As in the title, I would like to learn about Category Theory and Topos Theory.
I am a math undergraduate, at master level (laurea Magistrale in the Italian system). The master I am starting is mainly on Logic (model and set theory) Geometry (both differential and algebraic) and Mathematical Physics (from a rather geometric perspective) I will also study some Algebra and Algebraic Topology on the way.
Sadly my university does not offer a course in Category Theory. Category are used in the other courses but only to a limited extent. In some sense I think here in Italy the categorical viewpoint is somewhat neglected, at least in teaching.
I remained fascinated by Topos Theory, reading an introductory article on John Baez site, and I think it would particularly fit the three main areas I will be studying.
My knowledge of Category Theory is limited to the reading of Emily Riehl textbook Category Theory in context, up to the Yoneda Lemma.
What I ask is if someone could suggest me a study 'plan' to fulfill this interest.
Any suggestion of notes, choice of topics, books is what I am searchinh for: the area seem very broad and it seems to me that without some roadmap one gest lost easily.
The only constraint is that all of this would be an extra to my daily study of university subjects, hence I will only to dedicate some spare time to it (whence also the need to have 'smart' references to study from).
Nonetheless, I feel these are topics one should master if wanting to try to do research in his life, and what is more important, if one wants to experience Matematics from a different vantage point.
Thanks in advance
Best Answer
Given your interest in geometry as well, "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk is probably a good fit. It also seems to start roughly at your current knowledge of category theory. The nice thing here is that both logical and geometrical aspects are touched upon, so you really get to see toposes as both "universes of sets" (logical perspective) and "generalised topological spaces" (geometrical perspective).