Tobias Dyckerhoff is Lichtenberg Professor of Mathematics at the University of Hamburg, Germany. His research interests are centered on higher structures in algebra and geometry, and extend to mathematical physics.Mikhail Kapranov is Professor of Mathematics at the Kavli Institute for the Physics and Mathematics of the Universe in Japan. His research spans algebra, algebraic geometry and category theory, with a particular interest in using category theory to extend ideas at the interface of algebra and geometry. His previous books include Discriminants, Resultants, and Multidimensional Determinants, with I. M. Gelfand and A. V. Zelevinsky, a Modern Birkhäuser Classic.

• 1. Preliminaries.- 2. Topological 1-Segal and 2-Segal Spaces.- 3. Discrete 2-Segal Spaces.- 4. Model Categories and Bousfield localization.- 5. The 1-Segal and 2-Segal model structures.- 6. The path space criterion for 2-Segal Spaces.- 7. 2-Segal Spaces from higher categories.- 8. Hall Algebras associated to 2-Segal Spaces.- 9. Hall (∞,2)-Categories.- 10. An (∞,2)-categorical theory of Spans.- 11. 2-segal Spaces as monads in bispans App.- A: Bicategories.

Tobias Dyckerhoff is Lichtenberg Professor of Mathematics at the University of Hamburg, Germany. His research interests are centered on higher structures in algebra and geometry, and extend to mathematical physics.Mikhail Kapranov is Professor of Mathematics at the Kavli Institute for the Physics and Mathematics of the Universe in Japan. His research spans algebra, algebraic geometry and category theory, with a particular interest in using category theory to extend ideas at the interface of algebra and geometry. His previous books include Discriminants, Resultants, and Multidimensional Determinants, with I. M. Gelfand and A. V. Zelevinsky, a Modern Birkhäuser Classic.

• 1. Preliminaries.- 2. Topological 1-Segal and 2-Segal Spaces.- 3. Discrete 2-Segal Spaces.- 4. Model Categories and Bousfield localization.- 5. The 1-Segal and 2-Segal model structures.- 6. The path space criterion for 2-Segal Spaces.- 7. 2-Segal Spaces from higher categories.- 8. Hall Algebras associated to 2-Segal Spaces.- 9. Hall (∞,2)-Categories.- 10. An (∞,2)-categorical theory of Spans.- 11. 2-segal Spaces as monads in bispans App.- A: Bicategories.

Tobias Dyckerhoff is Lichtenberg Professor of Mathematics at the University of Hamburg, Germany. His research interests are centered on higher structures in algebra and geometry, and extend to mathematical physics.Mikhail Kapranov is Professor of Mathematics at the Kavli Institute for the Physics and Mathematics of the Universe in Japan. His research spans algebra, algebraic geometry and category theory, with a particular interest in using category theory to extend ideas at the interface of algebra and geometry. His previous books include Discriminants, Resultants, and Multidimensional Determinants, with I. M. Gelfand and A. V. Zelevinsky, a Modern Birkhäuser Classic.

• 1. Preliminaries.- 2. Topological 1-Segal and 2-Segal Spaces.- 3. Discrete 2-Segal Spaces.- 4. Model Categories and Bousfield localization.- 5. The 1-Segal and 2-Segal model structures.- 6. The path space criterion for 2-Segal Spaces.- 7. 2-Segal Spaces from higher categories.- 8. Hall Algebras associated to 2-Segal Spaces.- 9. Hall (∞,2)-Categories.- 10. An (∞,2)-categorical theory of Spans.- 11. 2-segal Spaces as monads in bispans App.- A: Bicategories.