ABSTRACT
The least one can say about the definition of a noncommutative Grothendieck topology in 1.3.10 is that it is somewhat "ad hoc". Lemma 1.3.11 motivated that definition but it does not make it clear to what extent the definition is natural. Indeed, does it make sense to restrict to words in an alphabet of Ore sets, to morphisms that do not take relations between single Ore sets into account, to covers induced by global covers? The simplified approach of Section 1.3 does have the advantage that no new "abstract nonsense" had to be introduced, moreover : it works ..., at least in the cases we wanted to study. That success was granted to us by the existence of a version of Serre's global sections theorem (see Theorem 2.1.4, Theorem 2.15) which seems to prove "a posteriori" that the definition of the topology is satisfactory. In fact that result even suggests that the choice of this kind of topology may be canonical to some extent. That really was a "Deus ex machina" to this author! But to what extent is "Deus" created by the "machina" ? In this chapter this should be clarified. We are about to define skew topologies in an abstract, one could say "axiomatic" way. At the moment when the lattice-type approach has to be combined with the idea of a Grothendieck topology it becomes clear how certain conditions and definitions are actually forced upon us. At first sight one may say that there is an almost unlimited number of different generalizations of topology to a noncommutative setting. Just for the elementary fun of it, the reader should try his hand at it .. , but it will turn out to be a crosswords-puzzle in four dimensions. Again, when one is stubborn enough to ask for a few properties that make the new
structure really into a kind of topology, also demanding that some example should exist, the collection of definitions possible is a very limited one indeed. The origin of the "Deus ex machina" is probably discovered here. The nature of this chapter is abstract but very basic; once the bridge has been built it is easy to ride on it.
