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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?

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    $\begingroup$ Why do you want to learn topos theory? Algebraic geometry is the branch of mathematics which has made the most use of topos theory, but you do not need to know any algebraic geometry to learn topos theory. $\endgroup$ – Steven Gubkin Feb 22 '14 at 16:54
  • $\begingroup$ Just reviewing here (that happens to everybody's first question) can you cite the titles of the books you refer to in full, (at least title and year ) $\endgroup$ – Willemien Feb 22 '14 at 17:03
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    $\begingroup$ Why do I want to learn topos theory? Why should I not want to learn it? I mean, there is no need to, but it seems like an area that puts a lot of category theory's ideas in practice. I agree that you don't need algebraic geometry to learn topos theory, but saying this is like saying that one does not need algebra and topology for learning category theory. Something which is true, but not effective in my opinion... $\endgroup$ – Rubert_Serent Feb 22 '14 at 17:32
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    $\begingroup$ I don't think that EGA covers topos theory. But SGA does. $\endgroup$ – Martin Brandenburg Feb 22 '14 at 18:57
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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...

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