On the topological characterization of gestures in a convenient category of spaces

ABSTRACT

In this paper, we reconsider the topological characterization of gestures in a convenient category of spaces mentioned by Mazzola in 2009, recovering Arias's 2018 result that the relevant equivalence is a homeomorphism. We also show the topological characterization of gestures extends to an adjunction between the category of gestures and the category of continuous maps whose domain is a one-dimensional CW complex. Our arguments utilize only basic tools from category theory and almost no point-set topology. With little generality lost from our restriction of spaces considered, the ease of conceptualization in this setting provides an advantageous entry point for researchers interested in studying or applying gestures but who may not have advanced knowledge of category theory, algebraic geometry, or point-set topology. Thus one may view this work, in part, as an advertisement for the topological interpretation of gestures developed within.

Keywords:

• Gesture
• hypergesture
• CW complex
• adjoint functor
• category
• topology
• mapping space

2010 Mathematics Subject Classification:

• 18A10
• 18A40
• 18B30
• 54B30
• 54C35

Acknowledgments

This work is heavily inspired by the work of Juan Sebastián Arias in Arias (2018), Guerino Mazzola and his collaborators in Mazzola et al. (2017), as well as Dmitri Tymoczko in Tymoczko (2011). The author is grateful for the constructive criticisms received from the referees and editors, which served to vastly improve the quality of the exposition.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Timothy L. Clark http://orcid.org/0000-0002-9954-5793

Notes

1 Strictly speaking, we form the free category (Mac Lane 1978, Section II.7, Theorem 1) on Γ, and consider functors from this category into a category of vector spaces.

2 Called the skeleton in Mazzola et al. (2017), but we will reserve this terminology for CW complexes.

1 The connection between these geometric ideas and gestures is not lost on Tymoczko, as indicated in a footnote on page 32 of Tymoczko (2011).