Related papers: Undecidability of Equality in the Free Locally Car…
Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-L\"of type…
The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 1-types, and Sigma-types are biequivalent to finitely complete categories, while adding Pi-types yields locally Cartesian closed…
We establish a DK-equivalence between the relative category of $\pi$-tribes and the relative category of locally cartesian closed quasicategories. From this follows one of the internal languages conjecture: Martin-L\"of type theory with…
This text summarizes and expands the content of a general audience talk given in 2018 at the University of Mainz. Motivated by recent developments in dependent type theory and infinity category theory, it presents a history of ideas around…
It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed…
We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin-L\"of type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We…
We prove an extensionality theorem for the "type-in-type" dependent type theory with Sigma-types. We suggest that the extensional equality type be identified with the logical equivalence relation on the free term model of type theory.
We investigate categories in which products distribute over coproducts, a structure we call doubly-infinitary distributive categories. Through a range of examples, we explore how this notion relates to established concepts such as…
The present paper gives a generalization of cartesian closed categories, called cartesian closed categories with dependence, whose strict version induces categories with families that support 1-, Sigma- and Pi-types in the strict sense.…
One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the…
We consider the canonical pseudodistributive law between various free limit completion pseudomonads and the free coproduct completion pseudomonad. When the class of limits includes pullbacks, we show that this consideration leads to notions…
We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-L\"of's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.
We prove that the quasicategories arising from models of Martin-L\"of type theory via simplicial localization are locally cartesian closed.
We give a type system in which the universe of types is closed by reflection into it of the logical relation defined externally by induction on the structure of types. This contribution is placed in the context of the search for a natural,…
In this note we remark on the problem of equality of objects in categories formalized in Martin-L\"of's constructive type theory. A standard notion of category in this system is E-category, where no such equality is specified. The main…
It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed…
It is proved that equalities between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equality in the language of free cartesian categories collapses a cartesian category into a preorder. An…