Extending homotopy type theory with strict equalityTools Altenkirch, Thorsten and Capriotti, Paolo and Nicolai, Kraus (2016) Extending homotopy type theory with strict equality. In: 25th EACSL Annual Conference on Computer Science Logic, 28 Aug  3 Sep 2016, Marseille, France.
AbstractIn homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semisimplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2level theory which features both strict and weak equality. This can essentially be represented as two type theories: an ``outer'' one, containing a strict equality type former, and an ``inner'' one, which is some version of HoTT. Our type theory is inspired by Voevodsky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we do not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the metatheoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of ``infinite structures'' which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a typetheoretic framework for working with ``schematic'' definitions in HoTT. As demonstrations, we define semisimplicial types and formalise constructions of Reedy fibrant diagrams.
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