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I recently came across the following expression regarding function composition:

\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 "$::$"?

Jerry
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2 Answers2

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In this context it means "defined by", but I've never seen it written like that before. I would have written it "$g \circ f: A \to C$ defined by $a \mapsto g(f(a))$".

Patrick Stevens
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I've actually seen this notation with either a single or double colon. Rather compressed notation that really needs you to key into the second arrow being "mapsto" and not just a right arrow. There are a lot of merits to be said for this way of expressing a function, particularly from a lambda calculus / category theory perspective. In addition, just consider the amount of ink and page-space necessary for defining a super messy function on a crazy domain:

Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined by $$f(x) = x^2.$$

versus

Let $f$ be the function $$f: \mathbb{R} \to \mathbb{R} ~~:~~x \mapsto x^2.$$

erfink
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  • Is it a notation commonly used in lambda calculus? I have never encountered this in some introductory category theory books I have read so far. Should it be read as "defined by" as suggested by @Patrick Stevens (and also as suggested in your answer)? – Jerry Mar 18 '17 at 18:48
  • Lambda calculus has rather different notation (c.f. http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf ); my comment was more towards the idea that a function is the rule rule of assignment. And yes, verbalizing as "defined by" is reasonable. – erfink Mar 18 '17 at 20:34