Related papers: External univalence for second-order generalized a…
In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our…
We discuss how canonical and universal constructions, properties and characterizations interact with equality in the framework of Homotopy Type Theory, comparing it with Grothendieck's use of equality and shedding further light on…
We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set…
A typoid is a type equipped with an equivalence relation, such that the terms of equivalence between the terms of the type satisfy certain conditions, with respect to a given equivalence relation between them, that generalise the properties…
We show that Voevodsky's univalence axiom for intensional type theory is valid in categories of simplicial presheaves on elegant Reedy categories. In addition to diagrams on inverse categories, as considered in previous work of the author,…
We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky's univalent foundations. We observe for the first…
Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in…
It is well known that univalence is incompatible with uniqueness of identity proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets having non-trivial automorphisms as soon as they are not h-propositions. A natural…
Connections between homotopy theory and type theory have recently attracted a lot of attention, with Voevodsky's univalent foundations and the interpretation of Martin-Lof's identity types in Quillen model categories as some of the…
In a recent paper, the second author and Joana Cirici proved a theorem that says that given appropriate hypotheses, $n$-formality of a differential graded algebraic structure is equivalent to the existence of a chain-level lift of a…
We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…
This PhD thesis deals with some new models of intensional type theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic fibration categories (the notion of…
We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is…
The non-standard identity concept developed in the Homotopy Type theory allows for an alternative analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities and accounts for the…
We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy…
The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an object $I$ in a topos to give such a path-based model of type theory in which paths are just functions with domain $I$.…
By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In…
Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point…
We present a concept of uniform encodability of theories and develop tools related to this concept. As an application we obtain general undecidability results which are uniform for large families of structures. In the way, we define…
This paper develops a version of dependent type theory in which isomorphism is handled through a direct generalization of the 1939 definitions of Bourbaki. More specifically we generalize the Bourbaki definition of structure from simple…