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I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to be a problem - at least until now.

As a (reasonable, I thought) progression to my studies, I am trying to teach myself Topos Theory. Until the notion of geometric morphisms, I thought it might be ok to just omit the Logic examples from my studying. And it worked fine, I could keep up with learning. But I have now arrived to the concept of a classifying topos. It feels that now my lack of knowledge in Logic is going to be a very big obstacle. It seems almost impossible to continue. I am worried that this was a big mistake from the beginning. So I am asking the Topos theorists that read this, for their opinion on how stupid they think I have been by thinking in this way, and maybe what is the best way/source to learn what is needed. Thanks for any help.

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I'm hardly a topos theorist. That said, I wouldn't worry too much - when there are several things to be learnt, one has to be first, and the choice may be arbitrary. Have you looked at the text of Moerdijk-MacLane? – Adam Epstein Nov 11 '13 at 8:04
One can definitely omit the logic examples and still get a lot out of the M-M book. However, it might require having an external source of motivation for studying topoi in the first place. – Marguax Nov 11 '13 at 8:23
Thanks for your replies! Of course, it is M-M that I am reading. But I cannot see how lack of knowledge in Logic would allow me to understand the notion of a classifying topos. I mean, the words model and theory have to appear in any definition of a classifying topos... – Muriel Bech Nov 11 '13 at 8:42
@MurielBech, as far as I remember, the book of M-M contains a detailed enough introduction to logic to understand what classifying topoi are. Even if you never had any formal background inlogic, I'm positive that you know what $\forall$, $\exists$ and $\implies$ mean, or how to write statements formally (any basic analysis or algebra course usually explains it). You don't really need more. – Anton Fetisov Nov 11 '13 at 11:14
@Muriel: Don't worry! You haven't been stupid. I did the same thing as you, and I have even taught topos theory as a graduate course. It's very possible to wrap your head around classifying topoi without a formal background in logic, by looking at examples. I would look first at the classifying topos for rings, e.g. That being said, if you choose to learn the logic, you will of course have a deeper understanding, but depending on what you want to do with topos theory, this may not be strictly necessary. – David Carchedi Nov 11 '13 at 11:36

My first advice is to stop worrying about what progression you should be learning stuff. Just read things you find interesting, and if you find yourself reading something that makes no sense to you then you can start backtracking and looking at prerequisites. You don't need to plan ahead so much. Anyway, I second the recommendation of the book of Mac Lane and Moerdijk.

Also, I don't think it sounds absurd to study category theory without having a background in set theory/logic. Most people who know a lot of category theory do not, either; I would say most applications of category theory are in topology/algebraic geometry and related areas. But there are also logicians (as you mention), computer scientists, mathematical physicists, and many others that apply category theory in their work.

Finally, there's a big risk your question is going to closed. Don't take it personally, it's just a bit too discussion-y to be a great fit for the MO "format".

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