Singular Homology of Hypergestures (original) (raw)

2017, The Topos of Music III: Gestures

In this chapter 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 Ò. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of topological spaces, and (2) to any sequence of digraph pΓ n q nPZ instead of the constant series of Ò. We then define the corresponding chain complexes, and prove the core boundary operator equation B 2 " 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.-Σ-63.1 An Introductory Example Let us give an introductory example of the Escher Theorem, which has musical relevance, before we embark on the homological theme. Let G be a topological group, X a topological space, and GˆX Ñ X a continuous group action. Denote by G X the topological category whose objects are the elements of X, and whose morphisms g : x Ñ y are the triples px, y, gq P X 2ˆG such that y " gx, the topology of G X being induced from the product topology on X 2ˆG. If the topologies are all indiscrete, a continuous curve F : ∇ Ñ G X is just a functor. A classical example for such a topological category G X from transformational music theory is the canonical action of the general affine group G " Ý Ñ GLpZ 12 q " T Z12¸Zˆo n the pitch class set X " Z 12 , together with the indiscrete topology. Recall that in Section 62.2.1, we have constructed special curves, so-called discrete gestures, OE pgq : ∇ Ñ G X for every morphism g : x Ñ y as follows: