Positive Opetopes with Contractions form a Test Category

Abstract: We show that the category of positive opetopes with contraction morphisms, i.e. all face maps and some degeneracies, forms a test category. The category of positive opetopic sets pOpeSet can be defined as a full subcategory of the category of polygraphs Poly. An object of pOpeSet has generators whose codomains are again generators and whose domains are non-identity cells (i.e. non-empty composition of generators). The category pOpeSet is a presheaf category with the exponent being called the category of positive opetopes pOpe. Objects of pOpe are called positive opetopes and morphisms are face maps only. Since Poly has a full-on-isomorphisms embedding into the category of omega-categories oCat, we can think of morphisms in pOpe as omega-functors that send generators to generators. The category of positive opetopes with contractions pOpe_iota has the same objects and face maps pOpe, but in addition it has some degeneracy maps. A morphism in pOpe_iota is an omega-functor that sends generators to either generators or to identities on generators. We show that the category pOpe_iota is a test category.