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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.
We show that the (positive) zoom complexes, here called tree complexes, with fairly natural morphisms, form a dual category to the category of positive opetopes with contraction epimorphisms. We also show how this duality can be slightly generalized to thicket complexes and opetopic cardinals.
We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link conditio
This preprint contains a part of the results of our earlier preprint arXiv:0907.3335v2 presented in a form suitable for journal publication. It covers a construction of a 2-fold monoidal structure on the category of tetramodules, with all necessary
We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up
When $mathbb C$ is a semi-abelian category, it is well known that the category $mathsf{Grpd}(mathbb C)$ of internal groupoids in $mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property