Editing Simplicial set (section)

==Motivation==

A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from [[simplex|simplices]] and their incidence relations.  This is similar to the approach of [[CW complex]]es to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.

To get back to actual topological spaces, there is a ''geometric realization'' [[functor]] which turns simplicial sets into [[compactly generated Hausdorff space]]s. Most classical results on CW complexes in [[homotopy theory]] are generalized by analogous results for simplicial sets.  While [[algebraic topology|algebraic topologists]] largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in [[algebraic geometry]] where CW complexes do not naturally exist.