Outline

Sheaves are versatile objects in topology and geometry. Cohomology of a constant sheaf recovers classical notions of cohomology (singular, cellular, de Rham, ...), while locally constant sheaves correspond to representations of the fundamental group. I will discuss the foundations of sheaf theory, and highlight some recent advances in the classification of sheaves on topological spaces and their counterparts in algebraic geometry.

Talks

Click the titles below for (handwritten) notes.

  1. Sheaves and cohomology in topology.
  2. Why étale cohomology?
  3. Constructible sheaves in topology: from monodromy to exodromy.
  4. Monodromy and exodromy for étale sheaves.

References

Here are some sources I used for writing the lecture notes, plus some introductory texts (where available).

Sheaves, sheaf cohomology, and Čech cohomology:

[Ive]
B. Iversen, Cohomology of sheaves.
[Bre]
G. E. Bredon, Sheaf theory.
[Ten]
B. R. Tennison, Sheaf theory.
[KS]
M. Kashiwara, P. Schapira, Sheaves on manifolds.
[Dim]
A. Dimca, Sheaves in topology.
[MM]
S. Mac Lane, I. Moerdijk, Sheaves in geometry and logic.
[BT]
R. Bott, L. W. Tu, Differential forms in algebraic topology.
[Voi]
C. Voisin, Hodge theory and complex algebraic geometry I.
[HTT]
J. Lurie, Higher topos theory.
[Stacks]
The Stacks project, sheaves and cohomology of sheaves.

Étale fundamental groups and étale cohomology:

[SGA1]
A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1).
[SGA4]
M. Artin, A. Grothendieck, J.-L. Verdier (eds), Théorie des topos et cohomologie étale des schémas (SGA 4). Part I, Part II, and Part III.
[SGA4½]
P. Deligne, Cohomologie étale (SGA 4½).
[SGA5]
L. Illusie (ed), Cohomologie ℓ-adique et fonctions L (SGA 5).
[Del]
P. Deligne, La conjecture de Weil, I.
[FK]
E. Freitag, R. Kiehl, Étale cohomology and the Weil conjecture.
[Mil]
J. S. Milne, Étale cohomology.
[MilLec]
J. S. Milne, Lectures on étale cohomology.
[Stacks]
The Stacks project, étale fundamental groups and étale cohomology.

Exodromy in topology:

[Tre]
D. Treumann, Exit paths and constructible stacks.
[CP]
J. Curry, A. Patel, Classification of constructible cosheaves.
[HA]
J. Lurie, Higher algebra, Appendix A.
[PT]
M. Porta, J.-B. Teyssier, Topological exodromy with coefficients.

Étale exodromy:

[BGH]
C. Barwick, S. Glasman, P. Haine, Exodromy.
[Wolf]
S. Wolf, The pro-étale topos as a category of pyknotic presheaves.
[vDdBW]
R. van Dobben de Bruyn, S. Wolf, work in progress.

A seminar picture