"Let us now consider toposes. ... Unlike schemes, toposes generate geometry without points. In fact, nothing prevents us from proposing an axiomatic framework for geometry in which points, lines, and planes would all be on the same footing. Thus we know axiomatic systems for projective geometry (George Birkhoff) in which the primitive notion is that of a plate (a generalization of lines and planes), and in which the fundamental relationship is that of incidence. In mathematics, we consider a class of partially ordered sets called lattices; each of these corresponds to a distinct geometry. ... In the geometry of a topological space, the lattice of open sets plays a starring role, while points are relatively minor. But Grothendieck’s originality was to reprise Riemann’s idea that multivalued functions actually live not on open sets of the complex plane, but on spread-out Riemann surfaces. The spread-out Riemann surfaces project down to each other and thus form the objects of a category. However, a lattice is a special case of a category, since it includes at most one transformation between two given objects. Grothendieck thus proposed replacing the lattice of open sets with the category of spread-out open sets. When adapted to algebraic geometry, this idea solves a fundamental difficulty, since there is no implicit function theorem for algebraic functions. Sheaves can now be considered as special functors on the lattice of open sets (viewed as a category), and can thus be generalized to étale sheaves, which are special functors of the étale topology. Grothendieck would successfully play many variations on this theme in the context of various problems of geometric construction (for example, the problem of modules for algebraic curves). His greatest success in this regard would be the étale “ℓ-adic” cohomology of schemes, the cohomological theory needed to attack the Weil conjectures."