[Alta-Logic] Stacks Seminar

[Ben MacAdam]

We’re very pleased to kick off the Stacks Seminar with a talk from Geoff Vooys. 

Speaker: Geoff Vooys

Time and Location: 12:00 Friday, September 14 in ICT 616

Title: Sites for Sore Eyes: An Introduction to Grothendieck Topologies and Sheaves

Abstract: In this talk we will discuss how to do geometry in a category (and in particular a category with fibre products) by defining Grothendieck pretopologies, which allow one to describe what it means to cover an object in a category-theoretic context. From there we will show that Grothendieck pretopologies are ambiguous in the sense that two different pretopologies can give rise to the same sheaves; define Grothendieck topologies, which are unambiguous in the sheaf-theoretic sense; and then show how a Grothendieck pretopology gives rise to a Grothendieck topology. After this we will define sheaves on a Grothendieck topology, and sketch how if the Grothendieck topology comes from a pretopology, then the sheaf can be determined on the covers of the pretopology. Finally, we will discuss how to turn presheaves into sheaves through the Associated Sheaf Functor, and then show that sheaf categories are Cartesian Closed by the use of the Day Reflection Theorem. We will also have some explicit examples throughout to motivate things and to provide some intuition and context for the abstract nonsense which we discuss.