Properadic Graphical Category

Abstract

We define the properadic graphical category Γ, whose objects are graphical properads. Its morphisms are called properadic graphical maps. To define such graphical maps, we first discuss coface and codegeneracy maps between graphical properads. We establish graphical analogs of the cosimplicial identities. The most interesting case is the graphical analog of the cosimplicial identity $$\displaystyle{d^{j}d^{i} = d^{i}d^{j-1}}$$ for i < j because it involves iterating the operations of deleting an almost isolated vertex and of smashing two closest neighbors together. Graphical maps do not have the bad behavior discussed in the examples in Chap. 5 In particular, it is observed that each graphical map has a factorization into codegeneracy maps followed by coface maps. Such factorizations do not exist for general properad maps between graphical properads. Finally, we show that the properadic graphical category admits the structure of a (dualizable) generalized Reedy category, in the sense of Berger and Moerdijk (Math. Z. 269(3–4), 977–1004, 2011).