In this note we interpret Voevodsky’s Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category Qt in which the mapping Hom(w)(Z × B,C) : Qt −→ Sets is functorial in Z and represented in Qt satisfies our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question repo...

More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of Amadio-Droste [1,6,7] is the largest cartesian closed full sub-category of the category of ω-algebraic meet-cpos with stable functions. A solution to this problem has two major steps: (1) Show that for any ω-algebraic meet-cpo D, if all higher-ord...

We introduce the category IVRel(H) consisting of interval-valued H-fuzzy relational spaces and relation preserving mappings between them and we study structures of the category IVRel(H) in the viewpoint of the topological universe introduced by Nel. Thus we show that IVRel(H) satisfies all the conditions of a topological universe over Set except the terminal separator property and IVRel(H) is C...

We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint universal binary coproducts.

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to obtain a differential calculus on groups, from which we then extract a Boolean differential calculus, in which both linearity and the product rule, also called th...

We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpo-enriched cartesian closed categories to have H∗, the maximal consistent sensible λ-theory, as their equational theory. Finally, we prove that our relational model fulfils thes...