Infinity Properads and Infinity Wheeled Properads

TL;DR

This paper develops a comprehensive theory of infinity properads and infinity wheeled properads, extending existing higher category and operad theories, and introduces new graphical and homotopical characterizations.

Contribution

It extends the theory of infinity properads to include wheeled structures, introduces graphical categories and nerve functors, and provides new characterizations of strict infinity properads.

Findings

Defined infinity properads via graphical sets satisfying horn extension properties.

Constructed symmetric monoidal closed structures for properads and graphical sets.

Characterized the fundamental properad using homotopy classes of 1-dimensional elements.

Abstract

A theory of $\infty$-properads is developed, extending both the Joyal-Lurie $\infty$-categories and the Cisinski-Moerdijk-Weiss $\infty$-operads. Every connected wheel-free graph generates a properad, giving rise to the graphical category $\Gamma$ of properads. Using graphical analogs of coface maps and the properadic nerve functor, an $\infty$-properad is defined as an object in the graphical set category $Set^{\Gamma^{op}}$ that satisfies some inner horn extension property. Symmetric monoidal closed structures are constructed in the categories of properads and of graphical sets. Strict $\infty$-properads, in which inner horns have unique fillers, are given two alternative characterizations, one in terms of graphical analogs of the Segal maps, and the other as images of the properadic nerve. The fundamental properad of an $\infty$-properad is characterized in terms of homotopy classes of $1$-dimensional elements. Using all connected graphs instead of connected wheel-free graphs, a parallel theory of $\infty$-wheeled properads is also developed.