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I've came across this expression regarding function composition in some notes by people from computer science, and I don't know what the notation "$::$" means.

\begin{equation} g \circ f : A \rightarrow C :: a \mapsto g(f(a)) \end{equation}

I get what this expression is trying to tell me, but I am pretty disturbed by the fact that I don't know what "$::$" means exactly and how to read it. Can anyone tell me what is "$::$"?

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    $\begingroup$ Any way you could post the notes? CS uses a lot of overloaded notation, so there's no way we can tell you what it means without some context. $\endgroup$
    – Joey Eremondi
    Commented Mar 18, 2017 at 21:57
  • $\begingroup$ It is just an example of a category, specifically the category of sets and functions. I've found this specific example in a PhD thesis, but similar notation is used the following notes as well:cs.ox.ac.uk/teaching/materials16-17/catsproofsprocs/… $\endgroup$
    – Jerry
    Commented Mar 18, 2017 at 22:01

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It stands for "given by" or "defined by". The function $g \circ f$ is a function from $A$ to $C$ which maps $a$ to $g(f(a))$.

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    $\begingroup$ Is this a common notation in computer science? This is the first time I encountered this notation. $\endgroup$
    – Jerry
    Commented Mar 19, 2017 at 0:17
  • $\begingroup$ @Jerry I've seen it before, though it's not common in algorithms or complexity. $\endgroup$
    – Yuval Filmus
    Commented Mar 19, 2017 at 6:02

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