From Concurrency to Algebraic Topology

Conference on Algebraic Topological Methods in Computer Science

The recognition that within the area of computational geometry, the methods of algebraic topology can provide qualitative and shape information which isn't available from other methods. Algebraic topology provides a tool for visualization and feature identification in high dimensional data. Algebraic topology is an extremely useful framework for analyzing problems in distributed computing, such...

Applications of Algebraic Topology to Concurrent Computation

All parallel programs require some amount of synchronization to coordinate their concurrency to achieve correct solutions. It is commonly known that synchronization can cause poor performance by burdening the program with excessive overhead. This chapter develops a connection between certain synchronization primitives and topology. This connection permits the theoretical study of concurrent com...

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

A Few Points on Directed Algebraic Topology

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A 'directed space' has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Applications have been mostly developed in the theory of concurrency. Unexpected links with noncomm...

Concurrency and Directed Algebraic Topology

Directed Algebraic Topology, Categories and Higher Categories

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applicatio...