Categorical and derived tools for analytic sheaves
An English course with 237 lessons and 948 fully worked exercises. Further topics will be added.
Universal properties and category theory; Ind and Pro constructions; monoidal categories and duality; modules, stacks and coherent descent; complexes, triangulated categories and derived functors; sheaf cohomology, local-to-global comparisons and gerbes.
- Universes and small categories
- Relations and cancellation in categories
- Zero maps, components and subobjects
- Natural transformations and composition of functors
- Equivalences and chosen representatives
- Products, disjoint unions and mixed functors
- Ind-objects through their elements
- Generators and small quotient families
- Coimages, images and composition of quotients
- Solution sets and universal representations
- Dense probes and reconstruction from colimits
- Finite duality and failure of opposite density
- Formal colimits and compact presentations
- Ideal-generated modules and nonstrict quotients
- Trace coreflections and balanced nonabelian categories
- Internal groups, limits and transport
- Recovering addition from products and coproducts
- Additive localization and projector decompositions
- Formal linear combinations and finite sums
- Bilinear functors and tensor products of categories
- Quotients by a factorization ideal
- Full subcategories and exact closure
- Testing exactness with doubled objects
- Coherence from finite probes
- Composition factors and uniform chain bounds
- Exact squares and endpoint tests
- Character duals and one-sided injectivity
- Recovering isomorphisms from filtered pieces
- Finite support and the limits that a category permits
- Splittings, support and sequential exactness
- Finite orbit spans and actions on formal unions
- Initial objects, universal colimits and pointed categories
- Locally nilpotent operators and coinduced duals
- Small modules with too many scalar operators
- Exact complexes, signs and lifting covers
- Extending maps, splittings and exact functors
- Pointwise abelian structure and natural splittings
- Kernels, cokernels and the abelian comparison
- Finite module tests and Banach quotients
- Hom, tensor and the exactness of module limits
- Testing injectivity on generators and constructing enough injectives
- Serre quotients and local saturation
- Extending right exact functors from generators
- Operators, matrix relations and internal tensor–Hom
- Natural scalars on modules and finite abelian groups
- From a generator to module presentations
- Two cokernels, controlled rows and abelian completion
- Finite diagrams and comma objects
- Limits and colimits of formal objects
- Retracts and stabilization of formal objects
- Finite-rank tests and finite generation
- Profinite spaces and finite-stage maps
- Localizing functors through formal objects
- Descent through replacement objects
- One-sided fractions and saturation
- Tensor actions and absolute algebra presentations
- Braidings, cocycles and cyclic power obstructions
- Tensor duality and coherent inverses
- Free tensor words and ordered algebra encoding
- Coherent integer actions from commuting equivalences
- Representing objects from local data
- Local module categories and Morita equivalence
- Twisted modules from coherent gluing
- Attachment maps from diagrams to costalks
- Points, fibres and universal representations
- Passing maps across an adjunction
- Cuts and endpoint maps in finite orders
- Testing equivalence with arrows and finite characters
- Building categories from familiar structures
- Compatible families and universal cones
- Universal forks and diagram constructions
- Extending diagrams by universal maps
- Colimits as connected components
- Change an index without changing its universal maps
- Formal colimits and their test objects
- Extending a functor from its representables
- Universal tests for forks and finite limits
- Fibres, slices and index adjunctions
- Reindexing, units and terminal probes
- Formal coproducts and economical indices
- Presheaf exactness and pullback comparisons
- Filtered stages and finite limits
- Local tests and independent product comparisons
- Incoming comma tests and assembly from finite pieces
- Exactness through comma categories and small Kan values
- Comma functors, iterated cocones and arrow categories
- Universal maps, finite probes and countable indexing
- Well-orders, cardinal bounds and regular successors
- Common arrows and bounded Set comparisons
- Small predecessors and maps into filtered colimits
- Bounded colimits, accessible sets and cardinal ind-objects
- Cardinal Ind presentations and their realization
- Coproduct covers and generator bounds
- Accessible objects from a single generator
- Chain homotopies, mapping cones and triangles
- Derived morphisms through quasi-isomorphism roofs
- Triangles after quasi-isomorphism localization
- Injective replacements and right derived functors
- Universal cohomology and acyclic resolutions
- Postnikov towers and Grothendieck spectral sequences
- Filtered complexes and Cartan–Eilenberg resolutions
- Unbounded injective replacements and derived functors
- Relative injectivity and small test families
- Injective cogenerators and faithful module realizations
- Essential extensions and filtered test categories
- Triangles, finite universal maps and the limits of cohomology
- Translations, coherent powers and universal triangle comparisons
- Connecting signs and opposite triangle structures
- Null systems, roofs and the retract kernel
- Localizing triangulated functors and signed bifunctors
- Extending cohomology through filtered arrow stages
- Product-preserving functors and countable Brown probes
- Brown towers, representability and exact adjoints
- Split triangles, idempotent telescopes and semisimple structures
- Orthogonal localization and the morphisms between pieces
- Representing filtered triangles by a finite stage
- Signed three-by-three diagrams and double cones
- Differential objects, curvature and cone homotopies
- Stabilizing an endofunctor through eventual maps
- Graded complexes, contravariant signs and stupid truncations
- Augmented finite orders and contracting coface complexes
- Signed bifunctors and sum and product total complexes
- Cohomology in abelian categories with translation
- Smart truncations and image boundary homotopies
- Determinant signs and subset Koszul complexes
- Intersection complexes and the full distributivity criterion
- Distributive sublattices, duality and simultaneous splittings
- Finite diagonal totals and signed double truncations
- Acyclic lines, Weil staircases and corner removal
- Exact bifunctors and the canonical field Künneth map
- Countable Mittag-Leffler towers and formal Pro derived limits
- Local tests and global maps in real-parameter module diagrams
- Real-parameter complexes and cohomology continuity
- Cohomological bounds, denominators and truncation adjunctions
- Hereditary categories and cohomology splittings
- Relative replacements and finite resolution bounds
- Compatible injective classes and exact refinements
- Universal derived functors and compatible composition
- Formal Pro products and classes that compute derived limits
- Derived Hom and bounded resolution pairs
- First derived cohomology and composition through truncation
- Injective tests and finite homological dimension
- Short exact extensions and the Baer group
- Higher extensions, roofs and Yoneda composition
- Thick abelian subcategories and derived comparison
- Functorial cones and derived connecting maps
- Two cohomology degrees, Ext classes and polynomial models
- Finite cohomology splittings and derived kernels
- Right derived tensor and torsion-free quotients
- Cosyzygies and the bounded injective quotient
- Hard-Lefschetz maps and cohomology splittings
- Weyl polynomial modules and their Koszul calculations
- Polynomial rings, finite resolutions and derived duality
- Hereditary cohomology and integral tensor splittings
- Nested Hom classes and bounded derived embeddings
- Tensor and Hom for twisted module stacks
- Bimodules and adjoints of module-stack functors
- Noncommutative gluing by invertible bimodules
- Dual and tensor products of locally module stacks
- Relative pullback of locally module stacks
- Injective objects in locally module stacks
- Flat covers and bounded tensor for locally module stacks
- Ind-completion and Serre quotients
- Derived sections of Ind Serre quotients
- Finitely presented generators and coherent Ind reconstruction
- Closed support and localization for locally module stacks
- Noetherian generators, filtered injectives and Ext
- Localizing subcategories and noetherian Serre pieces
- Derived perfect duality for twisted module complexes
- Separable algebra objects and absolute balanced tensor
- Separable monads and the Karoubi completion of free modules
- Flat algebra epimorphisms and exact localization of module stacks
- Faithfully flat scalar descent and Amitsur resolutions
- Finite projective descent and Morita generators
- Associated sheaves with general coefficients
- Finite-cover sheafification and locally nilpotent coefficients
- Flat modules as an Ind completion of finite free modules
- Coefficient morphism sheaves and local isomorphisms
- Modules and tensor functors over small preadditive categories
- Cauchy completion and Morita equivalence of many-object modules
- Ideals, quotients and torsion triples for many-object modules
- Projective probes and recollements of many-object modules
- Finite projective bimodules and both adjoints of tensor
- Dualizable sheaves, finite covers and global projectivity
- Finite presentations and Ind reconstruction on finite-cover sites
- Finite-poset sheaves and canonical homological resolutions
- Derived direct images and base change on finite posets
- Local systems, tensor duals and global projectivity on finite posets
- Ind representations and the locally finite coreflector
- Pro representations and continuous duality
- Completed tensors and continuous algebra objects
- Finite subcoalgebras and profinite algebra reconstruction
- Comodules, rational modules and cofree injective resolutions
- Godement resolutions and flabby sheaf cohomology
- Čech complexes, refinements and acyclic covers
- Torsors, first cohomology and line bundles
- Higher direct images and the Leray spectral sequence
- Sheaf Ext and the local-to-global spectral sequence
- Filtered colimits and sheaf cohomology on quasi-compact opens
- Cohomological dimension of noetherian spaces
- Sieves, local isomorphisms and presheaf localization
- Restriction, inverse images and morphisms of sieve sites
- Slice sites, dependent products and base change
- Ringed sieve sites and extension of scalars
- Quasi-injectives and bounded comparisons for Ind categories
- Derived Ind functors and filtered Ext comparisons
- Dense subsites and change of topology
- Coefficient sheafification on arbitrary sieve sites
- Stackification, local morphisms and ordinary inverse images
- Stackification by cartesian sections on covering sieves
- Functor stacks, cartesian currying and two-fibre products
- Finite closed covers and sheaf cohomology
- Locally constant module sheaves and transition maps
- Interval cohomology and homotopy invariance for sheaves
- Path transport and monodromy of locally constant sheaves
- Strict monoidal models and transport of algebra objects
- Monoidal adjunctions, mates and dual projection formulas
- Germ spaces, étalé maps and geometric sheafification
- Presentable probes and accessible adjoint functors
- Small orthogonality and accessible reflective localizations
- Small presentations and saturated reflection equivalences
- Coend coalgebras and reconstruction from finite fibres
- Monoidal fibres, bialgebras and duality reconstruction
- Scalar tensor automorphisms and affine group reconstruction
- Locally presentable coefficients and accessible sheafification
- Ind tensor products and compact duals
- Internal Hom in Ind categories and flat module obstructions
- Derived limits on small categories and cofinal restriction
- Cofiltered derived limits and morphisms of pro-objects
- Soft sheaves on paracompact Hausdorff spaces
- Fine module sheaves and central partitions of unity
- Closed subsets and continuity of sheaf cohomology
- Čech comparison and cohomological bounds on paracompact spaces
- Covering dimension and locally finite refinements
- Stacks over sheaves and epimorphic descent
- Abelian gerbes and second sheaf cohomology
- Adjoints of completed tensor products