Infinity Wheeled Properads

Abstract

We define the adjunction $L: \mathtt{Set}^{\Gamma _{\circlearrowright }^{\mathop{\mathrm{op}}\nolimits } } \rightleftarrows \mathtt{Properad}^{\circlearrowright }: N$ between wheeled properads and wheeled properadic graphical sets. Then we define ∞-wheeled properads as wheeled properadic graphical sets that satisfy an inner horn extension property. Next we give two alternative characterizations of strict ∞-wheeled properads, one in terms of the wheeled properadic Segal maps, and the other in terms of the wheeled properadic nerve. In the last section, we give an explicit description of the fundamental wheeled properad L K of an ∞-wheeled properad K in terms of homotopy classes of 1-dimensional elements.