Related papers: Classifying Types
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 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…
We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral…
We give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-L\"of type theory, without Higher Inductive Types…
This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of…
Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…
Homotopy type theory is a modern foundation for mathematics that introduces the univalence axiom and is particularly suitable for the study of homotopical mathematics and its formalization via proof assistants. In order to better comprehend…
This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the…
We bring a linkage from representation theory of Lie groups to homotopy theory for maps between flag manifolds. As applications we derive from representation theory abundant families of homotopy classes of maps between flag manifolds whose…
In this paper we lay the foundations of an $\infty$-categorical theory of Stokes data.
The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum…
We give a detailed exposition of the homotopy theory of equivalence relations, perhaps the simplest nontrivial example of a model structure.
Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course…
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by…
These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.
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…
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…
Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as…
In homotopy type theory, a natural number type is freely generated by an element and an endomorphism. Similarly, an integer type is freely generated by an element and an automorphism. Using only dependent sums, identity types, extensional…