Jacopo Emmenegger (Università di Genova) will give a talk on
Categories in fibred categories and the Benabou–Roubaud theorem
Abstract:
I will present a definition of category internal to a Grothendieck fibration (a.k.a. fibred category) that exploits the intrinsic indexing provided by the fibred structure. Interestingly, in this way enriched and internal categories both become instances of categories internal to a fibration. This has been already observed in the past, at least by Mike Shulman, but only for indexed categories rather than fibred ones. The fibred perspective makes the technical details much more manageable. As an indication of this fact, I will show how the monadicity of the category of M-sets over Sets (for M a monoid) generalises to actions of categories internal to a fibration. In fact, the flexibility of the fibred formalism allows us to recover several constructions of categories of actions. Finally, I will describe the connection between actions of groupoids and descent data, via a notion of “equality groupoid”, and show how the monad associated to “kernel groupoids” allows us to give a neat proof of the Benabou–Roubaud theorem for monadic descent.