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I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For example, say we have some collection of objects $A$ and $B$, and two maps $f:A\to B$ and $g:B\to A$.

  • If I say "compose $f$ with $g$", does this mean $g\circ f$ or $f\circ g$?
  • Similarly, if I say "precompose $f$ with $g$", does this mean $g\circ f$ or $f\circ g$?
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Both usages are ambiguous, and that is that; there is nothing more to be said. (You could say precompose $f$ to $g$ to eliminate the ambiguity in that case, though) – Mariano Suárez-Alvarez♦ Feb 14 at 21:45
"Precompose with $f$" usually means the operation $(-) \circ f$. "Pre" here refers to the order of application and not the order in which the symbols are printed on paper... – Zhen Lin Feb 14 at 21:50
I'm getting two different answers here. I'm not so sure there's a huge subtlety between "with" and "to" either. – Riem Feb 14 at 21:52
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Not sure why you're being so caustic. Your answer only confuses me more, as you argue that both usages are ambiguous and yet there is no ambiguity with "to". This hardly makes sense to me, leaving more questions than any answers you feel have "nothing more to be said." – Riem Feb 14 at 22:13
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Precompose $f$ to $g$ describes an operation on $f$ that places it before $g$ (in the relevant sense of before), so it is not ambiguous. While I would myself expect precompose $f$ with $g$ to mean the same thing, the fact is that it is also used to describe an operation on $f$ and $g$ jointly that results in one being precomposed to the other without clearly specifying which is which $-$ hence the ambiguity. In short, Mariano is right, though I suspect that precompose with is more often understood to be precompose to than to be the very ambiguous precompose and. – Brian M. Scott Feb 14 at 22:32
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