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Types are internal infinity-groupoids


By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of ∞-groupoid internal to type theory and we prove that the type of such ∞-groupoids is equivalent to the universe of types. That is, every type admits the structure of an ∞-groupoid internally, and this structure is unique.
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Dates and versions

hal-03133144 , version 1 (05-02-2021)
hal-03133144 , version 2 (05-01-2022)


  • HAL Id : hal-03133144 , version 2


Eric Finster, Antoine Allioux, Matthieu Sozeau. Types are internal infinity-groupoids. LICS 2021, Jun 2021, Rome, Italy. ⟨hal-03133144v2⟩
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