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In mainstream notation for composition, by $g \circ f$ we mean $\operatorname{dom} \left(g \circ f \right) = \operatorname{dom}\left(f\right)$ and $\operatorname{cod} \left(g \circ f \right) = \operatorname{cod}\left(g\right)$

However, when watching a story about categories, I prefer seeing paths going from domains to codomins, i.e. like a linear narrative. But when writing, I have to take care of a kind of flashback, when writing. Also, I know that many accomplished mathematicians do use the reversed notation of the mainstream.

I would like to know what are the pros and cons of these two notations and how harmful can be, getting used to a non-mainstream notation?

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Please use standard notation if you want other people to read your work. –  Adam Smith Nov 14 '12 at 18:53

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There are some prominent computer scientists that do prefere the notation $f;g$ over the usual mathematical notation $g \circ f$. I guess, the reason is that computer scientists care more about coherence between graphical $A \to^f B \to^g C$ and textual $f;g$ form.

The only problem with this notation is that the whole mathematical world is used to (an obviously wrong notation :-) and) writing $g \circ f$ for functional composition and $f(x)$ for application of $f$ to $x$. Let me racall, that by Yoneda lemma, we may think of morphisms $X \to^x A$ as generalised elements of $A$ parametrised by $X$, and write $x \in_X A$. Then $f \circ x$ becomes $f(x)$ for a generalised element $x \in_X A$. The calculus of generalised elements imitates the usual set-theoretic notion of elements of a set --- for example, a morphism $m$ is mono precisely when $m(x) = m(y) \Rightarrow x = y$ on generalised elements $x, y$. Therefore, to be consistent with the natural order of composition, you should use a smalltalk (message-passing) like notation $(x)f$ or just $x f$ for application of $f$ to $x$.

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