Consider that there are many instances where we "carry" data from representations in lower-level (or even higher-level, as in UI elements) parts of a program into other representations. Might there be appropriate cohomologies that would enable us to talk about (better yet, calculate) higher-dimensional things than just the following?
data reps 0-d vertices
functions 1-d edges
loops 2-d faces
?? 3-d solids