Abstract
From the point of view of musical performance, gestures are the movements of the body of the performer when playing an instrument. This vague idea can be modeled mathematically, by mixing category theory and topology, giving rise to the definition of a topological gesture with a given skeleton and body in a topological space. The skeleton represents the abstract configuration of the body's limbs and the topological space is a generalization of the three-dimensional space where the body's movements are usually modeled. The collection of all gestures with the same skeleton and body in a fixed space has a canonical topology, yielding a space of gestures. This article intends to show that the space of gestures is homeomorphic to the function space , endowed with the compact-open topology. The topology of this space is the most natural choice for a space of functions, in the sense that it is related to the universal property of exponentials in the category of topological spaces. In particular, when the skeleton has a suitable property of finiteness, we show that the function space becomes a true exponential.
Acknowledgments
I thank the referees for their helpful comments, and I thank Co-Editors-in-Chief Thomas Fiore and Jason Yust for corrections to the galley proof.
Disclosure statement
No potential conflict of interest was reported by the author.
ORCID
Juan Sebastián Arias http://orcid.org/0000-0001-6812-7639
Notes
1 Informally, an adjunction consists of a pair of functors that are sorts of generalized inverses of each other; adjunctions are dialectic relations more profound and dynamic than isomorphisms of categories. See CitationMac Lane (Citation1998, Chapter IV) for the formal definition.
2 The study of exponentiable spaces and their relation to local compactness goes back to CitationFox (1945). Later, CitationDay Citationand Kelly (Citation1970) showed the link between exponentiable spaces and continuous lattices of open sets. The existence of exponentials of spaces is an important condition for convenient categories of spaces; see CitationSteenrod (1967).
3 The definition of local compactness used throughout this article is the following: a topological space X is said to be locally compact if for each point and each open neighborhood , there is a compact neighborhood of x contained in U. In the case when X is a Hausdorff space, this definition is equivalent to saying that each point in X has a compact neighborhood. In this way, every compact Hausdorff space is locally compact.
4 Or core-compact, according to the terminology in CitationEscardó and Heckmann (Citation2001–02, Section 5). The lattice of opens of a topological space X is continuous if any given open neighborhood V of a point x contains an open neighborhood U of x with the property that every open cover of V has a finite subcover of U.
5 See (CitationMac Lane Citation1998, Sections III.4 and III.5) for the definition of colimit and limit in a category.
6 In signs, when , this corresponds to the bijection (see CitationMac Lane [Citation1998, Equation (3) on page 117]), which follows from the bijection (see CitationMac Lane [Citation1998, Equation (1) on page 117]) applied to the opposite category and then writing the resulting sentence, namely , in terms of the category by putting all inverted arrows in their original place.
7 For the definition of product and equalizer in a category, see CitationMac Lane (Citation1998, Section III.4). In the case of the category of all topological spaces, the equalizer of a pair of continuous maps is the subspace E of X consisting of all x in X such that .
8 The Tychonoff topology on a Cartesian product of topological spaces is the least topology with the property that the natural projections and are continuous. In other words, it is the topology whose subbasic opens are those of the form (for U open in X) or (for V open in Y ).
9 This disjoint union coincides, of course, with .
10 Intuitively, to build a CW-complex we start with a discrete space, then we attach inductively copies of disks of increasing dimensions along their boundaries. As we will see, CW-complexes have desirable properties (Hausdorff, special behavior of compact subspaces) for our purposes. See CitationFritsch and Piccinini (Citation1993) for a quick introduction to CW-complexes and the basic definitions used here.
11 Proof of this detail. The affirmation implies . But . This means that , that is, . On the other hand .
12 Note that the compact-open topology on is just the topology on X.
13 Given a category and an object C of , a generalized element of C is a morphism with codomain C; cf. the definition of generalized element in a topos in CitationMac Lane and Moerdijk (Citation1992, pp. 236–7).