topology

Let $\mathcal{C}$ be a minimal quasi-category equivalent to the $\infty$-category of spaces. (Such a $\mathcal{C}$ is unique up to isomorphism of simplicial sets.) In particular, equivalent objects of ...

The homotopy type of the space of invertible non-negative matrices $\DeclareMathOperator\NNGL{NNGL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Given that the homotopy type of the space of ...

This paper provides a structural analysis of a topological unification framework based on the universal Hopf fibration and the identification of $\mathbb{CP}^{\infty}$ as a physical base space. We establish explicit criteria for what constitutes a physically admissible theory and show that the analyzed construction fails to meet these conditions. The framework replaces dynamical derivation with t…

Jeff Giansiracusa is a professor at Swansea University in the Department of Mathematics. He has worked on homotopy theoretic aspects of moduli spaces, operads, topological field theory, and diffeomorphism groups, using topological techniques from algebraic K-theory to study the homotopy theory of moduli spaces arising in algebraic geometry and now includes aspects of topological data analysis, tr…

One may often read that mathematical knots (closed curves) can't exist in a 4D Euclidean space. I say that 2D manifolds in 4D can be knots. Except for the simple 2-sphere I couldn't imagine what such a knot would be like. Here's a video about how to visualize this. The... Read more

We develop a theory of ×-homotopy, fundamental groupoids and covering spaces that applies to non-simple graphs, generalizing existing results for simple graphs. We prove that ×-homotopies from finite graphs can be decomposed into moves that adjust at most one vertex at a time, generalizing the spider lemma of Chih & Scull (2021). We define a notion of homotopy covering map and develop a theory of…

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems In anabelian geometry one studies how much information about a space (specifically: an algebraic variety) is contained already in its first étale homotopy group (specifically: the algebraic fundamental group). The term “anabelia…

nLab focal point Contents Context Topology topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Introduction Basic concepts - open subset, closed subset, neighbourhood - topological space, locale - base for the topology, neighbourhood base - finer/coarser topology - closure, interior, boundary - …

Thom's theory of elementary catastrophes forms a system of topological morphogenesis which has already been used in various applications. But the semantics encountered in previous works has required more pregnant logos shapes. We could have got over the lack of generalizations of Thom's classification theorem, by considering infinitesimal morphogenetic changes. However, this would have broken the…

The h-cobordism theorem (due to Smale 1962) and the s-cobordism theorem provide sufficient conditions for an h-cobordism to be isomorphic to a cylinder. There are famous counterexamples where the h-cobordism theorem fails (in the smooth category) and h-cobordisms exist which are not isomorphic to a cylinder. These counterexamples arise specifically in the case of 5-dimensional cobordisms between …

Alexander Pieter Martijn Kupers On open-closed string topology operations in terms of HQFT with branes: On the h-principle and microflexible sheaves: On h-cobordism between smooth 4-manifolds:

representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenberg-Moore category, algebra over an operad, actegory, crossed module Rational equivariant stable homotopy theory is the study of equivariant spectra just on the level of their rationalization, hence concerning only…

On lifting the Witten genus of the heterotic string to topological modular forms: Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly (arXiv:2103.12211) Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies, Comm. Math. Phys. 402 (2023) 1585-1620 [arXiv:2108.13542, doi:10.1007/s00220-023-04761-2] and in re…

hom-set, hom-object, internal hom, exponential object, derived hom-space loop space object, free loop space object, derived loop space Given categories and , the functor category – written or – is the category whose morphisms are natural transformations between these functors. Discussion in homotopy type theory. Note: the HoTT book calls a internal category in HoTT a “precategory” and a univalent…

Let $$X = \overline{B}_{n + 1},$$ $$C = \{x \in X: (\forall i \in (n + 1) \setminus \{0\}: x_i = 0) \land x_0 \geq 0\},$$ $$A = S_n \cup C,$$ where $\overline{B}_{n + 1}$ is the closed unit $(n + ...

On scissors congruence via algebraic K-theory: Renee Hoekzema, Mona Merling, Laura Murray, Carmen Rovi, zolkariev?: Cut and paste invariants of manifolds via algebraic K-theory, Topology and its Applications 316 (2022), arXiv:2001.00176, doi:10.1016/j.topol.2022.108105; Mona Merling, Ming Ng, zolkariev?, Alba Sendón Blanco, Lucas Williams: Scissors congruence K-theory for equivariant manifolds, a…

The concept of a cylinder object in a category is an abstraction of the construction in Top which associates to any topological space the cylinder over , where is the standard topological interval. It is notably used to define the concept of left homotopy, say in a model category. The standard topological cylinder naturally comes equipped with a continuous map that identifies as the two ends and …

This paper examines the structural shape of any Theory of Theories of Everything (TOE of TOEs) and offers the most honest available account of whether such an architecture is achievable in principle. A single self-grounding ledge is identified at the level of the bare Root Axiom, where action-as-existence enacts itself in any act of denial or affirmation. Above this ledge, every architecture is b…

Let say that $\Omega\subset\mathbb{R}^N,N\geq 2$ be an open set, and $f:\Omega\to\mathbb{R}$ be a continuous function with the property that for any $x_0\in\partial \Omega$ we have that the following ...

Does there exist a compact Hausdorff space $X$ of weight $\mathfrak{c}$ with ccc such that $|\text{Homeo}(X)|>|M(X)|$, where $M(X)$ is the space of complex-valued regular Borel measures on $X$. If ...