Where does one learn the algebraic geometry needed for topos theory?

I am a masters student familiar with category theory. I have started learning topos theory from MacLane-Moerdijk's book "Sheaves in Geometry and Logic: A First Introduction to topos Theory". I get the feeling that topos theory has a "logic" part and an "algebraic geometry" part. I am not interested in logic and I am not familiar with algebraic geometry at all. I am though interested in abstract algebra and thus if it has to be one of the two directions, it must be algebraic geometry.

I would like to ask if there is any treatment in the literature of the algebraic geometry needed for a category theorist to become comfortable with notions like the étale and Zariski sites etc. I have heard many people mentioning Grothendieck's EGA when discussing topos theory. But I find this very difficult to read. What do I need to know in order to attempt studying it? Of course in the mentioned book (MacLane-Moerdijk) one can find the elements needed to get an idea of some constructions, but I think that it does not give the big picture. Or do I have to learn algebraic geometry from scratch?

Solution 1:

There is no need to know any algebraic geometry to study topos theory, unless you specifically wish to study the applications of topos theory to algebraic geometry. However, it is useful to know the basics of sheaf theory. This is adequately covered in Chapter II of [Mac Lane and Moerdijk]. Try rereading that chapter before moving on to more general ideas like Grothendieck sites or geometric morphisms – there is little chance of making sense of those things if you do not first understand what they are generalising!

If you are interested in the genesis of topos theory, it is probably enough to learn the basic definitions in Chapter I of [Hartshorne, Algebraic geometry], a few key results in Chapter III, and some algebraic topology (at least up to singular cohomology). Then you will understand what people mean when they say things like "the Zariski topology does not have enough open sets". But it will not help you understand topos theory per se...