Primo , the new concept isn't too large, in the sense that within these new "spaces", ( or, for the sake of overly delicate ears [1], "toposes") the most essential "geometric" intuitions [2] and constructions, familiar to us from the old traditional spaces, can be easily transposed in an evident manner. In other words, one has at one's disposal in these new objects the rich collection of images and mental associations, of ideas and certainly some techniques, that were formerly confined to objects of the earlier sort.
Secundo, the new concept is large enough to encapsulate a host of situations which, until now, were not considered capable of supporting intuitions of a "topologic-geometric" nature - those intuitions, indeed, which had been reserved in the past exclusively for the ordinary topological spaces ( and for good reason....)
What is crucial, from the standpoint of the Weil conjectures, is that the new ideas be ample enough to allow us to associate with every scheme such a "generalized space" or "topos" ( called the "étale topos" of the corresponding scheme) . Certain "cohomological invariants" of this topos ( nothing can be more "childishly simple" !) then appeared to furnish one with "what was needed" in order to bring out the full meaning of these conjectures, and perhaps (who knew then!) supply the means for demonstrating them.
It's in the pages that I'm in the process of writing at this very moment that , for the first time in my life as a mathematician, I can take the time needed to evoke ( if only for myself) the ensemble of the master-themes and motivating ideas of my mathematical work. It's lead me to an appreciation of the role and the extensions of each of these themes and the "viewpoints" they incarnate, in the great geometric vision that unite them and from which they've issued. It is through this work that the two innovative ideas of the first powerful surge of the new geometry first saw the light of day: that of schemes and that of the topos .
It's the second of these ideas, that of the topos, which at this moment impresses me as the more profound of the two of them. Given that I, at the end of the 50's, rolled up my sleeves to do the obstinate work of developing , through twelve long years, of a "schematic tool" of extraordinary power and delicacy, it is almost incomprehensible to me that in the ten or twenty years that have since followed, others besides myself have not carried through the obvious implications of these ideas, or raised up at least a few dilapidated "prefabricated" shacks as a contribution to the spacious and comfortable mansions that I had the heart to build up brick by brick and with my own bare hands.
At the same time, I haven't seen anyone else on the mathematical scene, over the last three decades, who possesses that quality of naivete, or innocence, to take ( in my place) that crucial step, the introduction of the virtually infantile notion of the topos, ( or even that of the "site"). And, granted that this idea had already been introduced by myself, and with it the timid promise that it appeared to hold out - I know of no-one else, whether among my former friends or among my students, who would have had the "wind", and above all the "faith", to carry this lowly notion [3] to term ( so insignificant at first sight, given that the ultimate goal appeared infinitely distant...) : since its first stumbling steps, all the way to full maturity of the "mastery of étale cohomology", which, in my hands, it came to incarnate over the years that followed.
[3] ( For the mathematical reader) When I speak of "wind" and of "faith", I'm referring to characteristics of a non-technical nature , although I consider them to be essentially necessary characteristics. At another level I might add that I have referred to the "cohomological flair", that is to say the sort of aptitude that was developed in me through the erection of theories of cohomology. I believed that I was able to transmit this to my students in cohomology. With a perspective of 17 years after my departure from the world of mathematics, I can say that not a one of them had developed it.