Alphabeta MathDerived Categories — Examples
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1 · Prerequisites
- Abelian Categories
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Derived Categories
- Derived Functors
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Finite Counting, Factorials and Binomial Coefficients
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Limits and Colimits
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Preadditive and Additive Categories and Biproducts
- Projective and Injective Resolutions
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor Products of Modules
- The Diagram Lemmas in an Abelian Category
- The ZFC Axioms and the Basic Set Constructions
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Properties, Representables and the Yoneda Lemma
- Yoneda Extensions and Homological Dimension
2 · Logical flowchart
3 · Items
Example
ex-a-roof-representing-an-ext-one-class.md
Example
ex-an-acyclic-complex-that-becomes-zero-in-d-but-not-in-k.md
Example
ex-inverting-a-quasi-isomorphism-by-a-reversed-roof.md
Example
ex-ext-one-as-a-derived-category-morphism.md
Example
ex-brutal-versus-canonical-truncation.md
Example
ex-a-canonical-truncation-triangle.md
Example
ex-derived-tensor-of-two-cyclic-abelian-groups.md
Example
ex-derived-hom-of-cyclic-abelian-groups.md
Counterexample
cex-an-unbounded-complex-of-projectives-that-is-not-k-projective.md
Sources
- 10.4.7 and 10.7.5, pp. 388, 400
- 13.11.1–13.11.6
- 12.15, all four chain and four cochain truncations
- Remark 13.12.4 and its three triangles
- 10.6.1–10.6.4 and Exercise 10.6.1, p. 395; elementary finite-diagonal replacement for spectral sequence proof
- 10.7.2–10.7.5 and Exercise 10.7.1, pp. 399–400
- Explicit specialization of the bounded models; see batch notes