splines on simplicial complexes in 1, 2, and 3 dimensions are well studied objects. as the dimension is raised, there is increased complexity of both the connectivity and geometric information. abstract simplicial complexes provide a means to separate connectivity from geometric information. they are well studied objects, and are a standard tool used to construct and study topologies. in this talk we present the theory of abstract simplicial complexes and splines necessary to understand a proof that the conditions for continuity on the lower order sub-simplices are contained in the conditions for continuity on the connected, higher order sub-simplices.