoid "left/right" terminology when it comes to composition and related things and instead use "pre-/post-composition" as this is unambiguous and not tied to notation. This would lead to terminology like "pre-/post-inverse" which I don't actually use, but maybe I'll start. I usually just remember that section is the mono and retraction the epi and then refigure out the how compositions should go. – 2017-01-28

0 Answers 0