Wild ω-Categories for the Homotopy Hypothesis in Type Theory

In classical homotopy theory, the homotopy hypothesis asserts that the fundamental ω-groupoid construction induces an equivalence between topological spaces and weak ω-groupoids. In the light of Voevodsky’s univalent foundations program, which puts forward an interpretation of types as topological spaces, we consider the question of transposing the homotopy hypothesis to type theory. Indeed such a transposition could stand as a new approach to specifying higher inductive types. Since the formalisation of general weak ω-groupoids in type theory is a difficult task, we only take a first step towards this goal, which consists in exploring a shortcut through strict ω-categories. The first outcome is a satisfactory type-theoretic notion of strict ω-category, which has hsets of cells in all dimensions. For this notion, defining the ‘fundamental strict ω-category’ of a type seems out of reach. The second outcome is an ‘incoherently strict’ notion of type-theoretic ωcategory, which admits arbitrary types of cells in all dimensions. These are the ‘wild’ ω-categories of the title. They allow the definition of a ‘fundamental wild ω-category’ map, which leads to our (partial) homotopy hypothesis for type theory (stating an adjunction, not an equivalence). All of our results have been formalised in the Coq proof assistant. Our formalisation makes systematic use of the machinery of coinductive types. 1998 ACM Subject Classification F.4.1 Mathematical logic