Modules for substitudes
Ross Street – 12 June 2002
While James Dolan was visiting Macquarie he convinced me that (two-sided) modules for the terminal monoid in the monoidal category Set/N, with substitution tensor product, are precisely monoidal functors from the opposite of the algebraic simplicial category (with ordinal sum) to Set (with cartesian product). Alexei Davydov rediscovered the result and asked me whether I had a conceptual reason for it. This topic grew out of that question.
The importance of considering such (two-sided) modules comes from Batanin's idea [appearing in "Monoidal globular categories as a natural environment for the theory of weak n-categories", Advances in Math 136 (1998) 39-103] that morphisms of weak omega-categories are definable using a deformation of the identity module of the higher operad K for weak omega-categories.
Monoids in the substitution monoidal category Set/N are of course (not-necessarily-symmetric) operads. [The terminal monoid is the operad for monoids.] An operad is a substitude structure on the terminal category 1 (see Day-Street "Abstract substitution in enriched categories" available on my publication web site). Generalizing the fact that each operad gives rise to a strict monoidal category (the PROP without the second "P" for permutations), each substitude B gives rise to a strict monoidal category B^*. There is also a standard convolution lax monoidal structure on the functor category [A, Set] for any substitude A. Recall from Day-Street "Lax monoids, pseudo-operads, and convolution, Contemporary Mathematics (to appear)" that substitudes are monads in an appropriate Kleisli bicategory, so the notion of "bimodule" between substitudes has a straightforward meaning. I claim that such a module from the substitude A to the substitude B is a morphism (B*)op-->[A,Set] of lax monoidal categories (where (B*)op has the lax monoidal structure coming from the strict monoidal B* while [A,Set] has the standard convolution structure. This generalizes Dolan's observation. A conceptual reason for this generalization begins with the observation that B* is obtained by the Kleisli construction for a certain monad on the free strict monoidal category on B in a suitable module bicategory of lax monoidal categories.