Natural models of homotopy type theory

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π and intensional identity types Id, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.

Type

Paper

Information

Mathematical Structures in Computer Science , Volume 28 , Issue 2 , February 2018 , pp. 241 - 286

DOI: https://doi.org/10.1017/S0960129516000268

Access options

Get access to the full version of this content by using one of the access options below.

References

Artin, M., Grothendieck, A. and Verdier, J.-L. (eds.) (1972/3). SGA4: Théorie des topos et cohomologie étale des schémas, 1963–1964. In: Lecture Notes in Mathematics 269, 270 and 305, Springer.Google Scholar

Awodey, S., Butz, C., Simpson, A. and Streicher, T. (2014). Relating first-order set theories, toposes and categories of classes. Annals of Pure and Applied Logic 165 (2) 428–502.CrossRef Google Scholar

Awodey, S. and Warren, M.A. (2009). Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 45–55.CrossRef Google Scholar

Bezem, M., Coquand, T. and Huber, S. (2014). A model of type theory in cubical sets. Unpublished preprint dated 3 May 2014.Google Scholar

Dybjer, P. (1996). Internal type theory. LNCS 1158 120–134.Google Scholar

Dyckhoff, R. and Tholen, W. (1987). Exponentiable morphisms, partial products and pullback complements. JPAA 7 (49) 103–116.Google Scholar

Fiore, M. (2012). Discrete generalised polynomial functors. Talk given at ICALP.Google Scholar

Fu, Y. (1997). Categorical properties of logical frameworks. Mathematical Structures in Computer Science 7 1–47.CrossRef Google Scholar

Gambino, N. and Garner, R. (2008). The identity type weak factorisation system. Theoretical Computer Science 409 (1) 94–109.CrossRef Google Scholar

Gambino, N. and Kock, J. (2013). Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society 154 153–192.CrossRef Google Scholar

Hofmann, M. (1994). On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L. and Tiuryn, J. (eds.) CSL, Springer 427–441.Google Scholar

Hofmann, M. (1997). Syntax and semantics of dependent types. In: Pitts, A.M. and Dybjer, P. (eds.) Semantics of Logics of Computation, Cambridge University Press.Google Scholar

Johnstone, P.T. (1992). Partial products, bagdomains and hyperlocal toposes. LMS Lecture Note Series 177 315–339.Google Scholar

Johnstone, P.T. (1994). Variations on the bagdomain theme. Theoretical Computer Science 136 3–20.CrossRef Google Scholar

Joyal, A. (2014). Categorical homotopy type theory. Slides from a talk at MIT dated 17 March 2014.Google Scholar

Kapulkin, C., LeFanu Lumsdaine, P. and Voevodsky, V. (2014). The simplicial model of univalent foundations. On the arXiv as 1211.2851v2, dated 15 April 2014.Google Scholar

LeFanu Lumsdaine, P. and Warren, M.A. (2015). The local universes model: An overlooked coherence construction for dependent type theories. ACM Transactions on Computational Logic 16 (3) Article No. 23.Google Scholar

Martin-Löf, P. (1975). An intuitionistic theory of types: Predicative part. In: Rose, H.E. and Shepherdson, J.C. (eds.) Studies in Logic and the Foundations of Mathematics, Logic Colloquium '73, Volume 80, Amsterdam, North-Holland Publishing Company 73–118.Google Scholar

Streicher, T. (2006). Identity Types and Weak Omega-Groupoids. Talks in Uppsala at a meeting on ‘Identity Types - Topological and Categorical Structure’.Google Scholar

Streicher, T. (2014). Semantics of type theory formulated in terms of representability. Unpublished note dated 26 February 2014.Google Scholar

The Stacks Project Authors (2016). Stacks Project. Available at: http://stacks.math.columbia.edu.Google Scholar

The Univalent Foundations Program, Institute for Advanced Study (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Available at: http://homotopytypetheory.org/book Google Scholar

van den Berg, B. and Garner, R. (2012). Topological and simplicial models of identity types. ACM Transactions on Computational Logic 13 1.CrossRef Google Scholar

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0

Total number of PDF views: 155 *

View data table for this chart

* Views captured on Cambridge Core between 17th November 2016 - 26th January 2021. This data will be updated every 24 hours.

4 Cited by

Article contents

Natural models of homotopy type theory

Abstract

Access options

References

Full text views

Send article to Kindle

Send article to Dropbox

Send article to Google Drive

Reply to: Submit a response

Your details

Conflicting interests