The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye.
It made me wonder if it is possible to describe the Beck-Chevalley condition geometrically, by means of simplices. In particular I mean e.g to describe the Beck-Chevalley condition in a similar way a Kan complex is defined in terms of a filling condition, i.e. somethings extend to simplices.
P.S. It seems the question took little of attention, should I try to reformulate it somehow? Any suggestions are welcome.
P.P.S. By somethings extend to simplices I mean something like: Kan condition (extension to tetrahedron): If three 2-dimensional triangles are such that any two share the 1-dimensional boundary, than there is a 3-dimensional simplex filling the gap extending them to a tetrahedron.
Beck-Chevalley condition: If two 2-dimensional triangles are such that they share the 1-dimensional boundary, then there is a 3-dimensional simplex filling the gap extending them to a tetrahedron.