Related papers: Towards Higher Topology

We give the site-theoretic account of the spectral construction as first introduced by Coste. We provide a detailed examination of the geometric properties of the spectrum, in particular what classes of topoi it produces when applied to the…

Assume that $f$ is a $C^r(r\geq 3)$ specially partially hyperbolic endomorphism on the 2-torus which is homotopic to an expanding linear endomorphism $A$ with irrational eigenvalues. We prove that $f$ and $A$ are topologically conjugate, if…

For a path connected, locally path connected and semilocally simply connected space $X$, let $\Pi_1(X)$ denote its topologised fundamental groupoid as established in the first article of this series. Let $\mathcal{E}$ be the category of…

The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit…

This belongs to a series of papers devoted to the study of the cohomology of classifying spaces of Lie groupoids. Our aim here is to introduce and study the notion of representation up to homotopy of Lie groupoids, the resulting derived…

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and…

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category.…

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective $T_0$ spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true…

To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with…

For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies…

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of…

In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces,…

Given a cohomological functor from a triangulated category to an abelian category, we construct under appropriate assumptions for any localization functor of the abelian category a lift to a localization functor of the triangulated…

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…

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated…

A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e.,…

With a frame $L$ as the truth value table, we study the topological representations for frame-valued domains. We introduce the notions of locally super-compact $L$-topological space and strong locally super-compact $L$-topological space.…

In this thesis we propose and study a theory of ordered locales, a type of point-free space equipped with a preorder structure on its frame of opens. It is proved that the Stone-type duality between topological spaces and locales lifts to a…

Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic…

It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between…