Simplicial sets are reached by a chain of deliberate weakenings starting from geometric simplicial complexes: drop the embedding, order the vertices to make face maps well defined, drop the requirement that a simplex be determined by its vertices to get Delta sets, then add degeneracy maps to get simplicial sets - equivalently, contravariant functors from the category of finite ordered sets to Set. Geometric realization sends a simplicial set to a CW complex with one cell per nondegenerate simplex and is left adjoint to the singular set functor. Simplicial homotopy theory then rests entirely on the Kan (horn-filling) condition, without which path-connectedness is not transitive and homotopy of maps is not an equivalence relation, and with which the simplicial homotopy groups can be defined and shown to be groups.
Greg Friedman (Texas Christian University) wrote this expository introduction to simplicial sets and simplicial homotopy theory; it appeared as Rocky Mountain Journal of Mathematics 42 (2012), no. 2,…