Non-linear contexts and dependent operads

Abstract

We study the category $\mathbf{Dep_{aff}}$ of finite posets and opfibrations between them, and prove that it is a perfect operator category in the sense of Barwick. Based on its additional properties, we propose a definition for a dependent operator category, and formulate a notion of dependent affine operad in a fibered way. In the particular case of $\mathbf{Dep_{aff}}$, we investigate the basic theory of such operads, establishing that the category of algebras for an operad is monadic over a canonical presheaf topos of "dependent sets". We finally extend this theory to a setting where we allow variable duplication, which we refer to as general dependent operads.