[Alta-Logic] Peripatetic Seminar: Series on Differentiable Stacks

[Ben MacAdam]

Hi everyone,

Speaker: Geoff Vooys
Time and Location: MS522, 2-3 PM, Friday, March 29, 2019
Title: An Introduction to the Classical Perspective on Stacks 2: Strongly Cartesian Morphisms and (Grothendieck) Fibrations
Abstract: In this talk we will begin our descent into (classical) stack theory a la SGA 1 by learning about Grothendieck Fibrations, which essentially allow us to talk about spaces which have categories of fibres above them instead of simply having spaces or sets of fibres. We will go about this by beginning with a study of Strongly Cartesian morphisms, which ``categorify'' what it means for an arrow in a category to be a fibre over an arrow in its base in a reasonable way.  This leads us to provide two distinct perspectives on Strongly Cartesian morphisms, and then introduce fibred categories (also called fibrations). From here we will describe some basic properties of the category of fibred categories over a fixed base and describe how to go about a change of base for fibred categories. After this we will describe cleavages of fibred categories, pullback functors between categories of fibres, and leave with a question of how far away from a functor into the category of categories we get if we do a ``category of fibres'' construction from a fixed fibred category with cleavage.