[Paper Review] Positive Opetopes with Contractions form a Test Category

Marek Zawadowski|arXiv (Cornell University)|Dec 16, 2017

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper establishes that the category of positive opetopes with contraction morphisms—face maps and selected degeneracies—forms a test category. By constructing a cylinder functor that becomes a product with the interval object in the presheaf category, the authors prove that this category models homotopy types, extending the foundational role of $Δ$ to higher-dimensional category theory.

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.

Motivation & Objective

• To establish that the category of positive opetopes with contraction morphisms is a test category.
• To extend the homotopical framework of opetopic sets to include degeneracy maps via $ι$-maps.
• To show that the presheaf category over this extended category models homotopy types via a cylinder functor.
• To verify that the cylinder construction in the presheaf category becomes a product with the interval object, satisfying test category axioms.
• To provide a categorical foundation for higher-dimensional algebra and homotopy theory using opetopes with contractions.

Proposed method

• Define the category ${\bf pOpe}_{ι}$ as an extension of ${\bf pOpe}$ by adding specific degeneracy maps ($\u03b9$-maps) that send generators to generators or identities on generators.
• Construct a cylinder functor ${\bf Cyl}_p$ on the presheaf category $\widehat{{\bf pOpe}}$ using flags, punctured flags, and successor operations on opetopes.
• Prove that ${\bf Cyl}_p(P)$ is a straight object, ensuring it satisfies the necessary conditions for a test category.
• Extend the cylinder functor to $\widehat{{\bf pOpe}_{ι}}$ via left Kan extension along the inclusion $\kappa: {\bf pOpe} \to {\bf pOpe}_{ι}$, showing it becomes isomorphic to the product with the interval $I$.
• Use the universal property of the cylinder and the full embedding of ${\bf pHg}_{ι}$ into $\widehat{{\bf pOpe}_{ι}}$ to verify that ${\bf Cyl}_{ι}(P)$ satisfies the product condition.
• Leverage results from prior work on positive opetopic cardinals and hypergraphs to establish technical lemmas on face maps, kernels, and domain preservation under $\u03b9$-maps.

Experimental results

Research questions

• RQ1 Does the category of positive opetopes with contraction morphisms satisfy the axioms of a test category?
• RQ2 Can the cylinder functor on $\widehat{{\bf pOpe}}$ be extended to $\widehat{{\bf pOpe}_{ι}}$ such that it becomes isomorphic to the product with the interval object $I$?
• RQ3 Do the $\u03b9$-maps (contraction morphisms) preserve the domain and codomain structure of opetopes in a way that supports homotopical modeling?
• RQ4 Is the left Kan extension of the cylinder functor along $\kappa: {\bf pOpe} \to {\bf pOpe}_{ι}$ sufficient to ensure that ${\bf Cyl}_{ι}(P)$ is a straight object?
• RQ5 Can the category ${\bf pOpe}_{ι}$ serve as a basis for modeling homotopy types via its presheaf category, analogous to $\Delta$?

Key findings

• The category ${\bf pOpe}_{\iota}$ of positive opetopes with contraction morphisms is a test category, as it satisfies the necessary conditions for modeling homotopy types.
• The cylinder functor ${\bf Cyl}_{\iota}(P)$ in $\widehat{{\bf pOpe}_{\iota}}$ is isomorphic to the product $I \times P$, where $I$ is the 1-dimensional opetope, confirming the product structure.
• The left Kan extension of the cylinder functor along $\kappa: {\bf pOpe} \to {\bf pOpe}_{\iota}$ preserves colimits, representables, and monomorphisms, ensuring the extension is well-behaved.
• The cylinder ${\bf Cyl}_p(P)$ is a straight object in $\widehat{{\bf pOpe}}$, which is a key requirement for the test category property.
• The map $H$ defined on faces of the cylinder preserves domains and codomains, and induces bijections on face sets, ensuring compatibility with the product structure.
• The embedding of ${\bf pHg}_{\iota}$ into $\widehat{{\bf pOpe}_{\iota}}$ is full on hom-sets, which allows the universal property of the cylinder to lift to the presheaf category.