
Partial Univalence in ntruncated Type Theory
It is well known that univalence is incompatible with uniqueness of iden...
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The univalence axiom in cubical sets
In this note we show that Voevodsky's univalence axiom holds in the mode...
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Relational Type Theory (All Proofs)
This paper introduces Relational Type Theory (RelTT), a new approach to ...
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Axioms for Modelling Cubical Type Theory in a Topos
The homotopical approach to intensional type theory views proofs of equa...
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Effective Kan fibrations in simplicial sets
We introduce the notion of an effective Kan fibration, a new mathematica...
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Homotopy canonicity for cubical type theory
Cubical type theory provides a constructive justification of homotopy ty...
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Constructing Higher Inductive Types as Groupoid Quotients
In this paper, we show that all finitary 1truncated higher inductive ty...
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Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional MartinLöf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is nontruncated, i.e. contains nontrivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing nontruncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
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