M Theory Lesson 90

How does one discuss a category of operads when the intention was to define categories using operads in the first place? Such self referential questions appear to lurk behind the mystery of weak n-categories. We want an $\omega$-category of $\omega$-operads such that the categories defined as algebras are precisely what we get when we take the algebras of a special Koszul monad. Gee, that’s already way too much mathematics. And duality won’t do for all (categorical) dimensions: in logos theory there are n-alities, so we need a concept of Koszul n-ality. Fortunately, the importance of 2 to topos theory (a la locales and schizophrenic objects) is just the place we thought about extending dualities. So we want an $\omega$-category of $\omega$-operads such that an $\omega$-monad of Koszul n-alities gives weak n-categories as n-algebras. Sigh. Maybe we should return to pictures of trees and discs.

Aside: I don’t like inventing new words for things, since there are so many definitions in mathematics as it is, so I will ignore all objections to the term schizophrenic object. After all, do dwarfs object to having stars named after them? If a term is descriptive, it is a good term.

2 Responses so far »

  1. 1

    kneemo said,

    Maybe we should return to pictures of trees and discs.

    Or even better, return to the work of Mulase & Waldron and try to figure out what kind of ribbon graphs correspond to the Gaussian Exceptional Ensemble. 😉

  2. 2

    Kea said,

    Hi kneemo. Ah, yes, we must not forget all that. Where were we? Non associativity is important, and the only real ideas there are from Bar-Natan (as far as I know). And we were thinking about Fano planes to motivate triple-ribbons, if I recall.

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