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Model structures on categories of models of type theories


Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory T has enough structure, then the category T-Mod of its models carries the structure of a model category. We also show that if T has Σ-types, then weak equivalences can be characterized in terms of homotopy categories of models.

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Avigad, J., Kapulkin, K. and Lumsdaine, P.L. (2015). Homotopy limits in type theory. Mathematical Structures in Computer Science 25 (special issue 5) 10401070.
Cartmell, J. (1986). Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209243.
Cisinski, D.-C. (2010). Invariance de la K-Théorie par équivalences dérivées. Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology 6 (3) 505546.
Clairambault, P. and Dybjer, P. (2011). The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories, Springer, Berlin Heidelberg, 91106.
Dybjer, P. (1996). Internal Type Theory, Springer, Berlin Heidelberg, 120134.
Hovey, M. (1999). Model Categories, Mathematical Surveys and Monographs, American Mathematical Society.
Isaev, V. (2013). On Fibrant Objects in Model Categories, unpublished, arXiv:1312.4327.
Isaev, V. (2016) Algebraic Presentations of Dependent Type Theories, unpublished, arXiv:1602.08504.
Kapulkin, C. (2015). Locally Cartesian Closed Quasicategories from Type Theory, unpublished, arXiv:1507.02648.
Kapulkin, C. and Lumsdaine, P.L. (2016). The homotopy theory of type theories, unpublished, arXiv:1610.00037.
Lumsdaine, P.L. and Warren, M.A. (2015). The local universes model: An overlooked coherence construction for dependent type theories. ACM Transactions on Computational Logic 16 (3) 23:123:31.
Palmgren, E. and Vickers, S.J. (2007). Partial horn logic and Cartesian categories. Annals of Pure and Applied Logic 145 (3) 314353.
Pitts, A.M. (2000). Categorical logic. In: Abramsky, S., Gabbay, D.M. and Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, Algebraic and Logical Structures, Volume 5, Oxford University Press, 39128.
Shulman, M. (2015). Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (special issue 5) 12031277.
Szumiło, K. (2014). Two Models for the Homotopy Theory of Cocomplete Homotopy Theories, Ph.D. thesis, University of Bonn.
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics,, Institute for Advanced Study.
Voevodsky, V. (2014). B-systems, unpublished, arXiv:1410.5389.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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