Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the ``lax algebras'' or ``Kleisli monoids'' relative to a ``monad'' on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.
Keywords: Enriched categories, change of base, monoidal categories, double categories, multicategories, operads, monads
2000 MSC: 18D05,18D20,18D50
Theory and Applications of Categories,
Vol. 24, 2010,
No. 21, pp 580-655.