Operadic Fibrations and Unary Operadic 2-categories

Abstract

We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction of a categorical O-operad, O being a unary operadic category. The construction is a fully faithful functor $\int_O$ which takes categorical O-operads to operadic functors over O, and we characterize its essential image by certain lifting properties. Such operadic functors are called operadic fibrations. Our theory is an extension of the discrete (unary) operadic case and, in some sense, of the classical Grothendieck construction of a categorical presheaf. For the terminal unary operadic category $\odot$, a categorical $\odot$-operad is a strict monoidal category V and its Grothendieck construction $\int_\odot V$ is connected to the `para' construction appearing in machine learning. The 2-categorical setting provides a characterization of O-operads valued in V as operadic functors $O \to \int_\odot V$. Last, we describe a left adjoint to $\int_\odot$.