Topos theory: sheaf cohomology

Seminar Details

đź•’ Time: 15:15 - 17 on Mondays

đź“Ť Location: BBG - 005

Supervisors: LĂ©onard Guetta & Ieke Moerdijk

Preliminary Syllabus

This is a preliminary list of topics that we might cover during the seminar. It will be adapted depending on the advancement throughout the semester.

Sheaves

  • Presheaves on small categories
  • Sites and Grothendieck topologies
  • Sheaves of Sets, abelian groups (or R-Mod)
  • Sheafification
  • Exactness properties on categories of sheaves

Topos

  • Definition of a Topos and characterizations
  • Examples of Topos
  • Morphisms of toposes and morphisms of sites
  • Points of a topos, flat functors
  • Topos with enough points

*Topology

  • *Open and closed subtoposes
  • *(Compact,) connected and locally connected toposes

Homological Algebra

  • Abelian categories
  • Homology of (co)chain complexes in Abelian categories
  • Short exact sequences and long induced sequence in (co)homology
  • Exact functors and derived functors
  • Injective and projective resolutions
  • F-acyclic resolutions and double chain complex lemma

Cohomology of Sheaves

  • Category of pre-sheaves always have enough projectives and injectives
  • Category of sheaves always have enough injectives
  • Cohomology of sheaves as a derived functor
  • *Homology in the case of pre-sheaves toposes
  • Locality of sheaf cohomology \((H^{\bullet}(U,F)\simeq H^{\bullet}(X/U,F|U))\)

Comparison with Other Cohomology Theories

  • Flabby, soft, and fine sheaves
  • De Rham co-homology for smooth manifolds
  • Singular co-homology for locally contractible top. spaces
  • (Co)homology of groups and small categories
  • (Co)homology of a simplicial set X as the (co)homology of \(\widehat{(\Delta/X)}\)

More Sheaf Cohomology

  • ÄŚech cohomology, comparison with sheaf cohomology
  • Hyper-coverings
  • Torsors and geometrical interpretation of \(H^1\)

References

Topos Theory

  • Johnstone, Topos Theory
  • Kashiwara & Shapira, Categories and Sheaves
  • MacLane & Moerdijk, Sheaves in Geometry and Logic
  • Artin, Grothendieck Topologies
  • Grothendieck et al., SGA4
  • de Jong, Stacks Project
  • Moerdijk, Classifying Spaces and Classifying Topoi

Sheaves on Topological Spaces

  • Dimca, Sheaves in Topology
  • Godement, ThĂ©orie des faisceaux et topologie algĂ©brique
  • Bredon, Sheaf Theory
  • Kashiwara & Shapira, Sheaves on Manifolds
  • Swan, The Theory of Sheaves

Homological Algebra

  • Moerdijk, Notes on homological algebra
  • Kashiwara & Shapira, Categories and Sheaves
  • Weibel, An Introduction to Homological Algebra
  • Gelfand & Manin, Methods of Homological Algebra

Weekly Topics

Week Date Speaker Topic
0 03/02 LĂ©onard Guetta Introduction and overview
1 11/02 Maarten de Groot Grothendieck topologies, sheaves on a site
2 17/02 Valentijn van der Land Sheafification and categories of sheaves
Lecture notes (with homework)
3 24/02 Anton Odina Definition of Topos and characterizations