2023
DOI: 10.1017/s0960129523000130
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Two-level type theory and applications

Abstract: We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. … Show more

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Cited by 8 publications
(4 citation statements)
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“…• Given an auto-internal implementation, we can still pretend that we have a meta-internal situation by restricting ourselves to a subset of the language. But we automatically get a two-level type theory [Voe13,ACKS23], where we have access to non-fibrant types from within. (This does not prove conservativity of two-level type theory over the object system.)…”
Section: Introduction and Related Workmentioning
confidence: 99%
“…• Given an auto-internal implementation, we can still pretend that we have a meta-internal situation by restricting ourselves to a subset of the language. But we automatically get a two-level type theory [Voe13,ACKS23], where we have access to non-fibrant types from within. (This does not prove conservativity of two-level type theory over the object system.)…”
Section: Introduction and Related Workmentioning
confidence: 99%
“…We describe models of 2LTT in the following. This is not the only possible way to present 2LTT; our approach differs from [ACKS19] in some ways. We will summarize the differences at the end of this section.…”
Section: Modelsmentioning
confidence: 99%
“…We digress a bit on a different way to present 2LTT. In the primary 2LTT reference [ACKS19], inner and outer layers are specified as follows. We have two different family structures on the base cwf, (Ty 0 , Tm 0 ) and (Ty 1 , Tm 1 ), and a morphism between them.…”
Section: Alternative Presentation For 2lttmentioning
confidence: 99%
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