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.