Topological Methods in Algebra
Anthony Bak · 2020
This chapter introduces a general concept of deformation and homotopy theory into pure algebra. Homotopy theory has a long tradition in analysis and geometry and is grounded in the concepts of topological space, continuous map, path, and deformation of continuous maps. Wherever these notions occur in a natural and meaningful way, one can endevour to apply homotopical concepts and methodology to formulate and solve problems. The chapter begins by defining a global action and giving a few examples including that of a particular line. Various concepts of morphism between global actions are introduced. A natural, intuitive notion of path in the global action is described and then formalized as a certain kind of morphism. The chapter defines a global action structure on the set of all morphisms from one global action to another and establishes the exponential law. The global action on morphism spaces plays the role of the compact open topology on function spaces in topology.