ACTIONS, SHEAVES and CATEGORIES

a book project

Part 0: Introduction and contexts

• Introduction

• Few words on formal background; logics, categories, universes

• Simplicial, globular and cyclic sets, nerves and geometric realization • Abelian categories and old homological algebra

• Noncommutative geometry – a general overview

• Algebraic theories

• Internalization in category theory

Part 1: Actions

• Monoids, categories, spans and groupoids and their actions

• Monads

• Operads

• Hopf algebras, actions and coactions

• Monoidal categories, bicategories and double categories

• Enriched categories

• Distributive laws

• Simplicial groups and groupoids

• Actions in topological, smooth and algebraic contexts

• Actions in the context of operator algebras

Part 2: Presheaves, limits and fibrations

• Yoneda lemma and universal constructions • Pseudofunctors and Yoneda lemma for bicategories • Basic types of limits in categories

• Limits in 2-categorical context

• Interplay of functors and limits

• Theorems of Barr and Beck

• Fibered categories

• Regular and Barr-exact categories

• Sketches

Part 3: Topologies, sheaves and stacks

• Grothendieck topologies

• Sheaves

• Stacks

• Grothendieck’s topoi and Guraud’s theorem

• Subobject classifier and elementary topoi

• Lawvere-Tierney topologies

Part 4: Morita theory

• Algebraic Morita theory

• Azumaya theory for groupoids

• Azumaya complex (after Duskin)

• Geometric Morita theory and Poisson geometry • Bimodules and noncommutative geometry

• Distributors

Part 5: Galois descent and torsors

• Classical Galois theory

• Grothendieck Galois theory

• Torsors

• Equivariant objects

• Descent along torsors

• Hopf-Galois extensions

• Galois theory for corings

• Grothendieck Galois group

Part 6: Localization

• Localization of categories

• Localization of bicategories

• Simplicial localization

• Localization in abelian categories

• Bousfeld localization

• Localization in noncommutative geometry

Part 7: Homotopy theory and higher categories

• Homotopy theory of simplicial sets

• Model categories

• Triangulated categories, derived category

• Classifying spaces and topoi

• Moduli spaces

• Higher operads and multicategories

• Batanin’s definition of higher categories • Algebras up to homotopy

• A_{∞}-categories

• dg-categories

• Loop spaces

• Stable homotopy

• Spectra

• Abstract obstruction theory

• Rational homotopy theory and formality

• Simplicial sheaves

Part 8: Lie theories

• Formal groups over operads

• L_{∞}-algebras

• Lie algebroids, courant algebroids and Lie Rinehardt algebras • Generalized complex geometry (after Hitchin)

• Maurer-Cartan forms in Lie theory

• Atiyah class

• Deformation theory

• Koszul duality

• Resolutions of diagonal and de Rham groupoid

Part 9: Nonabelian cohomology

• Bitorsors, group of bitorsors

• Schreier’s theory

• Gerbes • Tannakian duality

• K(π, n)-torsors

• Crossed modules and 2-groups

• Crossed complexes

• Connections on 2-bundles and gerbes

• 2-fibered 2-categories

• Gerbes and CFT

• Quasihopf algebras and Drinfeld associators

Part 10: Noncommutative geometry and homotopic algebraic geometry

• Flat descent in abstract context

• Noncommutative schemes

• Quasitopologies

• Q-categories, sheaves and stacks (after Rosenberg)

• Segal categories

• Algebraic geometry over model categories

• Hopf-Galois extensions of S-algebras (after Rognes)

• Derived moduli spaces

• Derived categories of coherent sheaves

• Topological B-model and Hochschild homology

• Overview of dualities via categories

• Secondary structures in homotopical algebra