Operadic categories as a natural environment for Koszul duality

TL;DR

This paper establishes foundational properties of operadic categories to develop Koszul duality theory for operads, focusing on the operadic category of graphs and its variants, and sets the stage for further duality and Koszulity results.

Contribution

It introduces additional properties of operadic categories necessary for Koszul duality, analyzes the operadic category of graphs, and connects to Loday's question about encoding operad types.

Findings

Operadic categories can be equipped with properties enabling Koszul duality.

The operadic category of graphs satisfies these properties.

The work addresses how to encode types of operads.

Abstract

This is the first paper of a series which aims to set up the cornerstones of Koszul duality for operads over operadic categories. To this end we single out additional properties of operadic categories under which the theory of quadratic operads and their Koszulity can be developped, parallel to the traditional one by Ginzburg and Kapranov. We then investigate how these extra properties interact with discrete operadic (op)fibrations, which we use as a powerful tool to construct new operadic categories from old ones. We pay particular attention to the operadic category of graphs, giving a full description of this category (and its variants) as an operadic category, and proving that it satisfies all the additional properties. Our present work provides an answer to a question formulated in Loday's last talk in 2012:``What encodes types of operads?''. In the second and third papers of our series we continue Loday's program by answering his second question: ``How to construct Koszul duals to these objects?'', and proving Koszulity of some of the most relevant operads.