Homotopy Type Theory

Semantics of (homotopy) type theory

(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.

Contextual Categories

John Cartmell, Generalised algebraic theories and contextual categories, 1986; Thomas Streicher, Semantics of Type Theory, 1991.

Very elementarily defined, and stratified by definition.

Categories with Attributes, aka Type-categories

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.

Categories with Families

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.

Comprehension Categories

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.

Coherence for interpretation in comprehension categories

Joyal’s Categorical Homotopy Type Theory

Andr’e Joyal’s approach based on the notion of a “tribe”. Slides from a talk available here Joyal's MIT slides. Later PDF.

Natural Models

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.

Introductions and surveys

Bart Jacobs book; Martin Hofmann 1997 paper; PLL (Peter LeFanu Lumsdaine) thesis. Sec 1.2; nLab page

Last revised on June 2, 2018 at 08:10:00. See the history of this page for a list of all contributions to it.