Motivation
A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces which can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology (this will become clear in the formal definition).
To get back to actual topological spaces, there is a geometric realization functor available which takes values in the category of compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory have analogous versions for simplicial sets which generalize these results. While 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.
Read more about this topic: Simplicial Set
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