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?

veryinteresting. Will you allow me to put a little friendly pressure on you in order that you transform them into genuine answers? – Georges Elencwajg Oct 31 '11 at 13:38