Kan complex has nice pictorial interpretation in the simplicial complex, from the other side there is standard $n$-simplex. I am trying to understand how simplicial complex (and Kan horn in it and Kan complex in these terms) can be mapped, represented by standard $n$-simplex? How the vertices in simplicial complex be mapped to the elements of mappings $([m],[n])$ and how points/line segments/triangles can be mapped to the sets of such mappings and how the definition of horn looks into standard n-simplex?
I guess that such correspondence/mapping between simplicial complex standard $n$-simplex should exist, because many definitions (e.g., the definition of the Kan complex itself) involves standard $n$-simplices and their horns. So, standard $n$-simplices as some kind of quite universal structures.