Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-z9m8x Total loading time: 0.389 Render date: 2022-10-04T14:54:41.946Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Model structures on categories of models of type theories

Published online by Cambridge University Press:  28 September 2017

VALERY ISAEV*
Affiliation:
Department of Mathematics and Information Technology, Saint Petersburg Academic University, Saint Petersburg, Russia Email: valery.isaev@gmail.com

Abstract

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.

Type
Paper
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avigad, J., Kapulkin, K. and Lumsdaine, P.L. (2015). Homotopy limits in type theory. Mathematical Structures in Computer Science 25 (special issue 5) 10401070.CrossRefGoogle Scholar
Cartmell, J. (1986). Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209243.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Clairambault, P. and Dybjer, P. (2011). The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories, Springer, Berlin Heidelberg, 91106.Google Scholar
Dybjer, P. (1996). Internal Type Theory, Springer, Berlin Heidelberg, 120134.Google Scholar
Hovey, M. (1999). Model Categories, Mathematical Surveys and Monographs, American Mathematical Society.Google Scholar
Isaev, V. (2013). On Fibrant Objects in Model Categories, unpublished, arXiv:1312.4327.Google Scholar
Isaev, V. (2016) Algebraic Presentations of Dependent Type Theories, unpublished, arXiv:1602.08504.Google Scholar
Kapulkin, C. (2015). Locally Cartesian Closed Quasicategories from Type Theory, unpublished, arXiv:1507.02648.Google Scholar
Kapulkin, C. and Lumsdaine, P.L. (2016). The homotopy theory of type theories, unpublished, arXiv:1610.00037.Google Scholar
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.CrossRefGoogle Scholar
Palmgren, E. and Vickers, S.J. (2007). Partial horn logic and Cartesian categories. Annals of Pure and Applied Logic 145 (3) 314353.CrossRefGoogle Scholar
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.Google Scholar
Shulman, M. (2015). Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (special issue 5) 12031277.CrossRefGoogle Scholar
Szumiło, K. (2014). Two Models for the Homotopy Theory of Cocomplete Homotopy Theories, Ph.D. thesis, University of Bonn.Google Scholar
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics, https://homotopytypetheory.org/book, Institute for Advanced Study.Google Scholar
Voevodsky, V. (2014). B-systems, unpublished, arXiv:1410.5389.Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Model structures on categories of models of type theories
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Model structures on categories of models of type theories
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Model structures on categories of models of type theories
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *