There is a functor |•|: S → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category of compactly-generated Hausdorff topological spaces.
This larger category is used as the target of the functor because, in particular, a product of simplicial sets
is realized as a product
of the corresponding topological spaces, where denotes the Kelley space product. To define the realization functor, we first define it on n-simplices Δn as the corresponding topological n-simplex |Δn|. The definition then naturally extends to any simplicial set X by setting
- |X| = limΔn → X |Δn|
where the colimit is taken over the n-simplex category of X. The geometric realization is functorial on S.
Read more about this topic: Simplicial Set
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