1. algebraic geometry
  2. algebraic topology
  3. analysis of PDEs
  4. category theory
  5. classical analysis and ODEs
  6. combinatorics
  7. commutative algebra
  8. complex variables
  9. differential geometry
  10. dynamical systems
  11. functional analysis
  12. general mathematics
  13. general topology
  14. geometric topology
  15. group theory
  16. information theory
  17. K-theory and homology
  18. logic
  19. mathematical physics
  20. metric geometry
  21. number theory
  22. numerical analysis
  23. operator algebras
  24. optimization and control
  25. probability
  26. quantum algebra
  27. representation theory
  28. rings and algebras
  29. spectral theory
  30. statistics theory
  31. symplectic geometry

Fun example. Empty colimit does not commute with empty limit

One of the important properties of filtered colimits is that they commute with finite limits in the category of sets.

Theorem: Let F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets} be a functor, where \mathcal{C} is a filtered small category and \mathcal{D} is a finite category. Then the natural mapping

    \[\mathrm{colim}_{\mathcal{C}} \lim_{\mathcal{D}} F (c, d) \to \lim_{\mathcal{D}} \mathrm{colim}_{\mathcal{C}} F(c, d)\]

is an isomorphism.

This statement is, for example, useful to check that a continuous morphism of sites \mathcal{D} \to \mathcal{C} commuting …