Plus-constructions I have known

Steve Lack – 28 May 1997

For a small category C, I exhibited monoidal right-closed structures on the following categories [C^op x C, Set] [m(C)^op x C, Set] [sm(C)^op x C, Set] [fl(C)^op x C, Set] where m(C) is the free monoidal category on C, where sm(C) is the free symmetric monoidal category, and fl(C) the free category with finite limits. In each case, considering monoids in the monoidal category in question gives a notion of operad. Case (3) gives precisely the C-operads of Baez-Dolan (see their paper Higher-Dimensional Algebra III: n-dimensional categories and the algebra of opetopes, while case (2) gives those C-operads which they call planar (or non-permutative). Case (1) gives monads on [C,Set] which have a right adjoint, and case (4) gives finitary monads on [C,Set]. In each case (except maybe (2), which I haven't checked properly) one can imitate the ``plus-construction'' of Baez-Dolan; given an operad M and an algebra A for the operad, it exhibits M-algebras over A as algebras for another operad called A+. The next step is to exhibit C-operads as algebras for some other operad, and carry out the ``plus-construction'' on this latter operad. Baez and Dolan also want to carry out this process for operads in a restricted sense in which C is required to be discrete; it was precisely in order to repeat this for the case of finitary monads (i.e. for the operads of case (4) above) that I came to prove the results in my talk On the monadicity of finitary monads.