This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of ...

This paper contributes towards establishing the category QCB, of topological quotients of countably based spaces, and its subcategory TP, of topological predomains, as a flexible framework for denotational semantics of programming languages. In particular, we show that both categories have free algebras for arbitrary countable parametrised equational theories, and are thus, following ideas of P...

We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...

A common theme in algebraic topology is to study functors from the homotopy category of topological spaces to various categories of algebraic objects, such as the classical homology functor, the cohomology functor and the homopoty functor. In this talk, I will introduce some functors from topological spaces to Lie algebras and algebraic varieties, and explore their properties. We can use these ...

Every nontrivial countably productive coreflective subcategory of topological linear spaces is κ-productive for a large cardinal κ (see [10]). Unlike that case, in uniform spaces for every infinite regular cardinal κ, there are coreflective subcategories that are κ-productive and not κ-productive (see [8]). From certain points of view, the category of topological groups lies in between those ca...

Based on a complete Heyting algebra, we modify the definition of lattice-valued fuzzifying convergence space using fuzzy inclusion order and construct in this way a Cartesian-closed category, called the category of L−ordered fuzzifying convergence spaces, in which the category of L−fuzzifying topological spaces can be embedded. In addition, two new categories are introduced, which are called th...

In the classical paper [A-H] Adams-Hilton constructed a free chain algebra which is an important algebraic model of a simply connected homotopy type. We show that this chain algebra (endowed with an additional structure given by a “height function”) yields actually an invariant of a proper homotopy type. For this we introduce the homotopy category of locally finite chain algebras without using ...

We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...

We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be...