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I NEVER imagined matmul as a 2D -> 3D transformation. Reminds me of that scene in the movie, CONTACT, where the aging mathematician solves the alien primer by

• taliesinb 3y ago
  Yeah, I thought exactly the same thing when I watching Contact again a few months ago! There are all kinds of fascinating places where you can gain mental leverage by thinking in higher dimensions. For example, the definition of a monoidal category, which includes various equivalences (or for a strict monoidal category, equalities), can be seen as telling you about the existence of certain 3-dimensional "sheets", 2-dimensional slices of which are equivalent (or equal) ordinary functorial string diagrams[0]. This is just a higher dimensional extension of the fact that chaining 1-dimensional slices of functorial string diagrams give you particular paths in an ordinary commutative diagram. see Marsden [1] for more on that. Unfortunately the computer tools for generating and manipulating these kinds of topological constructs are in their infancy, which is probably why they aren't used much by mathematicians. [0]: https://twitter.com/nathanielvirgo/status/1262019641720832001 https://twitter.com/nathanielvirgo/status/126201964172083200... [1]: https://arxiv.org/abs/1401.7220v2 https://arxiv.org/abs/1401.7220v2