(Co)fibrations, fiber bundles and simplicial sets: a dive into some topological tools

Abstract

We look into some tools for analyzing topological spaces and computing their homotopy groups. Fiber bundles are a way of constructing a new space out of two original spaces, and it induces a long exact sequence that relates their homotopy groups. If some of these groups are known, one can use the properties of a long exact sequence to compute others. Simplicial sets are useful for translating topological problems into category theory, while preserving information on their homotopy groups. There are adjoint functors between the categories of simplicial sets and topological spaces. The categories of Kan complexes (a specific type of simplicial set) and CW complexes are equivalent.