Abstract
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 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.
BibTeX  Entry
@InProceedings{altenkirch_et_al:LIPIcs:2016:6561,
author = {Thorsten Altenkirch and Paolo Capriotti and Nicolai Kraus},
title = {{Extending Homotopy Type Theory with Strict Equality}},
booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
pages = {21:121:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770224},
ISSN = {18688969},
year = {2016},
volume = {62},
editor = {JeanMarc Talbot and Laurent Regnier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6561},
URN = {urn:nbn:de:0030drops65612},
doi = {10.4230/LIPIcs.CSL.2016.21},
annote = {Keywords: homotopy type theory, coherences, strict equality, homotopy type system}
}
Keywords: 

homotopy type theory, coherences, strict equality, homotopy type system 
Collection: 

25th EACSL Annual Conference on Computer Science Logic (CSL 2016) 
Issue Date: 

2016 
Date of publication: 

29.08.2016 