Algebraic models in the homotopy theory of classifying spaces

Classically, algebraic topology refers to the study of the homotopy type of topological spaces by assigning algebraic invariants in a functorial way, but in the last decades homotopical methods have influenced other areas of mathematics. In this context, some topological spaces are more suitable for those kinds of interactions since they already encode algebraic information. This is the case of classifying spaces for groups. The classification of geometrical structures reduces to the understanding the homotopy type of maps into classifying spaces.