Introduction to Topos Theory

In an effort to keep myself well-rounded as a mathematician, I plan to spend some time this summer learning about topos theory. In particular, I’d like to learn about topoi, their internal logic, and the application of these tools to algebraic geometry. One long term goal is to gain a better conceptual understanding of Grothendieck’s functorial approach to algebraic geometry. Another is to develop a solid foundation from which to develop an understanding of Lurie’s Higher Topos Theory. Whether or not these are reasonable goals is unclear, but then again, such is the case for any academic endeavor…

The Plan

For now, I have roughly two short term goals in mind:

  • learn the basics of topos theory, and
  • learn about the Mitchell-B\’enabou language and its interpretation via the Kripke-Joyal semantics.

The main resource I will be using is Mac Lane & Moerdijk’s Sheaves in Geometry and Logic: A First Introduction to Topos Theory. I plan to do this over the course of a month, writing weekly posts to keep myself on track.

Beyond that, I am not sure where I will go. One possibility is to read Ingo Blechschmidt’s Masters Thesis, which develops algebraic geometry from the internal point of view. Another possibility is Zhen Lin Low’s PhD Thesis, which looks more more generally at constructing a category consisting of spaces built from out of some model space(s). A third possibility is Anders Kock’s “Universal Projective Geometry via Topos Theory”, which sheds light on the connection between projective geometry over a local ring and intuitionistic pure projective geometry over an intuitionistic field. I will probably defer this decision until the time comes.