General Operads and Multicategories

Notions of operad' and multicategory’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term (S,*)-multicategory', subject to certain conditions on S and *. Different choices of S and * give some of the existing notions of operad and multicategory. We then describe the algebras’ for an (S,*)-multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan’s opetopes and Batanin’s operads; both of these are included.