I’m not sure when it was I first noticed the relationship between manifolds and semiotics but I distinctly recall the passage in Serge Lang’s Differential and Riemannian Manifolds which brought the triadic character of tangent vectors into high relief.  I copied out a set of excerpts highlighting the point and shared it with the Inquiry, Ontology, and Peirce lists.

Excerpts from Serge Lang, Differential and Riemannian Manifolds,
Springer‑Verlag, New York, NY, 1995.

• Inquiry Archives • Part 1 • Part 2

Chapter 2.  Manifolds

• 2.1. Atlases, Charts, Morphisms • (1) • (2) • (3) • (4) • (5) • (6)

• 2.2. Submanifolds, Immersions, Submersions • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18)

Using the concepts and terminology from Lang’s text, I explained the connection between manifold theory and semiotics in the following way.

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