This reading seminar is devoted to Grothendieck's formalism of Galois categories and étale
fundamental groups. Starting from the analogy between classical Galois theory, finite topological
covering spaces and finite étale covers of schemes, we will develop the abstract theory of
Galois categories and their fundamental groups, and then apply it to the construction and study of
the étale fundamental group of a scheme. The aim is to build a conceptual bridge between
algebra, topology, and algebraic geometry: after setting up the general Galois category formalism,
we will interpret the category of finite étale covers of a scheme as a Galois
category, identify its fundamental group with the étale fundamental group, and discuss its
basic properties such as homotopy exact sequences, the relation between geometric and arithmetic
fundamental groups, comparison with the topological fundamental group over complex field, and
specialization in families of schemes
