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1-2-3 Seminar: Sheaves and Toposes in Logic, or “What’s Bigger Than 2? A Topological Space”

Nelson Niu, University of Washington
Monday, April 10, 2023 - 4:00pm to 4:50pm
PDL C-401

In math, every statement is either true or false; an element is either in a set or out of it. But real life is not so simple: the statement “it is raining” is only ever true on a subset of the 2-sphere on which we live, and I am in Padelford only at certain times of the day on certain days of the week. Can all this be formalized mathematically? It turns out that by leaving the world of sets behind, we’ll enter an alternate world at once strange and familiar, in which there are many more truth values possible than simply “true” and “false”—as many, in fact, as there are open sets in a given topology. Each such world is a special category called a topos, in which sets are replaced by sheaves. No prior knowledge of category theory, sheaves, or toposes necessary: we’ll build everything up from sets and functions.

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