TY - GEN

T1 - Extending homotopy type theory with strict equality

AU - Altenkirch, Thorsten

AU - Capriotti, Paolo

AU - Kraus, Nicolai

PY - 2016/9/1

Y1 - 2016/9/1

N2 - In 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 semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level 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 meta-theoretical 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 type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

AB - In 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 semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level 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 meta-theoretical 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 type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

KW - coherences

KW - homotopy type system

KW - homotopy type theory

KW - strict equality

UR - http://www.scopus.com/inward/record.url?scp=85012918859&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CSL.2016.21

DO - 10.4230/LIPIcs.CSL.2016.21

M3 - Conference contribution

AN - SCOPUS:85012918859

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 21:1-21:17

BT - 25th EACSL Annual Conference on Computer Science Logic 2016 (CSL 2016)

A2 - Talbot, Jean-Marc

A2 - Regnier, Laurent

PB - Schloss Dagstuhl

T2 - 25th EACSL Annual Conference on Computer Science Logic, CSL 2016 and the 30th Workshop on Computer Science Logic

Y2 - 29 August 2016 through 1 September 2016

ER -