Introduction Mathieu Anel and Gabriel Catren; Part I. Differential geometry: 1. An Introduction to diffeology Patrick Iglesias-Zemmour; 2. New methods for old spaces: synthetic differential geometry Anders Kock; 3. Microlocal analysis and beyond Pierre Schapira; Part II. Topology and algebraic topology: 4. Topo-logie Mathieu Anel and André Joyal; 5. Spaces as infinity-groupoids Timothy Porter; 6. Homotopy type theory: the logic of space Michael Shulman; Part III. Algebraic geometry: 7. Sheaves and functors of points Michel Vaquié; 8. Stacks Nicole Mestrano and Carlos Simpson; 9. The geometry of ambiguity: an introduction to the ideas of derived geometry Mathieu Anel; 10. Geometry in dg categories Maxim Kontsevich.
In this graduate-level book, leading researchers explore various new notions of 'space' in mathematics.
Mathieu Anel is a Visiting Assistant Professor at Carnegie Mellon University. His research interests include higher category theory, algebraic topology, and topos theory. Gabriel Catren is Permanent Researcher in philosophy of physics at the French National Centre for Scientific Research (CNRS). His research interests include the foundations of classical and quantum mechanics, and the foundations of gauge theories.
'The essays are self-contained, providing motivation to read selectively. Examples in each chapter illustrate the usefulness of these new notions of space … Recommended.' M. Clay, Choice Magazine