Seminario de Categorías

Expositor: Martín Szyld, Dalhousie University

Resumen: We consider the notions of Fibration of categories, (pseudo)Filtered category, and the axioms for a category of Fractions. A basic fact involving them is: given a Fibration, if the arrows of the base category are (pseudo)coFiltered, then the cartesian arrows satisfy Fractions. This is a Proposition in SGA 4 (Exp. VI, Prop. 6.4) whose proof is left to the reader as an exercise, and I want to start this talk by solving this exercise. Let me tell you why.

Each of the three "F" notions above has been considered for bicategories, or at least for 2-categories. I will start with what may be the easiest one to understand, that of Filtered: in a Filtered bicategory, in addition to asking for cones for two objects and for two parallel arrows, we add a third axiom asking for cones for parallel 2-cells. I will present the definitions of Filtered and pseudoFiltered bicategory, a set of axioms for a bicategory of Fractions, and some properties of Fibrations of bicategories that all fit this same pattern. We arrived at these notions when proving a "bicategory version" of the Proposition in SGA 4, in fact a small generalization that I will present.

This result is part of an ongoing collaboration with P. Bustillo and D. Pronk, we're working on showing some basic properties of the bicategorical localization by fractions which are known in dimension 1. If time permits, I hope to mention how we ended up here within our current work and how this result can be applied here.

Inscripción en Página:

https://sites.google.com/im.unam.mx/seminario-de-categorias-unam/inicio

Youtube

https://www.youtube.com/watch?v=D3XoAldxBOY