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When was the "arrow notation" $f: X \to Y$ for functions first introduced? Who introduced it and with which motivation?

I ask this question in order to understand whether it was, in part, this notation to suggest that there could be "higher morphisms" (in analogy with oriented paths and homotopies between them, and homotopies between homotopies and so on), or if it went the other way around (with category theorists first realizing that many constructions involving paths and homotopies thereof in Homotopy Theory could be generalized to other more abstract settings, and then setting up a notation that suggested the analogy "1-morphisms $\sim$ paths").

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Despite the claims reported from Wikipedia and the “Earliest Uses” site, this notation certainly started much before Hurewicz-Steenrod (1940; 1941) or Ore (1935, p. 416; 1936) for, respectively, Domain-to-Codomain and Argument-to-Value arrows.

1. Domain $\to$ Codomain (as in the question proper):

Scholz (2008, pp. 883-884) describes manuscripts and lecture notes of Hausdorff (1933):

In these manuscripts he made extensive use of the arrow symbolism for maps. Until that time this was by no means common. Mapping arrows were used only sporadically in the contemporary literature: sometimes (element-to-element) for boundary operators in homology theory [Alexander (1926), Alexandroff (1928), Čech (1932), Pontrjagin (1931)], occasionally also for general homomorphisms [van der Waerden (1930)]. For maps of entire groups, H. Weyl had used them to symbolize representation homomorphisms (1925).

I’m not finding any set-to-set arrows in Weyl (1925). But he has at least one in (1931, p. 267), and van der Waerden (1930; reviewed by Ore in 1932) has indeed, e.g.

(p. 33) $\dots$ eine homomorphe Abbildung $\smash{\mathfrak G\to\overline{\mathfrak G}}\dots$
(p. 87) $\dots$ Die Homomorphie $C\to\mathfrak P\dots$
(p. 190) $\dots$ in der Homomorphie $\smash{\mathfrak R\to\overline{\mathfrak R}}\dots$
(p. 203) $\dots$ eine Zuordnung $\smash{\mathsf P(\mathfrak A)\to \mathsf P(\mathfrak A')}\dots$ nämlich die Zuordnung $\alpha\to\alpha'\dots$

McLarty (2006, p. 200) argues that the first one at least was only “a prescient typographical error” for a Nœther tilde $\sim$. But I’m not sure I buy this, as there are more and this is unchanged in later editions (1955, 1971, 1993).

2. Argument $\to$ Value (nowadays written $\mapsto$, with other antecedents):

These are much older. All references above have many, but the earliest I’ve seen are shaped $\rightarrowtail$ and occur in perhaps the earliest commutative diagram, by Eduard Study (1891, p. 508). The next are again in papers of Study, (1905):

(p. 432) $\dots$ eine umkehrbahre analytische Zuordnung $(E\to E')$ $\dots$
(p. 435) $\dots$ die Zuordnung der Strahlen $(S\to S')$ $\dots$
(p. 438) $\dots$ die Transformation $(S\to S')$ kann zu einer orientierten $\dots$
(p. 438) $\dots$ Berührungstransformation $(E\to E')$ erweitert werden $\dots$

and (1906, pp. 493-496, 511-516). Blaschke (1910a, 1910b) adopts them throughout, as do books by Blaschke-Study (1911/13, pp. 15, 17, 23, etc.), Weyl (1913, e.g. pp. 32, 50, 54, 139, 159), Speiser (1923, e.g. pp. 38, 88, 103), or again Study (1923). There, §1 “Grundbegriffe und Zeichen” (p. 16) sounds much like he’s claiming the notation, while crediting Wiener (1890) for another:

mapping arrow

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  • $\begingroup$ Do you know when the colon in $f \colon X \to Y$ first appeared? A related question about the colon in type theory (as in $t\colon T$) made me wonder if there was any relation. $\endgroup$ – Michael Bächtold May 27 at 12:59
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    $\begingroup$ @MichaelBächtold I don’t know. Out of the above references, maybe Weyl (1913, p. 50)? $\endgroup$ – Francois Ziegler May 27 at 13:30
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The Wikipedia article on History of mathematical notation says:

The arrow, e.g., →, was developed for function notation in 1936 by Øystein Ore to denote images of specific elements. Later, in 1940, it took its present form, e.g., f: X → Y, through the work of Witold Hurewicz.

But I have no access to the references.

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Regarding element-to-element arrows, Emili Bifet writes on The Math Forum that Riemann used the notation $a \to b$ as follows:

Ist $a$ ein Verzweigunspunkt der Loesung einer linearen Differentialgleichung zweiter Ordnung und geht, waehrend $x$ sich im positiven Sinn um $a$ bewegt, $z_1$ ueber in $z_3$ und $z_2$ in $z_4$, was kurz durch $z_1 \to z_3$ und $z_2 \to z_4$ angedeutet werden soll...

II. Die Integrale einer linearen Differentialgleichung zweiter Ordnung in einem Verzweigungspunkt. (Aus einer Vorlesung Wintersemester 1856/57.)

Or in translation by Michael Bächtold:

If $a$ is a branching point of the solution to a linear differential equation of second order and, as $x$ moves clockwise around $a$, $z_1$ goes over to $z_3$ while $z_2$ goes over to $z_4$, what shall be denoted with $z_1 \to z_3$ and $z_2 \to z_4$ for short...

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  • $\begingroup$ I haven’t really figured this out, but to me it sounds like he’s talking about paths or homotopies having the $z_i$ as endpoints, so doesn't necessarily have maps sending $z_1\mapsto z_3$ and $z_2\mapsto z_4$. $\endgroup$ – Francois Ziegler Jul 1 '17 at 8:12
  • $\begingroup$ @FrancoisZiegler Bächtold says it's apparently the monodromy map. I can't judge whether Riemann's notation is about points moving to points, or the function assignment one gets for monodromy (ie something like fibres mapping to fibres) $\endgroup$ – David Roberts Jul 1 '17 at 8:20
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    $\begingroup$ Ooh, I see. Yes, there is a map ("action $z_1\mapsto z_3$ of a generator of $\pi_1$ on some covering's fiber") and it is interesting to see $\to$ already used to conceivably denote it. But: that map is many degrees of abstraction deeper than the paths which are also there and conceivably denoted $z_1\to z_3$. Which it is, depends on how much modern thinking we read into Riemann. Also: this is only about very special maps, and unlikely to have had more than a very confidential influence before publication (1902). $\endgroup$ – Francois Ziegler Jul 2 '17 at 1:10

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