
Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional MartinLöf t...
read it

On Higher Inductive Types in Cubical Type Theory
Cubical type theory provides a constructive justification to certain asp...
read it

The Scott model of PCF in univalent type theory
We develop the Scott model of the programming language PCF in constructi...
read it

A Cubical Language for Bishop Sets
We present XTT, a version of Cartesian cubical type theory specialized f...
read it

The Integers as a Higher Inductive Type
We consider the problem of defining the integers in Homotopy Type Theory...
read it

Constructing Higher Inductive Types as Groupoid Quotients
In this paper, we show that all finitary 1truncated higher inductive ty...
read it

Bijective proofs for Eulerian numbers in types B and D
Let ⟨n k⟩, ⟨B_n k⟩, and ⟨D_n k⟩ be the Eulerian numbers in the types A, ...
read it
Partial Univalence in ntruncated Type Theory
It is well known that univalence is incompatible with uniqueness of identity proofs (UIP), the axiom that all types are hsets. This is due to finite hsets having nontrivial automorphisms as soon as they are not hpropositions. A natural question is then whether univalence restricted to hpropositions is compatible with UIP. We answer this affirmatively by constructing a model where types are elements of a closed universe defined as a higher inductive type in homotopy type theory. This universe has a path constructor for simultaneous "partial" univalent completion, i.e., restricted to hpropositions. More generally, we show that univalence restricted to (n1)types is consistent with the assumption that all types are ntruncated. Moreover we parametrize our construction by a suitably wellbehaved container, to abstract from a concrete choice of type formers for the universe.
READ FULL TEXT
Comments
There are no comments yet.