Sites and Grothendieck toposes: an introduction

TL;DR

This book provides a comprehensive, step-by-step introduction to topos theory, highlighting its connections to geometry, logic, and category theory, suitable for readers with basic mathematical background.

Contribution

It offers a structured, accessible pathway from foundational concepts to advanced topics in topos theory, emphasizing dual perspectives and canonical examples.

Findings

Unifies various mathematical disciplines through topos theory

Connects geometric intuition with logical frameworks via classifying toposes

Prepares readers for advanced topics like relative toposes and toposes as bridges

Abstract

Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction that moves from familiar ground - groups and their actions, topological spaces, categories, and sheaves - to Grothendieck topologies and sites, then to the axiomatic and categorical foundations of toposes (via Giraud's theorem), to their geometry (morphisms, points, subtoposes, localizations) and finally to their deep ties with geometric logic through classifying toposes. Aimed at readers with a basic familiarity with algebra, general topology, and category theory, the book emphasizes reversible viewpoints - external/internal, local/global, syntactic/semantic - guided by canonical examples, key theorems, and universal constructions. It equips the reader with a workable language in which spaces become categories of sheaves and theories become places classified by toposes. Future chapters will present "toposes as bridges" and relative toposes.