Categorical and derived tools for analytic sheaves

Growing course with 86 lessons and 344 graded exercises with full solutions. Written or adapted and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort. Independent model reviews cover two exact unchanged lesson versions; thirteen historical reviews retain their original source bindings. Separate scoped model checks cover specified new arguments, retained prerequisite interfaces and four exercise solutions per new lesson; these checks add no whole-lesson or whole-course review. Original exposition uses CC0 1.0. The whole Tom Leinster adaptation retains CC BY-NC-SA 4.0. Further lessons, course-wide coverage, source reconciliation and review remain in progress.

Sources, authorship and component terms

  1. Universes and small categories
  2. Relations and cancellation in categories
  3. Zero maps, components and subobjects
  4. Natural transformations and composition of functors
  5. Equivalences and chosen representatives
  6. Products, disjoint unions and mixed functors
  7. Ind-objects through their elements
  8. Generators and small quotient families
  9. Coimages, images and composition of quotients
  10. Solution sets and universal representations
  11. Dense probes and reconstruction from colimits
  12. Finite duality and failure of opposite density
  13. Formal colimits and compact presentations
  14. Ideal-generated modules and nonstrict quotients
  15. Trace coreflections and balanced nonabelian categories
  16. Internal groups, limits and transport
  17. Recovering addition from products and coproducts
  18. Additive localization and projector decompositions
  19. Formal linear combinations and finite sums
  20. Bilinear functors and tensor products of categories
  21. Quotients by a factorization ideal
  22. Full subcategories and exact closure
  23. Testing exactness with doubled objects
  24. Coherence from finite probes
  25. Composition factors and uniform chain bounds
  26. Exact squares and endpoint tests
  27. Character duals and one-sided injectivity
  28. Recovering isomorphisms from filtered pieces
  29. Finite support and the limits that a category permits
  30. Splittings, support and sequential exactness
  31. Finite orbit spans and actions on formal unions
  32. Initial objects, universal colimits and pointed categories
  33. Locally nilpotent operators and coinduced duals
  34. Small modules with too many scalar operators
  35. Exact complexes, signs and lifting covers
  36. Extending maps, splittings and exact functors
  37. Pointwise abelian structure and natural splittings
  38. Kernels, cokernels and the abelian comparison
  39. Finite module tests and Banach quotients
  40. Hom, tensor and the exactness of module limits
  41. Testing injectivity on generators and constructing enough injectives
  42. Serre quotients and local saturation
  43. Extending right exact functors from generators
  44. Operators, matrix relations and internal tensor–Hom
  45. Natural scalars on modules and finite abelian groups
  46. From a generator to module presentations
  47. Two cokernels, controlled rows and abelian completion
  48. Finite diagrams and comma objects
  49. Limits and colimits of formal objects
  50. Retracts and stabilization of formal objects
  51. Finite-rank tests and finite generation
  52. Profinite spaces and finite-stage maps
  53. Localizing functors through formal objects
  54. Descent through replacement objects
  55. One-sided fractions and saturation
  56. Tensor actions and absolute algebra presentations
  57. Braidings, cocycles and cyclic power obstructions
  58. Tensor duality and coherent inverses
  59. Free tensor words and ordered algebra encoding
  60. Coherent integer actions from commuting equivalences
  61. Representing objects from local data
  62. Local module categories and Morita equivalence
  63. Twisted modules from coherent gluing
  64. Points, fibres and universal representations
  65. Passing maps across an adjunction
  66. Cuts and endpoint maps in finite orders
  67. Testing equivalence with arrows and finite characters
  68. Building categories from familiar structures
  69. Compatible families and universal cones
  70. Universal forks and diagram constructions
  71. Extending diagrams by universal maps
  72. Colimits as connected components
  73. Change an index without changing its universal maps
  74. Formal colimits and their test objects
  75. Extending a functor from its representables
  76. Universal tests for forks and finite limits
  77. Fibres, slices and index adjunctions
  78. Reindexing, units and terminal probes
  79. Formal coproducts and economical indices
  80. Presheaf exactness and pullback comparisons
  81. Filtered stages and finite limits
  82. Local tests and independent product comparisons
  83. Incoming comma tests and assembly from finite pieces
  84. Exactness through comma categories and small Kan values
  85. Comma functors, iterated cocones and arrow categories
  86. Universal maps, finite probes and countable indexing