Related papers: A Foundation for Synthetic Stone Duality
The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized…
In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction…
We give rigorous foundations for parametrized homotopy theory in this monograph. After preliminaries on point-set topology, base change functors, and proper actions of non-compact Lie groups, we develop the homotopy theory of equivariant…
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the…
What makes two computational systems equivalent? Topos theory answers with classifying toposes: a system's semantic content is encoded in the geometric theory it classifies, and two presentations are equivalent when their classifying…
We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology…
We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin.…
We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points…
The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids. The problem of higher structures is that of constructing the homotopy types needed…
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…
We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…
This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
This paper continues investigations in "synthetic homotopy theory": the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory We present a mechanized proof of the Blakers-Massey connectivity…
A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a…
We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be…
The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of…
We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for locally small spaces with bounded continuous mappings.
The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory (HoTT) has been recognized as important during the Year of Univalent Foundations at the Institute of Advanced Study. According to the interpretation of HoTT in…
We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure,…