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Semantics of higher inductive types

Part of: General logic Homological algebra Categories and theories

Published online by Cambridge University Press:  17 June 2019

PETER LEFANU LUMSDAINE  and
MICHAEL SHULMAN
PETER LEFANU LUMSDAINE
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, SE-10691, Sweden. e-mail: p.l.lumsdaine@gmail.com
MICHAEL SHULMAN
Affiliation:
Department of Mathematics, University of San Diego, 5998 Alcala Park, Sandiego, CA 92110, U.S.A. e-mail: shulman@sandiego.edu
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Abstract

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.

We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has weakly stable typal initial algebras for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.

Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

MSC classification

Primary: 18C50: Categorical semantics of formal languages
Secondary: 03B15: Higher-order logic and type theory 18G55: Homotopical algebra

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 159 - 208
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

This material is based on research sponsored by The United States Air Force Research Laboratory under agreement number FA9550-15-1-0053. The U.S. Government is authorised to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force Research Laboratory, the U.S. Government, or Carnegie Mellon University. The research presented here was also partly funded by the Swedish Research Council (VR) Grant 2015-03835 Constructive and category-theoretic foundations of mathematics.

References

REFERENCES

Altenkirch, T., Capriotti, P., Dijkstra, G. and N. Forsberg, F.. Quotient inductive-inductive types. CoRR (2016). arXiv:1612.02346.Google Scholar
Awodey, S., Gambino, N. and Sojakova., K.Homotopy-initial algebras in type theory. Extended abstract in LICS’12 (2015). arXiv:1504.05531.Google Scholar
Altenkirch, T. and Kaposi, A.. Type theory in type theory using quotient inductive types. ACM SIGPLAN Notices 51(1) (2016), 18–29.CrossRefGoogle Scholar
Awodey, S. and Warren, M. A.. Homotopy theoretic models of identity types. Math. Proc. Camb. Phil. Soc. 146(45) (2009), 45–55. arXiv:0709.0248.CrossRefGoogle Scholar
Bauer, A., Gross, J., Lefanu Lumsdaine, P., Shulman, M., Sozeau, M. and Spitters, B.. The HoTT Library: a formalisation of homotopy type theory in Coq. Certified Programs and Proofs (2017). arXiv:1610.04591.Google Scholar
Blass, A.. Words, free algebras and coequalisers. Fundamenta Mathematicae 117(2) (1983), 117–160. URL: http://eudml.org/doc/211359.CrossRefGoogle Scholar
Brunerie, G.. On the homotopy groups of spheres in homotopy type theory. PhD thesis, Université de Nice (2016). arXiv:1606.05916.Google Scholar
Cartmell, J.. Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32(0) (1986), 209 – 243.CrossRefGoogle Scholar
Coquand, T., Huber, S. and Mörtberg., A.On higher inductive types in cubical type theory. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ‘18, (New York, NY, USA, 2018), pages 255–264. ACM. doi:10.1145/3209108.3209197.CrossRefGoogle Scholar
Cisinski, D.-C.. Théories homotopiques dans les topos. J. Pure Appl. Algebra 174 (2002), 43–82.CrossRefGoogle Scholar
Cisinski, D.-C.. Les préfaisceaux comme modèles type d’homotopie. Astérisque, vol. 308 (Soc. Math. France), 2006.Google Scholar
Cisinski, D.-C.. Blog comment on post “The mysterious nature of right properness”. http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306 (May 2012).Google Scholar
Cisinski, D.-C.. Univalent universes for elegant models of homotopy types (2014). arXiv:1406.0058.Google Scholar
Garner, R.. Understanding the small object argument. Appl. Categ. Structures 17(3) (2009), 247–285. arXiv:0712.0724.CrossRefGoogle Scholar
GITIK, M.. All uncountable cardinals can be singular. Israel J. Math. 35(1–2) (1980), 61–88. doi:10.1007/BF02760939.CrossRefGoogle Scholar
Gepner, D. and Kock, J.. Univalence in locally cartesian closed ∞-categories. Forum Math. 29(3) (2017), 617–652. arXiv:1208.1749.CrossRefGoogle Scholar
Grandis, M. and Tholen, W.. Natural weak factorisation systems. Arch. Math. (Brno). 42(4) (2006), 397–408.Google Scholar
HoTT Project. The homotopy type theory Agda library. http://github.com/HoTT/HoTT-Agda/ (2013).Google Scholar
HoTT Project. The homotopy type theory Coq library. http://github.com/HoTT/HoTT/ (2015).Google Scholar
Hofmann, M. and Streicher, T.. The groupoid interpretation of type theory. In Twenty-five years of constructive type theory (Venice, 1995). Oxford Logic Guides, vol. 36, pages 83–111 (Oxford University Press, New York, 1998).Google Scholar
Kelly, G. M.. Doctrinal adjunction. In Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 257–280. Lecture Notes in Math., vol. 420 (Springer, Berlin, 1974).CrossRefGoogle Scholar
Kelly, G. M.. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22(1) (1980), 1–83.CrossRefGoogle Scholar
Kapulkin, C. and Lumsdaine, P. Lefanu. The simplicial model of univalent foundations (after Voevodsky) (2019) J. Eur. Math. Soc. arXiv:1211.2851.Google Scholar
Kraus, N.. Constructions with non-recursive higher inductive types. LICS’16 (2016).Google Scholar
Lurie, J.. Higher topos theory. Annals of Mathematics Studies, no. 170 (Princeton University Press, 2009). arXiv:math.CT/0608040.CrossRefGoogle Scholar
Lefanu Lumsdaine, P. and Warren, M. A.. The local universes model: An overlooked coherence construction for dependent type theories. ACM Trans. Comput. Logic 16(3) (July 2015), 23:1–23:31. arXiv:1411.1736.Google Scholar
NLAB Authors. Transfinite construction of free algebras. https://ncatlab.org/nlab/show/transfinite+construction+of+free+algebras.Google Scholar
Riehl, E.. Algebraic model structures. New York J. Math., 17 (2011), 173–231. arXiv:0910.2733.Google Scholar
Riehl, E.. Monoidal algebraic model structures. J. Pure Appl. Alg., 217(6): (2013), 1069–1104. http://doi.org/10.1016/j.jpaa.2012.09.029.CrossRefGoogle Scholar
Rijke, E.. Localisation at maps between omega-compact types via a small object argument in HoTT. http://www.fields.utoronto.ca/talks/localizationmaps-between-omega-compact-types-small-object-argument-hott (May 2016).Google Scholar
Rijke, E.. The join construction (2017). arXiv:1701.07538.Google Scholar
Rijke, E., Shulman, M. and Spitters., B. Modalities in homotopy type theory. arXiv:1706.07526 (2017).Google Scholar
Shulman, M.. Higher inductive-recursive univalence and type-directed definitions. http://homotopytypetheory.org/2014/06/08/hiru-tdd/ (June 2014).Google Scholar
Shulman, M.. The univalence axiom for elegant Reedy presheaves. Homology, Homotopy Appl. 17(2) (2015), 81–106. arXiv:1307.6248.CrossRefGoogle Scholar
Shulman, M.. Univalence for inverse diagrams and homotopy canonicity. Math. Structures Comput. Sci., 25 6 2015, 1203–1277. arXiv:1203.3253.CrossRefGoogle Scholar
Shulman, M.. Univalence for inverse EI diagrams. Homology, Homotopy Appl. 19(2) (2017), 219–249. arXiv:1507.03634.CrossRefGoogle Scholar
Shulman, M. (mathoverflow.net/users/49). Enriched cartesian closed categories. MathOverflow (2019). URL: http://mathoverflow.net/questions/322085.Google Scholar
Streicher, T.. Semantics of type theory: correctness, completeness and independence results. Progress in Theoretical Computer Science. (Birkhäuser, 1991).Google Scholar
Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. http://homotopytypetheory.org/book/, first edition (2013).Google Scholar
Van Doorn, F.. Constructing the propositional truncation using non-recursive HITs. Certified Programs and Proofs ‘16 (2016). arXiv:1512.02274.Google Scholar
Van Den Berg, B. and Moerdijk, I.. W-types in homotopy type theory. Math. Structures Comput. Sci., 25(6) (2015), 1100–1115. doi:10.1017/S0960129514000516.CrossRefGoogle Scholar