, Remark that the triangular identities of ? ? {?} give pr ?{ } { } = id { } ; however

S. Awodey and M. A. Warren, Homotopy theoretic models of identity types, Math. Proc. Cambridge Philos. Soc, vol.146, issue.1, pp.45-55, 2009.

C. Berger and I. Moerdijk, On an extension of the notion of reedy category, Mathematische Zeitschrift, vol.269, issue.3-4, pp.977-1004, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01296573

D. Cisinski, Les préfaisceaux comme modèles des types d'homotopie. Société mathématique de France, 2006.

P. Curien, Substitution up to isomorphism, Fundamenta Informaticae, vol.19, issue.1-2, pp.51-85, 1993.

G. William, D. Dwyer, and . Kan, Calculating simplicial localizations, Journal of Pure and Applied Algebra, vol.18, issue.1, pp.17-35, 1980.

M. Jeffrey and . Egger, Quillen model categories without equalisers or coequalisers, 2016.

T. Ehrhard, A categorical semantics of constructions, Proceedings of the Third Annual IEEE Symposium on Logic in Computer Science (LICS 1988), pp.264-273, 1988.

R. Garner, Combinatorial structure of type dependency, Journal of Pure and Applied Algebra, vol.219, issue.6, pp.1885-1914, 2015.

P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol.99, 2003.

M. Hofmann, On the interpretation of type theory in locally cartesian closed categories, Lecture Notes in Computer Science, vol.933, pp.427-441, 1994.

M. Hofmann, Extensional concepts in intensional type theory, 1995.

M. Hovey, Model categories, Mathematical Surveys and Monographs, vol.63, 1999.

M. Hyland and A. Pitts, The theory of constructions: Categorical semantics and topos-theoretic models, Contemporary Mathematics, vol.92, pp.137-199, 1989.

Y. Harpaz and M. Prasma, The Grothendieck construction for model categories, Advances in Mathematics, vol.281, pp.1306-1363, 2015.

M. Hofmann and T. Streicher, The groupoid interpretation of type theory, Venice Festschrift, pp.83-111, 1996.

B. Jacobs, Comprehension categories and the semantics of type dependency, Theoretical Computer Science, vol.107, issue.2, pp.169-207, 1993.

, André Joyal. Notes on clans and tribes, vol.7, p.2017

F. and W. Lawvere, Functorial Semantics of Algebraic Theories, 1963.

F. and W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics, vol.17, pp.1-14, 1970.

G. Maltsiniotis, Le théorème de Quillen, d'adjonction des foncteurs dérivés, revisité. Comptes Rendus de l'Académie des Sciences de Paris, vol.344, pp.549-552, 2007.

J. , P. May, and K. Ponto, More Concise Algebraic Topology: Localization, Completion, and Model Categories, Chicago Lectures in Mathematics, 2012.

P. Melliès and N. Zeilberger, Functors are type refinement systems, Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp.3-16, 2015.

P. Melliès and N. Zeilberger, A bifibrational reconstruction of lawvere's presheaf hyperdoctrine, Proceedings of the Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science, pp.555-564, 2016.

P. North, Type theoretic weak factorization systems, 2017.

D. Gray-quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol.43, 1967.

C. Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology and its Applications, vol.119, issue.1, pp.65-94, 2002.

A. Roig, Model category structures in bifibred categories, Journal of Pure and Applied Algebra, vol.95, issue.2, pp.203-223, 1994.

M. Shulman, Ternary factorization systems, 2010.

M. Shulman, Reedy categories and their generalizations, 2015.

A. E. Stanculescu, Bifibrations and weak factorisation systems, Applied Categorical Structures, vol.20, issue.1, pp.19-30, 2012.

C. Strachey, Towards a formal semantics, Formal Language Description Languages for Computer Programming, pp.198-220, 1966.

R. Street, Fibrations in bicategories. Cahiers de toplogie et géométrie différentielle catégoriques, vol.21, pp.111-160, 1980.

T. Streicher, Dependence and independence results for (impredicative) calculi of dependent types, Mathematical Structures in Computer Science, vol.2, issue.1, pp.29-54, 1992.

P. Taylor, Recursive domains, indexed category theory and polymorphism, 1986.