Singular homology on hypergestures
Guerino B. Mazzola · Journal of Mathematics and Music · 2012
In this paper, we interpret the basic cubic chain spaces of singular homology in terms of hypergestures in a topological space over a series of copies of the arrow digraph ↑. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of to topological spaces and (2) to any sequence of digraph (Γ n ) n∈ℤ instead of to the constant series of ↑. We then define the corresponding chain complexes and prove the core boundary operator equation ∂2 Its objects are the gestures with skeleton Γ and body in K, while its morphisms are continuous natural transformations between such gestures. =0, enabling the associated homology modules over a commutative ring R. We discuss some geometric examples and a musical one, interpreting contrapuntal rules in terms of singular homology.