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 Geometric realization of simplicial sets

  • March 1st, 2011
  • 9:27 pm

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This post is an expansion of all the details I did not have a good feeling for when I started with for page 7 of Goerss-Jardine, where the geometric realization of simplicial sets is introduced.

The construction works by constructing a few helpful categories, and using their properties along the way. Especially after unpacking the categorical results G-J rely on, there are quite a few categories floating around. I shall try to be very explicit about which category is which, and how they work.

As we recall, simplicial sets are contravariant functors from the category \mathbf{\Delta} of ordinal numbers to the category of sets. We introduce the simplex category \mathbf{\Delta}\downarrow X of a simplicial set X with objects (simplices) given by maps \sigma:\Delta^n\to X and a map from \sigma to \tau being given by a map f in \mathbf{\Delta} such that \sigma = \tau f.