Multicategories and operads in Cat-operads

Ross Street – 8 March 2000

A higher-dimensional version of the formal theory of monads could proceed in several directions. In particular, monads in a bicategory could generalize to substitudes in tricategories. Substitudes are multicategories with several objects. This talk defines multicategories in an arbitrary monoidal bicategory and describes their coherence. With local coproducts in the monoidal bicategory, we go on to show that even these multicategories (as in the case considered by Burroni/Hermida/Leinster) are monads in a suitably constructed bicategory. Recently Michael Batanin has been interested in the microcosmic phenomenon of operads within Cat-operads. We discuss the relationship between operads in Cat-operads and multicategories in monoidal bicategories.