Spaces of gestures are function spaces

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 $\mathrm{\Gamma }@X$ is homeomorphic to the function space $\mathbf{T}\mathbf{o}\mathbf{p}(|\mathrm{\Gamma }|,X)$, 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.

Keywords:

• gestures
• hypergestures
• geometric realization
• function spaces
• categories
• exponentials

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 Mac Lane (1998, Chapter IV) for the formal definition.

2 The study of exponentiable spaces and their relation to local compactness goes back to Fox (1945). Later, Day and Kelly (1970) 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 Steenrod (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 $x\in X$ and each open neighborhood $U\ni x$, 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 Escardó and Heckmann (2001–02, Section 5). The lattice of opens $\mathcal{O}\left(X\right)$ 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 (Mac Lane 1998, Sections III.4 and III.5) for the definition of colimit and limit in a category.

6 In signs, when $\mathcal{C}=\mathit{\text{Digraph}}$, this corresponds to the bijection $\mathcal{C}(ColimF,C)\cong Lim\mathcal{C}\left(F\right(-),C)$ (see Mac Lane [1998, Equation (3) on page 117]), which follows from the bijection $\mathcal{C}(C,LimF)\cong Lim\mathcal{C}(C,F(-\left)\right)$ (see Mac Lane [1998, Equation (1) on page 117]) applied to the opposite category ${\mathcal{C}}^{\mathrm{op}}$ and then writing the resulting sentence, namely ${\mathcal{C}}^{\mathrm{op}}(C,LimF)\cong Lim{\mathcal{C}}^{\mathrm{op}}(C,F(-\left)\right)$, in terms of the category $\mathcal{C}$ by putting all inverted arrows in their original place.

7 For the definition of product and equalizer in a category, see Mac Lane (1998, Section III.4). In the case of the category of all topological spaces, the equalizer of a pair of continuous maps $f,g:X⟶Y$ is the subspace E of X consisting of all x in X such that $f\left(x\right)=g\left(x\right)$.

8 The Tychonoff topology on a Cartesian product $X\times Y$ of topological spaces is the least topology with the property that the natural projections ${\text{π}}_1:X\times Y⟶X$ and ${\text{π}}_2:X\times Y⟶Y$ are continuous. In other words, it is the topology whose subbasic opens are those of the form ${\text{π}}_1^{-1}\left(U\right)$ (for U open in X) or ${\text{π}}_2^{-1}\left(V\right)$ (for V open in Y ).

9 This disjoint union coincides, of course, with $(I\times A)\cup V$.

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 Fritsch and Piccinini (1993) for a quick introduction to CW-complexes and the basic definitions used here.

11 Proof of this detail. The affirmation $K\subseteq q\left(\right(I\times A_0)\cup V_0)$ implies $K=K\cap q\left(\right(I\times A_0)\cup V_0)$. But $K\cap q\left(\right(I\times A_0)\cup V_0)=K\cap q\left(q^{-1}\right(K)\cap ((I\times A_0)\cup V_0\left)\right)=K\cap q\left(S\right)$. This means that $K=K\cap q\left(S\right)$, that is, $K\subseteq q\left(S\right)$. On the other hand $q\left(S\right)\subseteq q\left(q^{-1}\right(K\left)\right)=K$.

12 Note that the compact-open topology on $X^{\left\{z\right\}}$ is just the topology on X.

13 Given a category $\mathcal{C}$ and an object C of $\mathcal{C}$, a generalized element of C is a morphism with codomain C; cf. the definition of generalized element in a topos in Mac Lane and Moerdijk (1992, pp. 236–7).