Topos theory: sheaf cohomology

Seminar Details

🕒 Time: 15:15 - 17 on Mondays

📍 Location: BBG - 005

Supervisors: Léonard 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

Čech 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
Moerdijk, Introduction to the Language of Stacks and Gerbes

Sheaves on Topological Spaces

Dimca, Sheaves in Topology
Godement, Théorie des faisceaux et topologie algébrique
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

Others

Simplicial cheat-sheet

Weekly Topics

Week	Date	Speaker	Topic
0	03/02	Léonard 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 Homework
4	03/03	Nicholas Puthu	Morphisms of Toposes Homework
5	10/03	Jonathan Grube	Homological Algebra I Homework
6	17/03	Marcel Masque Salgado	Homological Algebra II Lecture notes (with homework)
7	24/03	Tom Sinke	Cohomology of sheaves Lecture notes (with homework)
8	31/03	Wouter Fransen	Comparison with other cohomology theories Lecture Notes (very unpolished!) Homework
9	07/04	Rashiqa Dawood	(Co)homology of groups, categories and simplicial sets Homework
10	14/04	Splinter Suidman	Čech cohomology Lecture notes (with homework)
11	28/04	Ningyi Li	Hypercoverings Lecture notes (with homework)
12	12/05	Chris Vos	Torsors, non-abelian H¹, stacks part I Lecture Notes Homework
13	19/05	Rob Schellingerhout	Torsors, non-abelian H¹, stacks part II A note on the automorphism sheaves Homework
14	26/05	Tigo Ruijs	Connected and locally connected topos, pro-π₀
15	02/06	Deik van der Horst	Pro-fundamental group of a topos, part I
16	15/06	Benjamin Mason	Pro-fundamental group of a topos, part II