Wheeled Properads and Graphical Wheeled Properads

Abstract

We first recall from Yau and Johnson (A Foundation for PROPs, Algebras, and Modules. Mathematical Surveys and Monographs, vol. 203, Am. Math. Soc., Providence, 2015) the biased and the unbiased definitions of a wheeled properad. There is a symmetric monoidal structure on the category of wheeled properads. Then we define graphical wheeled properads as free wheeled properads generated by connected graphs, possibly with loops and directed cycles. With the exception of the exceptional wheel, a graphical wheeled properad has a finite set of elements precisely when the generating graph is simply connected. So most graphical wheeled properads are infinite. In the rest of this chapter, we discuss wheeled versions of coface maps, codegeneracy maps, and graphical maps, which are used to define the wheeled properadic graphical category \(\Gamma _{\circlearrowright }\). Every wheeled properadic graphical map has a decomposition into codegeneracy maps followed by coface maps.