What would be a roadmap to learning Sheaf Theory and Topos Theory for a mathematics graduate?

The main thing you'll need to start with is getting familiar with category theory. There's plenty of books and online material out there (e.g., "Category Theory" by Awodey; the nLab is also an excellent resource, but less so for a complete beginner); just start reading and thinking and ask questions when necessary. After that, well, there's plenty of books on topos theory as well (e.g., "Introduction to Higher-Order Categorical Logic" by Lambek and Scott, "Sheaves in Geometry and Logic" by MacLane and Moerdijk), though which would be best for you depends on what exactly your motivations and background are. Again, just read and think and ask questions when necessary. It'll be easier if you're surrounded by other people who are versed in the same material, but it's certainly possible to learn things on your own. This may not be a very enlightening answer, but there's not very much to say. The only real prerequisite to studying topos theory is first obtaining comfort with category theory; after that, dive right in. Of course, if your interest is for reasons having to do with algebraic geometry, or intuitionistic logic, or topology, or semantics of programming languages, or this or that or the other thing, then you'll want to read up on those as well. But you can do things in many different orders, or not at all, depending on what it is you care about...

Here are a few things you could use as guiding lights: http://www.math.nmsu.edu/~gasparim/Coh.pdf http://math.stanford.edu/~vakil/725/course.html http://arxiv.org/abs/1303.3255 http://homepages.mcs.vuw.ac.nz/~rob/books.html http://www.cs.bham.ac.uk/~sjv/TopPLVN.pdf http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf browse the http://ncatlab.org/nlab/show/Toposes,+Triples,+and+Theories all the time The way I work is to start with the thing I want to learn and work backwards. Instead of grabbing a dictionary when I don't understand a word, I might need to instead read an entire book, or a chapter, or a paper. A few gists: http://math.ucr.edu/home/baez/topos.html http://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf for sheaves: think about projections, inclusions, and restrictions, for example you can read the chapter in Lawvere & Schanuel's Conceptual Mathematics or it's in Vakil. This is the "old" ⊂ operation just thinking much more carefully about it and drawing much more from sapping things in and out than I would have thought is productive. Agreement Triple-overlap A∩B∩C is what you need for sheaves to "work right". (Helly's theorem --- can't find the explanation that worked for me in top google results, sorry) for topoi: think about how statements in logic are either true or false. That's a map from statements to {T, F}. In Kleene logic maybe we map onto {true, false, indeterminate}. There are other things you could reasonably map to. Gasparim paper http://en.wikipedia.org/wiki?title=Talk%3ASheaf_(mathematics)#Some_visualization http://xorshammer.com/2010/03/13/topology-and-first-order-modal-logic/ http://www.cs.bham.ac.uk/~sjv/GeoFuzzy.pdf http://isomorphism.es/post/44153627461/wiggly-numbers (the concept of a Germ https://en.wikipedia.org/wiki/Local_ring#Ring_of_germs#ring of germs) Jack Huizenga's answer says to go to graduate school first. I am more in your boat, interested in learning sheaf theory and topos theory without graduate school. I agree with his timeframe (2-3 years of fulltime or whatever that multiplies to for how much time per week you're going to spend on it), and that it's difficult (including emotionally difficult), but you don't need graduate school.http://www.math.cornell.edu/~hatcher/AT/ATpage.html of the things you'll want to learn from are available for free, or under $100 or maybe you can find a university library. The really good authors have already given you what you need to understand it whether or not you want to participate in academia. Puro pasión; good luck.