Abstract

In this talk the construction of a polynomial 2-monad from an operad is given which is different from the standard construction of a monad from an operad, in that the algebras of our associated 2-monad are categorified algebras of the original operad. In this way one can characterise operads and clubs as categorical polynomial monads in a canonical way. The standard construction of a monad from an operad is recovered in a 2-categorical way from our associated 2-monad, and one can understand the Set-valued algebras of both as weak morphisms of operads into a Cat-operad of categories. Algebras of operads within general symmetric monoidal categories are easily expressed in our framework, and the associated PROP of an operad is recovered as a codescent object from the polynomial 2-monad data. For a sigma-free operad, we establish a Quillen equivalence with respect to the Lack model structures, between the strict algebras of our associated 2-monad, and those of the standard one.