**Contemporary Mathematics**

Volume: 643;
2015;
323 pp;
Softcover

MSC: Primary 14; 18; 55; 57; 81;

**Print ISBN: 978-1-4704-1557-0
Product Code: CONM/643**

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**Electronic ISBN: 978-1-4704-2568-5
Product Code: CONM/643.E**

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# Stacks and Categories in Geometry, Topology, and Algebra

Share this page *Edited by *
*Tony Pantev; Carlos Simpson; Bertrand Toën; Michel Vaquié; Gabriele Vezzosi*

This volume contains the proceedings of the CATS4 Conference on Higher
Categorical Structures and their Interactions with Algebraic Geometry,
Algebraic Topology and Algebra, held from July 2–7, 2012, at CIRM in
Luminy, France.

Over the past several years, the CATS conference series has brought
together top level researchers from around the world interested in
relative and higher category theory and its applications to classical
mathematical domains.

Included in this volume is a collection of articles covering the
applications of categories and stacks to geometry, topology and
algebra. Techniques such as localization, model categories, simplicial
objects, sheaves of categories, mapping stacks, dg structures,
hereditary categories, and derived stacks, are applied to give new
insight on cluster algebra, Lagrangians, trace theories, loop spaces,
structured surfaces, stability, ind-coherent complexes and
1-affineness showing up in geometric Langlands, branching out to many
related topics along the way.

#### Readership

Graduate students and research mathematicians interested in category theory and its applications in algebraic geometry, algebraic topology, representation theory, symplectic geometry, and mathematical physics.

# Table of Contents

## Stacks and Categories in Geometry, Topology, and Algebra

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Lagrangian structures on mapping stacks and semi-classical TFTs 110 free
- Cluster categories for topologists 2534
- Crossed simplicial groups and structured surfaces 3746
- Introduction 3746
- I. Crossed simplicial groups and planar Lie groups 4049
- II. Crossed simplicial groups and generalized orders 6170
- III. Structured surfaces 7887
- IV. Structured graphs 8695
- V. 2-Segal Δ\Gen-objects and invariants of \GG-structured surfaces 98107
- Appendix A. The tessellation complex and the Teichmüller space 103112
- Acknowledgements 107116
- References 107116

- A model structure on relative dg-Lie algebroids 111120
- Multiple derived Lagrangian intersections 119128
- Sheaves of categories and the notion of 1-affineness 127136
- Introduction 128137
- 1. Quasi-coherent sheaves of categories 135144
- 2. Statements of the results 140149
- 3. Direct and inverse images for sheaves of categories 143152
- 4. The case of formal completions 149158
- 5. Algebraic stacks: preparations 153162
- 6. Algebraic stacks: criteria for 1-affineness 159168
- 7. Classifying stacks of algebraic groups 165174
- 8. Algebraic stacks: proof of \thmref{t:alg} 168177
- 9. DG indschemes 182191
- 10. Classifying prestacks 188197
- 11. Groups with a rigid convolution category 193202
- 12. De Rham prestacks 197206
- 13. Infinitesimal loop spaces 201210
- 14. Classifying prestacks of (co)-affine group-prestacks 205214
- Appendix A. Descent theorems 208217
- Appendix B. Quasi-affine morphisms 212221
- Appendix C. Beck-Chevalley conditions 214223
- Appendix D. Rigid monoidal categories 217226
- Appendix E. Commutative Hopf algebras 223232
- References 225234

- Trace theories and localization 227236
- Non-semistable exceptional objects in hereditary categories: some remarks and conjectures 263272
- 1. Introduction 263272
- 2. The two examples in \space of regularity preserving categories with Ext-nontrivial couples 268277
- 3. The differential \space 269278
- 4. Remarks about \space in quivers without loops 273282
- 5. Remarks about \space in star shaped quivers 275284
- 6. Remarks about \space in Dynkin and in some extended Dynkin quivers 282291
- 7. Some directions of future research 285294
- Acknowledgements 285294
- References 286295

- Ind-coherent complexes on loop spaces and connections 289298
- Back Cover Back Cover1334