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This is just a question about notation - probably useless, but it's always baffled me:

Why was $\Omega$ chosen to denote the based loop functor?

I once heard someone speculate: "It's because $\Omega$ looks like... a loop?"

Is it really as simple as that? (If so, clever!) Can anyone verify and/or know more?

Thanks in advance!

(PS. I originally asked this question on StackExchange, but it seems Overflow is a better fit.)

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    $\begingroup$ Wouldn't surprise me. Note also that $\Delta$ is used to denote the simplex category and objects therein, and $\Lambda$ a horn of a simplex. $\endgroup$
    – Todd Trimble
    Commented Apr 24, 2017 at 10:52
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    $\begingroup$ It would help if we knew who chose that notation, and when they did. I don't know the answer. After poking around on MathSciNet, the earliest use of $\Omega$ for a loop space I can come up with is (MR0055683) G. Whitehead, "On the Freudenthal theorems" (Annals, 1953). He actually uses "$\Omega^{n+1}$" for what we call $\Omega S^{n+1}$. $\endgroup$
    – Charles Rezk
    Commented Apr 24, 2017 at 14:31
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    $\begingroup$ Of course, I should have checked Serre first: he uses $\Omega_x$ for "based loops of $X$ at $x$" in his 1951 Annals paper about the Serre spectral sequence, for instance. $\endgroup$
    – Charles Rezk
    Commented Apr 24, 2017 at 14:37
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    $\begingroup$ The way he talks about it in the first paragraph makes me suspect he took the notation from Morse and/or Hurewicz, but I can't follow those references. $\endgroup$
    – Charles Rezk
    Commented Apr 24, 2017 at 14:43
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    $\begingroup$ Morse's book "The Calculus of Variations in the Large" from 1934 uses $\Omega$ for the based loops in a manifold. $\endgroup$
    – Mark Grant
    Commented Apr 26, 2017 at 19:39

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