1. Introduction This article is intended to provide some new insights about concurrency theory using ideas from geometry, and more speciically from algebraic topology. The aim of the paper is twofold: we justify applications of geometrical methods in concurrency through some chosen examples and we give the mathematical foundations needed to understand the geometric phenomenon that we identify. ...

L. Lambán et al. () ACL2 and Algebraic Topology 1 / 23 Introduction Kenzo symbolic computation system: a Common Lisp program devoted to Algebraic (Simplicial) Topology. ◮ A research tool: used to obtain relevant results in the field, neither confirmed nor refuted by any other means. The following question makes sense: Is it Kenzo correct? Our goal: we want to formally prove correcteness propert...

This talk aims at presenting a new way of approaching topological structures, induced by recent developments in the eld of lattice-valued topology, and deemed to incorporate in itself both crisp and many-valued settings. Based on category theory and universal algebra, the new framework is called categorically-algebraic (catalg) [9-12], to underline its motivating theories, on one hand, and to d...

Part 1. The Fundamental Group 3 2. Basic Constructions 3 2.1. Paths and Homotopy 3 2.2. The Fundamental Group of the Circle 4 2.3. Induced Homomorphisms 6 3. Van Kampen’s Theorem 7 3.1. Free Products of Groups 7 3.2. The Van Kampen Theorem 7 3.3. Application to Cell Complexes 9 4. Covering Spaces 11 4.1. Lifting Properties 11 4.2. The Classification of Covering Spaces 12 4.3. Deck Transformatio...

This paper develops the introductory theory of equivariant algebraic topology. We first define G-CW complexes and prove some basic homotopy-theoretic results Whitehead’s theorem, cellular and CW approximation, and the Freudenthal suspension theorem. We then define ordinary (Bredon) homology and cohomology theories and give an application to Smith theory. Our treatment of this material closely f...

Grothendieck Toposes are often considered as generalized spaces; indeed, every space gives rise to a topos of sheaves, and various invariants and constructions from (algebraic) topology can be generalized to the level of toposes. In this talk, I will introduce a newly discovered invariant called the isotropy group of a topos and illustrate by considering special cases such as continuous group a...

© Algebraic topology for computer vision Daniel Freedman, Chao Chen HP Laboratories HPL-2009-375 algebraic topology, persistent homology, computer vision, image processing Algebraic topology is generally considered one of the purest subfields of mathematics. However, over the last decade two interesting new lines of research have emerged, one focusing on algorithms for algebraic topology, and t...

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation f...

Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn’t have to worry about the Zariski topology and its many pathologies, because they already had a better-behaved topology to work with: the analytic topology inherited from the usual topology on the complex numbers themselves. In this note, we introduce the analytic topology, and explore...