On positive face structures and positive-to-one computads

We introduce the notion of a positive face structure. The positive face structures to positive-to-one computads are like simple graphs, c.f. [MZ], to free ω-categories over ω-graphs. In particular, they allow to give an explicit combinatorial description of positive-to-one computads. Using this description we show, among other things, that positive-to-one computads form a presheaf category with the exponent category being the category of principal positive face structures. We also present the Harnik argument in this context showing that the ω-categories are monadic over positive-to-one computads with the ’free functor’ being the inclusion Comp → ωCat.