This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.

Introduction
Part I. Toposes and Differential Geometry: 1. Topos theory
2. Synthetic differential geometry
Part II. Topics in SDG: 3. The Ambrose–Palais–Singer theorem in SDG
4. Calculus of variations in SDG
Part III. Toposes and Differential Topology: 5. Local concepts in SDG
6. Synthetic differential topology
Part IV. Topics in SDT: 7. Stable mappings and Mather's theorem in SDT
8. Morse theory in SDT
Part V. SDT and Differential Topology: 9. Well-adapted models of SDT
10. An application to unfoldings
Part VI. A Well-Adapted Model of SDT: 11. The Dubuc topos G
12. G as a model of SDT
References
Index.

Subject Areas: Algebraic topology [PBPD], Differential & Riemannian geometry [PBMP], Set theory [PBCH], Mathematical logic [PBCD]