Skip to main content Accessibility help

Identity types and weak factorization systems in Cauchy complete categories

  • Paige Randall North (a1)


It has been known that categorical interpretations of dependent type theory with Σ- and Id-types induce weak factorization systems. When one has a weak factorization system $({\cal L},{\cal R})$ on a category $\mathbb{C}$ in hand, it is then natural to ask whether or not $({\cal L},{\cal R})$ harbors an interpretation of dependent type theory with Σ- and Id- (and possibly Π-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class ${\cal D}$ of morphisms of $\mathbb{C}$ such that the retract closure of ${\cal D}$ is the class ${\cal R}$ and the pair $(\mathbb{C},{\cal D})$ forms a display map category modeling Σ- and Id- (and possibly Π-) types. In this paper, we show, with the hypothesis that $\cal{C}$ is Cauchy complete, that there exists such a class $\cal{D}$ if and only if $(\mathbb{C},\cal{R})$ itself forms a display map category modeling Σ- and Id- (and possibly Π-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.


Corresponding author

*Corresponding author. Email:


Hide All

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-16-1-0212.



Hide All
Awodey, S. and Warren, M.A. (2009). Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1) 4555.
Borceux, F. (1994). Handbook of Categorical Algebra, vol. 1. Cambridge, UK, Cambridge University Press.
Emmenegger, J. (2014) A Category-Theoretic Version of the Identity Type Weak Factorization System. arXiv:1412.0153 [math.LO].
Gambino, N. and Garner, R. (2008). The identity type weak factorisation system. Theoretical Computer Science 409 (1) 94109.10.1016/j.tcs.2008.08.030
Hovey, M. (1999). Model Categories. Providence, RI, American Mathematical Society.
Joyal, A. (2017). Notes on Clans and Tribes. arXiv:1710.10238 [math.CT].
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) Art. 23, 31.10.1145/2754931
May, J.P. and Ponto, K. (2012). More Concise Algebraic Topology. Chicago, University of Chicago Press.
Moss, S. (2018). The Dialectica Models of Type Theory. PhD thesis, University of Cambridge.10.1145/3209108.3209207
Nordström, B. Petersson, K. and Smith, J.M. (2000). Martin-Löf’s Type Theory. In: Handbook of Logic in Computer Science, Oxford, UK, Oxford University Press, 132.
North, P.R. (2017). Type Theoretic Weak Factorization Systems. PhD thesis, University of Cambridge.
Shulman, M. (2015) Univalence for inverse diagrams and homotopy canonicity. Mathematical Structures in Computer Science 25 (5) 12031277.
Streicher, T. (1993). Investigations into Intensional Type Theory. Habilitationsschrift. Ludwig Maximilian University of Munich.
Taylor, P. (1999). Practical Foundations of Mathematics. Cambridge, UK, Cambridge University Press.
van den Berg, B. and Garner, R. (2012). Topological and simplicial models of identity types. ACM Transactions on Computational Logic 13 (1) 3:13:44.
Warren, M.A. (2008). Homotopy Theoretic Aspects of Constructive Type Theory. Phd thesis, Carnegie Mellon University.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed