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We construct a univalent universe in the sense of Voevodsky in some suitablemodel categories for homotopy types (obtained from Grothendieck's theory oftest categories). In practice, this means for instance that, appart from thehomotopy theory of simplicial sets, intensional type theory with the univalentaxiom can be interpreted in the homotopy theory of cubical sets (withconnections or not), or of Joyal's cellular sets.

Univalent universes for elegant models of homotopy types