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Description
Quid of topos theory?
If you spend too much time reading discussions about category theory, nLab pages, or Grothendieck, you will eventually hear about the mysterious topic known as topos theory. In this talk, I will try to finally demystify topoi by giving you a coherent narrative on what they are and what they can do, using some examples that you have most likely all met before and some others you can learn about through this talk.I will start with a brief introduction to categories and the idea that an object can be faithfully recovered from the data of all maps into it (known as the Yoneda lemma), then talk about "local-to-global" constructions in mathematics which are known as sheaves, and generalize them on categories to give you the definition of a topos. In particular, we will see how a topological space, the collection of manifolds in differential geometry, the collection of schemes in algebraic geometry, or a logical theory all give rise to natural topoi. This will allow us to see in what sense, depending on the way you look at it, a topos can be seen simultaneously as a collection of spaces, or a single generalized space, or a universe for mathematics similar to sets, or a universal model of some logical theory.At the end, I will very briefly mention homotopical versions of topoi known as higher topoi using stacks, and their role in recent branches of geometry and more specifically in my own field known as derived algebraic geometry and in the emerging brand new field of derived differential geometry.