Operads without numbers
Ross Street – 11 March 1998
I want a conceptual machine which, when you input a decent category E, outputs a mathematical structure M(E). Input Set and you should get "monoid". Input the category of globular sets and you should get "omega-category".
For this purpose, the mathematical structure M(E) can be identified with an endofunctor of (monad on) E. When the machine is applied to the category Set, the endofunctor should be "words in" (that is, t(x) = the set of words in the alphabet x). When the category is globular sets, the endofunctor should be "trees labelled in" (that is, Batanin's functor D_s).
We review characterizations of the subcategories of End(Set) equivalent to the categories of (i) sequences of sets, (ii) set-based clubs, (iii) Joyal species, (iv) Lawvere theories. Note that "words in" corresponds to a terminal object of (i). Motivated by this, we define analytic endofunctors on E relative to a given endofunctor h : E --> E. We show that, if h is parametrically representable then so is any analytic endofunctor relative to it. We can then reproduce the characterizations (i) - (iv) for decent E.
Our desired machine takes M(E) to be the terminal object of the type (i) subcategory of End(E).
Operads can be identified with monads in categories of analytic endofunctors. Our machine gives us a way of obtaining operads directly. Natural numbers, trees, etc, can be obtained later as the value of the endofunctor M(E) at the terminal object 1 of E.