Motivation of Simplicial Sets

A simplicial set is a purely algebraic model representing topological spaces that can be built up from simplices and their incidence relations. This is similar to the method of CW complexes to modeling topological spaces, with the critical difference that simplicial sets are purely algebraic and do not carry any actual topology.

To return back to topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. A topological space $X$ is said to be compactly generated if it satisfies the condition: A subspace $A$ is closed in $X$ if and only if $A\cap K$ is closed in $K$ for all compact subspaces $K\subseteq X$ . A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff.