On rectification and enrichment of infinity properads

TL;DR

This paper develops a comprehensive theory of infinity properads enriched in symmetric monoidal infinity categories, introduces new graph categories, and explores rectification limitations compared to infinity operads.

Contribution

It introduces a new framework for enriched infinity properads using a novel category of level graphs and analyzes rectification issues in this context.

Findings

Infinity properads cannot always be rectified to strict forms.

A new category of level graphs is introduced for defining enriched infinity properads.

Rectification theorems are established for infinity dioperads and output properads.

Abstract

We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition, over certain categories of graphs. In particular, we introduce a new category of level graphs which also allow us to give a framework for algebras over an enriched infinity properad. We show that one can vary the category of graphs without changing the underlying theory. We also show that infinity properads cannot always be rectified, indicating that a conjecture of the second author and Robertson is unlikely to hold. This stands in stark contrast to the situation for infinity operads, and we further demarcate these situations by examining the cases of infinity dioperads and infinity output properads. In both cases, we provide a rectification theorem that says that each up-to-homotopy object is equivalent to a strict one.