(References, and some notes, for PLL’s tutorial on the topic, Fri 28 Sep.)
There are various different categorical models for dependent type theory available in the literature. Most are equivalent, or almost so. Here we collect the various definitions, together with (hopefully definitive) references. If you have trouble tracking down any of these refs, email me [PLL]. All equivalences asserted are literal equivalences of 1-categories. Another overview can be found at categorical model of dependent types.
John Cartmell, Generalised algebraic theories and contextual categories, 1986; Thomas Streicher, Semantics of Type Theory, 1991.
Very elementarily defined, and stratified by definition.
John Cartmell, PhD thesis, Generalised Algebraic Theories and Contextual Categories, 1978; Eugenio Moggi, A category-theoretic account of program modules, 1989/1991; Andy Pitts, Categorial Logic, 2000 (as type-categories).
Fairly elementarily defined. Stratified CwA’s are equivalent to contextual categories.
Classically equivalent to CwA’s, but formulated slightly differently to be better-suited to formalisation. It can also be seen as a variable-free presentation of Martin-Lof’s substitution calculus.
More abstractly formulated than the others above, and more general by default. Full split comprehension categories are equivalent to CwA’s. The extra generality of comprehension cats is very useful for understanding the relationship with natural categorical models, where e.g. substitution may not be functorial on the nose.
Pierre-Louis Curien, Substitution up to isomorphism, 1993.
Pierre-Louis Curien, Richard Garner and Martin Hofmann, Revisiting the categorical interpretation of dependent type theory, 2013.
Andr’e Joyal’s approach based on the notion of a “tribe”. Slides from a talk available here Joyal's MIT slides.
Steve Awodey’s functorial reformulation of category-with-families semantics, including a theorem providing natural models in many homotopical categories. Draft paper from 28 May 2014 here.
Bart Jacobs book; Martin Hofmann 1997 paper; PLL thesis. Sec 1.2; nLab page