David Roberts geometric realization

Geometric realisation of a simplicial space X =X:Δ opTopX_\bullet = X:\Delta^{op} \to Top is the coend …

Concretely this is the space

( n|Δ n|×X n)/ \left(\coprod_n |\Delta^n| \times X_n\right)/\sim

where the equivalence relation \sim is …, and |Δ n||\Delta^n| is the topological nn-simplex.

This clearly gives the geometric realization of simplicial sets when X:Δ opSetX:\Delta^{op} \to Set.

Unless there is some control over the degeneracy maps of X X_\bullet, this is not homotopically well-behaved. For example, if all the degeneracy maps are cofibrations, a level-wise (weak) homotopy equivalence of simplicial spaces induces a (weak) homotopy equivalence on geometric realization, but in general this is not the case. The fix is to pass to the fat realization.

Last revised on April 5, 2009 at 04:08:49. See the history of this page for a list of all contributions to it.