Homological Algebra Constructions

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Abstractions that describe exactness and resolution in homological algebra, including chain-complex and exact-sequence machinery such as cokernels, exact sequences and the zig-zag lemma, derived and spectral-sequence tools like derived functors and the Lyndon-Hochschild-Serre spectral sequence, and categorical resolution frameworks such as Koszul algebras and Waldhausen categories.

19 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• AB5 category — An abelian category with arbitrary coproducts in which filtered colimits of exact sequences remain exact; adding a generator yields a Grothendieck category.
• Bar complex — A canonical chain complex built from iterated tensor products to resolve an algebra, group or related object.
• Beck's monadicity theorem — A categorical criterion determining when a functor is equivalent to the forgetful functor from algebras for the monad induced by its adjunction.
• Chain complex — A graded sequence of modules or abelian groups connected by boundary homomorphisms whose consecutive composition is zero.
• Cokernel — The universal quotient of a morphism's codomain that makes the morphism vanish, realized for linear maps as codomain modulo image.
• Derived functor — A functor obtained by resolving objects relative to an exactness-deficient functor and taking homology, systematically measuring its failure to preserve exact sequences.
• Exact sequence — A sequence of morphisms in which the image of every map is exactly the kernel of the next, encoding that each stage contains no unexplained residue between arrival and annihilation.
• Five-term exact sequence — The low-degree exact sequence extracted from a first-quadrant spectral sequence, linking edge terms, an early differential and the first two groups of the abutment.
• Hall algebra — An associative algebra whose basis represents isomorphism classes of objects and whose multiplication counts extensions or subobjects with specified quotient and subobject types.
• Koszul algebra — A graded algebra whose ground field admits a minimal graded free resolution that is linear in every homological degree.
• Koszul–Tate resolution — A differential graded commutative algebra resolution of a quotient ring that generalizes the Koszul complex by adding generators to kill successive homology.
• Lyndon–Hochschild–Serre spectral sequence — A spectral sequence that computes or constrains the homology or cohomology of a group from a normal subgroup, the quotient group and the quotient action on the subgroup's (co)homology.
• Morava K-theory — A prime- and height-indexed family of periodic generalized homology theories that isolates chromatic layers of stable homotopy theory.
• Nine lemma — A diagram lemma stating conditions under which exactness of rows and columns in a commutative three-by-three diagram forces exactness of the remaining row or column.
• Six operations — The six-functor formalism relating pullback, pushforward, extraordinary pullback and pushforward, tensor product and internal Hom across geometric categories.
• T-structure — A pair of subcategories of a triangulated or stable infinity category satisfying shift, orthogonality and truncation axioms, whose intersection forms an abelian heart.
• Waldhausen category — A category equipped with designated cofibrations and weak equivalences satisfying gluing axioms so its algebraic K-theory spectrum can be constructed by the S-construction.
• Weak dimension — The least upper bound of the flat dimensions of all modules over a ring, measuring how far the ring is from making every module flat.
• Zig-zag lemma — The homological-algebra result that a short exact sequence of chain complexes induces a natural long exact sequence in homology.