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Simplicial sets and their homotopy theory

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We can think of a simplicial set as a sort of space built out of the standard geometric simplices, or perhaps as some kind of infinitary directed multigraph. They are fundamental in modern homotopy theory and have recently been shown to provide a model of a univalent type theory.

I will give a beginner’s introduction to simplicial sets, focussing on their homotopy theory (which is the same as the homotopy theory of topological spaces). The homotopy theory can be bundled into the structure of a ‘model category’, which I will explain briefly. I will describe Kan’s ‘Ex-infinity’ functor and sketch how this allows one to give a purely combinatorial (and somewhat constructive) presentation of the model structure.

This talk is part of the Junior Category Theory Seminar series.

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