TITLE: Did Grothendieck introduce vertical arrows that denote morphisms?
QUESTION [16 upvotes]: It is usual in algebraic geometry to represent morphisms by vertical arrows pointing downwards, like that :
$$\begin{matrix} X \\\\ \downarrow  \\\\ S \end{matrix}$$
I suppose this stemmed from Grothendieck's amazingly original idea that a morphism of schemes should always be considered as some sort of fibre bundle, even in cases apparently very distant from the  bundles considered in topology.
Many geometers  have since adopted these vertical arrows, which they find   suggestive and psychologically helpful.
My question is simply whether anybody had drawn maps vertically before Grothendieck a) in topology b) in algebraic geometry.     
While on the subject I can't resist telling an anecdote I heard, according to which in some seminar led by Grothendieck, a joker ( Serre?) always drew the vertical morphism above on the blackboard just before Grothendieck arrived. So an auxiliary question might be: c) is this true?

REPLY [9 votes]: In Hasse's school of number theory, it was quite common to represent an extension of fields by writing the bigger field above the smaller one and drawing a line segment between them, without an arrowhead. This is one possible source of Grothendieck's notation.

REPLY [2 votes]: Vertical arrows appear everywhere in the book by Cartan and Eilenberg "Homological Algebra" (1956). 
For instance, at page 5 one finds the statement of the $5$-lemma, with the usual commutative diagram, and a page 6 there is the definition of projective module, again with a diagram containing a vertical arrow.  
I do not know where vertical arrows originated from. My guess is that they arise naturally in Topology when one tries to prove some kind of lifting theorem.