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Conference Papers Year : 2015

Wild omega-Categories for the Homotopy Hypothesis in Type Theory

Abstract

In classical homotopy theory, the homotopy hypothesis asserts that the fundamental omega-groupoid construction induces an equivalence between topological spaces and weak omega-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 omega-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 omega-categories. The first outcome is a satisfactory type-theoretic notion of strict omega-category, which has hsets of cells in all dimensions. For this notion, defining the 'fundamental strict omega-category' of a type seems out of reach. The second outcome is an 'incoherently strict' notion of type-theoretic omega-category, which admits arbitrary types of cells in all dimensions. These are the 'wild' omega-categories of the title. They allow the definition of a 'fundamental wild omega-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.
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Dates and versions

hal-01178301 , version 1 (18-07-2015)

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André Hirschowitz, Tom Hirschowitz, Nicolas Tabareau. Wild omega-Categories for the Homotopy Hypothesis in Type Theory. Typed Lambda Calculi and Applications, 2015, Varsovie, Poland. pp.226-240, ⟨10.4230/LIPIcs.TLCA.2015.226⟩. ⟨hal-01178301⟩
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