Diagrammatic Algebra, Part 1

J. Scott Carter (University of South Alabama)

This talk is based upon joint work with Seiichi Kamada. Abstract tensor notation for multi-linear maps uses boxes with in-coming and out-going strings to represent structure constants. A judicious choice of variables for such constants leads to the diagrammatic representation: boxes are replaced by glyphs. One of the most simple examples of a multi-category is the diagrammatic representation of embedded surfaces in 3-space that we all grew up learning. The standard drawing of a torus (an oval, a smile and a moustache) is a representation based upon drawings of surface singularities. We start from a two object category with a pair of arrows that relate them. Cups and caps are constructed easily. From these, births, deaths, saddles, forks, and cusps are created as triple arrows. At the top level, there are critical cancellations, lips, beak-to-beak, horizontal cusps, and swallow-tails. Interestingly, the structure extends inductively to describe many relations about handles in higher dimensions.

CKVK* Recommendation: The talk has many beautiful figures that are best viewed on Youtube in HD.

2 thoughts on “Diagrammatic Algebra, Part 1

  1. Thank you for a great talk! If you would like to ask a question of Prof. Carter, please reply to this comment below.

    1. To compose arrows in a category, I just continue from one to the next and complete the triangle. How do I compose arrows globularly? Thank you!

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