Spencer Breiner: "Polynomial Interfaces"

[@veztron]

Expressing my interest in the full lecture. Also, great questions all around and really nice to hear questions from Will and Slim.

[@davidspivak8343]

This is great; thanks Spencer! I just wanted to mention a typo at 25:06, where the combinatorial should say 12-->4, because there was a slight mistake in the computation on the right.

[@fbkintanar]

Interesting. There is a common intuition that a UI provides a specific language, a visual language. I wonder if a similar approach to categories and polynomials might serve to characterize other visual languages, and reductively some spoken or written languages or notations. In particular, I am thinking about sign language, where there are a number of parallel carriers of information (Dominant and Nondominant handshapes, their movement, a signing space, and a collection of distinctive facial expressions that play a syntactic or phonemic role (other facial expressions are more prosodic, like intonation rather than like phonemes or morphemes). I am thinking that individual lexical signs are polynomials, carrying some kind of lexical content. Functional morphemes may be functors on signs.

[@asdfghyter]

after watching the polynomial morphisms section like 10 times over several days and thinking about it in between I think I understood it! if we interpret elements of the polynomial as functions, then the downwards arrows are just the domain morphisms, which maps a function to its domain. the pullback is defined by the family of *Q'_{f_0(a)}* which will point to `a` in `P_0`. Each function `g_f : a -> r` in *Q'_f* is defined by a corresponding `g : f_0(a) -> r` in *Q'*. The rightward morphism in the pullback is just applying this definition. the main confusion that remains is what category the commuting diagram is in, where both sets of arrows and sets of dots are objects? How can both `P_0` and `P'` be objects in the same category?