Category Theory For Beginners: Fibrations and Lenses

We introduce fibrations intuitively by starting with discrete op fibrations. We describe connections and applications involving the category of elements, and then generalize to get to Grothendieck constructions and op fibrations. Then we dualize to obtain descriptions of contravariant Grothendieck constructions and fibrations. Then we illustrate applications of fibrations to logic, indexed sets, lenses and dynamical systems. We also connect with Kan extensions. Resources: Unlisted video on factorizing a functor https://youtu.be/fb1PZr1UX2c Categorical Logic and Type Theory Bart Jacobs https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf Coend calculus Fosco Loregian https://arxiv.org/abs/1501.02503 Generalized Lens Categories via functors Cop→Cat David I. Spivak https://arxiv.org/abs/1908.02202 Framed bicategories and monoidal fibrations Michael A. Shulman https://arxiv.org/abs/0706.1286 In this folder I have a handwritten proof that the projection functor associated with the covariant Grothendieck construction is an opfibration. https://drive.google.com/drive/folders/1bhqxC2hCf-517l6WCVUHYcDroUe9TZii In this folder one can find a sketch describing how to go from an opfibration to the corresponding Grothendieck construction https://drive.google.com/drive/folders/1cNz1pb9C9GzfxPkHvP8l1boggOq4yz4W?usp=drive_link This folder has more interesting ideas https://drive.google.com/drive/folders/1hsZSXL1URezncZ6jeiHCL7WjNhr_kHkW