History of Topos Theory - in The School of Grothendieck

In The School of Grothendieck

During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of étale cohomology.

With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems for a long time. The first was to do with its points: back in the days of projective geometry it was clear that the absence of 'enough' points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold). There was also the difficulty, that was clear as soon as topology took form in the first half of the twentieth century, that the topology of algebraic varieties had 'too few' open sets.

The question of points was close to resolution by 1950; Alexander Grothendieck took a sweeping step (appealing to the Yoneda lemma) that disposed of it — naturally at a cost, that every variety or more general scheme should become a functor. It wasn't possible to add open sets, though. The way forward was otherwise.

The topos definition first appeared somewhat obliquely, in or about 1960. General problems of so-called 'descent' in algebraic geometry were considered, at the same period when the fundamental group was generalised to the algebraic geometry setting (as a pro-finite group). In the light of later work (c. 1970), 'descent' is part of the theory of comonads; here we can see one way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.

There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules. An abelian category is supposed to be closed under certain category-theoretic operations — by using this kind of definition one can focus entirely on structure, saying nothing at all about the nature of the objects involved. This type of definition can be traced back, in one line, to the lattice concept of the 1930s. It was a possible question to ask, around 1957, for a purely category-theoretic characterisation of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work (the Tohoku paper).

Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier (see Verdier's Bourbaki seminar Analysis Situs). The characterisation was by means of categories 'with enough colimits', and applied to what is now called a Grothendieck topos. The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning, since it involved a Grothendieck topology.

The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate as a bold pun on the two senses of Riemann surface. Technically speaking it enabled the construction of the sought-after étale cohomology (as well as other refined theories such as flat cohomology and crystalline cohomology). At this point — about 1964 — the developments powered by algebraic geometry had largely run their course. The 'open set' discussion had effectively been summed up in the conclusion that varieties had a rich enough site of open sets in unramified covers of their (ordinary) Zariski-open sets.