Natural Examples of Simplicial Sets

This lecture introduces two fundamental examples of simplicial sets: the nerve of a small category and the singular simplicial set of a topological space. The slides cover the definition of the functor N : Cat → sSet, which maps a small category C to the simplicial set Fun([n], C). The lecture explores the structure of these examples, including the maps di and si, and their relationship to order-preserving functors. It also delves into the topological n-simplex, cosimplicial topological spaces, and the functor S : Top → sSet, which maps a topological space X to the simplicial set Top(A^n, X). The lecture concludes with a sketch of the proof for the functor S. The examples provided illustrate the concepts discussed.