Homological Algebra & Derived Structure

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Abstractions about chain complexes, exact sequences, derived functors, spectral sequences, categorical correspondences, and diagram lemmas.

12 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.

• Bar complex — A canonical chain complex built from iterated tensor products to resolve an algebra, group or related object.
• Chain complex — A graded sequence of modules or abelian groups connected by boundary homomorphisms whose consecutive composition is zero.
• 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.
• Dold–Kan correspondence — An equivalence between simplicial abelian groups and nonnegatively graded chain complexes, matching homotopy groups with homology groups and simplicial homotopy with chain homotopy.
• 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.
• 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.
• 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.
• Zig-zag lemma — The homological-algebra result that a short exact sequence of chain complexes induces a natural long exact sequence in homology.