I want to connect my knowledge of category theory and type theory to geometry, and I am wondering which theories I should learn. I know category/topos/type theory, but little other abstract algebra. I want advice about which theories/literature to study. Below I list some potential options I am considering studying:
- Toposes of smooth spaces.
- Chen Spaces
- Is there an easy route from homotopy type theory to differential or projective geometry ?
- Is there a way to describe synthetic projective geometry in a category theoretic way ?
Advice about which field I should focus on would be much appreciated. I am looking for a way to describe elementary geometry in terms of well behaved categories. I am hoping the theory sheds light on foundational issues, without relying too much on ideas from ring theory or advanced differential geometry. At the moment my plan is to study toposes of smooth spaces, like those described in Bell and Mclarty, but I welcome any advice about what I should look at.