Operadic categories and their skew monoidal categories of collections

TL;DR

This paper generalizes operadic categories and constructs skew monoidal categories of collections where monoids correspond to operads, providing new frameworks for understanding operadic structures.

Contribution

It introduces a broader class of operadic categories and constructs two skew monoidal categories of collections with specific properties, linking monoids to operads.

Findings

Two skew monoidal categories constructed for each operadic category

The first allows recovering the operadic category from collections

The second has an invertible right unit constraint

Abstract

I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions, for which an operad is just a symmetric operad in the usual sense, the first skew monoidal category has underlying category [N, Set], and the second is the usual monoidal category of collections [P, Set] with the substitution monoidal structure.