Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to bicategorical structures.
Keywords: Bicategories, equipments, formal theory of monads, generalized multicategories, lax categorification, tricategories}
2010 MSC: 18C15, 18D05, 18D50
Theory and Applications of Categories, Vol. 30, 2015, No. 10, pp 332-386.