Related papers: Computational Higher Type Theory IV: Inductive Typ…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard…
In the Calculus of Dependent Lambda Eliminations (CDLE), a pure Curry-style type theory, it is possible to generically {\lambda}-encode inductive datatypes which support course-of-values (CoV) induction. We present a datatype subsystem for…
The paper introduces a generalization for known probabilistic models such as log-linear and graphical models, called here multiplicative models. These models, that express probabilities via product of parameters are shown to capture…
We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While…
The language of homotopy type theory has proved to be appropriate as an internal language for various higher toposes, for example with Synthetic Algebraic Geometry for the Zariski topos. In this paper we apply such techniques to the higher…
We construct combinatorial model category structures on the categories of (marked) categories and (marked) pre-additive categories, and we characterize (marked) additive categories as fibrant objects in a Bousfield localization of…
We study the number of planes for four dimensional projective hypersurfaces which has so-called inductive structure. We also determine transcendental lattices for cubic fourfolds of this type.
To define enumerative invariants in geometry, one often needs orientations on moduli spaces of geometric objects. This monograph develops a new bordism-theoretic point of view on orientations of moduli spaces. Let $X$ be a manifold with…
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…
A well-known problem in the theory of dependent types is how to handle so-called nested data types. These data types are difficult to program and to reason about in total dependently typed languages such as Agda and Coq. In particular, it…
In recent years, languages like Haskell have seen a dramatic surge of new features that significantly extends the expressive power of their type systems. With these features, the challenge of kind inference for datatype declarations has…
Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann, we define canonical computability models over categories and appropriate Set-valued functors…
Formal deductive systems are very common in computer science. They are used to represent logics, programming languages, and security systems. Moreover, writing programs that manipulate them and that reason about them is important and…
We present new induction principles for the syntax of dependent type theories, which we call relative induction principles. The result of the induction principle relative to a functor F into the syntax is stable over the codomain of F. We…
Axiomatic type theory is a dependent type theory without computation rules. The term equality judgements that usually characterise these rules are replaced by computation axioms, i.e., additional term judgements that are typed by identity…
Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate…
Escard\'o and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy…
As quantum computers become real, it is high time we come up with effective techniques that help programmers write correct quantum programs. In classical computing, formal verification and sound static type systems prevent several classes…
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…
Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalization of the theory of canonical extension to the setting of first order logic. We define a notion of…