Operadic Fibrations and Unary Operadic 2-categories

Abstract: We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction for categorical $\mathbb{O}$-operads, $\mathbb{O}$ being a unary operadic category. The construction is a fully faithful functor $\int_\mathbb{O}$ which takes categorical $\mathbb{O}$-operads to operadic functors over $\mathbb{O}$, and we characterise its essential image by certain lifting properties. Such operadic functors are called operadic fibration. 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 $\mathscr{V}$ and its Grothendieck construction $\int_\odot \mathscr{V}$ is connected to the `Para' construction appearing in machine learning. The 2-categorical setting provides a characterisation of $\mathbb{O}$-operads valued in $\mathscr{V}$ as operadic functors $\mathbb{O} \to \int_\odot \mathscr{V}$. Last, we describe a left adjoint to $\int_\odot$.