Infinity Properads and Infinity Wheeled Properads

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.