相关论文: Brouwer's fixed-point theorem in real-cohesive hom…

Homotopy Brouwer theory is a tool to study the dynamics of surface homeomorphisms. We introduce and illustrate the main objects of homotopy Brouwer theory, and provide a proof of Handel's fixed point theorem. These are the notes of a…

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 investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a…

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer,…

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…

Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected…

Shulman's spatial type theory internalizes the modalities of Lawvere's axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber's modal approach to differential cohomology to be carried out…

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…

We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected type $(n \geq 1)$ and $A$ an abelian group, there is a natural isomorphism $\pi_n(X)^{ab} \otimes A \cong \tilde{H}_n(X; A)$ relating the…

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff…

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this…

The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes…

We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge…

It is investigated in what sense the Brouwer fixed point theorem may be viewed as a corollary of the Lawvere fixed point theorem. A suitable generalisation of the Lawvere fixed point theorem is found and a means is identified by which the…

Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…

This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric…

Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can…

Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple…

This paper defines homology in homotopy type theory, in the process stable homotopy groups are also defined. Previous research in synthetic homotopy theory is relied on, in particular the definition of cohomology. This work lays the…