natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
under construction
Homotopy type theory has categorical semantics in suitable homotopical categories which in turn present certain (infinity,1)-categories.
The concept of type-theoretic model category (Shulman 12a) or type-theoretic fibration category (Shulman 12b, def. 2.1) is one particular way of making this precise. Specifically every locally presentable locally cartesian closed (∞,1)-category has a presentation by a type-theoretic model category, hence provides higher categorical semantics for homotopy type theory (without possibly univalence). For more on this see also the respective sections at relation between type theory and category theory.
under construction
By decomposing the structure in homotopy type theory in layers as
with identity types
A 1-category whose internal logic can interpret this needs to
equipped with a weak factorization system with stable path objects, such that acyclic cofibrations are preserved by pullback along fibrations between fibrant objects.
(needs to be finished)
Function extensionality holds in the internal type theory of a type-theoretic fibration category precisely if dependent products (i.e. right base change) along fibrations preserve acyclic fibrations.
The condition in prop. 1 holds in particular in right proper Cisinski model structures, since in these right base change along fibrations is a right Quillen functor (see e.g. the proof here).
Notice that every presentable locally Cartesian closed (∞,1)-category (by the discussion there) has a presentation by a right proper Cisinski model category. Accordingly we may say that every presentable locally Cartesian closed (∞,1)-category interprets HoTT+FunExt.
The definition originates in and a discussion of categorical semantics of homotopy type theory in a type-theoretic model category appears in
An exposition is in
Mike Shulman, Minicourse on Homotopy Type Theory part 3, Categorical models of homotopy type theory, April 2012 (pdf)
Denis-Charles Cisinski, Univalent universes for elegant models of homotopy types (arXiv:1406.0058)
Similar conisderations (using the term “typos” for something similar to a type-theoretic model category) are presented in