Stacks and catetories in geometry, topology, and algebra : CATS4 Conference Higher Categorical Structures and Their Interactions with Algebraic Geometry, Algebraic Topology and Algebra, July 2-7, 2012, CIRM, Luminy, France / Tony Pantev [and four others].

Format:

Book

Contributor:

Pantev, Tony, 1963- editor.

Series:

Contemporary mathematics (American Mathematical Society) ; 643.

Contemporary Mathematics, 1098-3627 ; 643

Language:

English

Subjects (All):

Algebraic stacks--Congresses.

Algebraic stacks.

Algebraic topology--Congresses.

Algebraic topology.

Geometry--Congresses.

Geometry.

Algebra--Congresses.

Algebra.

Physical Description:

1 online resource (325 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2015.

Language Note:

English

Summary:

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.

Contents:

Cover

Title page

Contents

Preface

Lagrangian structures on mapping stacks and semi-classical TFTs

Introduction

Previous works

Motivational conjectures and main results

Description of the paper

1. Recollection on shifted symplectic structures

1.1. Definitions

1.2. Examples of shifted symplectic structures

2. Lagrangian structures

2.1. Recollection

2.2. Examples of Lagrangian structures

2.3. Symplectic structures on mapping stacks with boundary conditions

3. Recovering usual symplectic and Lagrangian moduli stacks

3.1. Topological context

3.2. Algebro-geometric context

4. Application: topological field theories from mapping stacks

4.1. Classical TFTs from mapping stacks

4.2. Semi-classical TFTs from mapping stacks with -symplectic target

Concluding remarks

TFTs with boundary conditions

References

Cluster categories for topologists

1. Introduction

2. Triangulated orbit categories

3. Algebraic triangulated categories

4. Topological triangulated categories

Crossed simplicial groups and structured surfaces

I. Crossed simplicial groups and planar Lie groups

II. Crossed simplicial groups and generalized orders

III. Structured surfaces

IV. Structured graphs

V. 2-Segal Δ\Gen-objects and invariants of \GG-structured surfaces

Appendix A. The tessellation complex and the Teichmüller space

Acknowledgements

A model structure on relative dg-Lie algebroids

2. Relative dg-Lie algebroids

Multiple derived Lagrangian intersections

2. Derived symplectic geometry

3. Multiple derived Lagrangian intersections

4. Examples

Sheaves of categories and the notion of 1-affineness

Introduction.

1. Quasi-coherent sheaves of categories

2. Statements of the results

3. Direct and inverse images for sheaves of categories

4. The case of formal completions

5. Algebraic stacks: preparations

6. Algebraic stacks: criteria for 1-affineness

7. Classifying stacks of algebraic groups

8. Algebraic stacks: proof of \thmref{t:alg}

9. DG indschemes

10. Classifying prestacks

11. Groups with a rigid convolution category

12. De Rham prestacks

13. Infinitesimal loop spaces

14. Classifying prestacks of (co)-affine group-prestacks

Appendix A. Descent theorems

Appendix B. Quasi-affine morphisms

Appendix C. Beck-Chevalley conditions

Appendix D. Rigid monoidal categories

Appendix E. Commutative Hopf algebras

Trace theories and localization

Introduction.

1. Categorical preliminaries.

2. Trace functors

3. Normalization and denormalization

4. DG algebras

5. DG categories

Non-semistable exceptional objects in hereditary categories: some remarks and conjectures

2. The two examples in \space of regularity preserving categories with Ext-nontrivial couples

3. The differential \space

4. Remarks about \space in quivers without loops

5. Remarks about \space in star shaped quivers

6. Remarks about \space in Dynkin and in some extended Dynkin quivers

7. Some directions of future research

Ind-coherent complexes on loop spaces and connections

2. Derived completions and Koszul duality

3. Ind-coherent complexes

4. Regularizing -structures

5. Localization theorems

6. Applications: The Borel-Moore version of the Theorem of Feigin-Tsygan

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references at the end of each chapters.

Description based on print version record.

ISBN:

1-4704-2568-8