Related papers: Two-Level Type Theory and Applications
We adjust the notion of typicality originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class…
Recent experimental results showing untypical nonlinear absorption and marked deviations from well known universality in the low temperature acoustic and dielectric losses in amorphous solids prove the need for improving the understanding…
In this short note, we construct a class of models of an extension of homotopy type theory, which we call homotopy type theory with an interval type.
Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. We recall some of the connections between the past and the present developments. Higher homotopies were isolated…
We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language…
We present the type theory CaTT, originally introduced by Finster and Mimram to describe globular weak $\omega$-categories, and we formalise this theory in the language of homotopy type theory. Most of the studies about this type theory…
Large language models (LLMs) have shown strong performance across natural language reasoning tasks, yet their reasoning processes remain brittle and difficult to interpret. Prompting techniques like Chain-of-Thought (CoT) enhance…
The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-L\"of Type Theory to asymmetric hom-types. We articulate…
We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of…
Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or…
Martin-L\"of's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive…
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models…
In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational…
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent…
Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping…
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract…
This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known…
We explore the symmetry structure of Type II Little String Theories and their T-dualities. We construct these theories both from the bottom-up perspective starting with seed Superconformal Field Theories, and from the top-down using…
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…
Scientific fields are often mapped using citations and metadata, despite knowledge being transmitted primarily through content. We introduce an 'inside-out' approach that reconstructs field structure directly from text by representing each…