Abstract:

We consider and develop the axioms introduced by A. Joyal that define an abstract notion of an etal class A of arrows in a Grothendieck topos E. The axioms are intended to be sufficient in the sense that the category of objects etal over the terminal object (and etal maps between them) should be a topos, the 'Etal' Topos Et. It can be shown that Et ⊂ E is a full subcategory, closed under all colimits and finite limits, and so it is almost a topos. However, the problem of the existence of generators for Et and the existence of a right adjoint for the inclusion making it the inverse image of a geometric morphism E → Et is still open. We introduce an additional axiom that we call the etal topology condition or ETC, and for a topos E equipped with such a class, we develop a general construction of germs which yields a new point associated to any given point F → E. A particular case of this furnishes a right adjoint for the inclusion Et ⊂ E, and it follows that in this case Et is a subtopos of E, the center of a local geometric morphism of topoi. E → Et. Also, we introduce a new general theory of Spectrum where etal classes take care of the role assigned to the admissible morphisms, and prove a general theorem of existence based in the construction of germs. This theorem includes all the known results in the theory of Cole's spectrum. This theory is more general since it is associated to any geometric morphism rather than only to the inclusion of subtopoi, and conceptually it is independent from the notion of geometric theory. © 2000 Elsevier Science B.V. All rights reserved.