Locally effective objects and algebraic topology

Algebraic topology consists of associating invariants are of an algebraic nature, describing certain topological properties. For example, since Poincaré, it is known how to associate the group π₁(X,x₀ to a topological space X and to one of its points x₀; this group is called the Poincaré group or the first homotopy group of the space X based on x₀. This group is null if and only if the space X is simply connected at x₀; in another case, the group measures the lack of simple connectivity. Many other groups can be associated to a topological space, evaluating certain properties of this space: homology groups,K-theory groups,etc.