homological

Homological is a term used across several areas of mathematics to describe concepts related to homology and the methods that study them. In topology and geometry, it refers to invariants that capture the shape of spaces by measuring holes of different dimensions. In algebra, it denotes the broader framework of homological algebra, which uses chain complexes and derived functors to study algebraic structures.

In topology, homology assigns a sequence of abelian groups or modules H_n(X) to a space X, together with maps between them that arise from continuous mappings. These groups are built from chain complexes, consisting of chains, boundary maps, and the property that composing two consecutive boundary maps yields zero. The nth homology group H_n(X) roughly counts n-dimensional holes: H_0 tracks connected components, H_1 encodes loops, H_2 voids, and so on. Homology is functorial: a continuous map f: X → Y induces homomorphisms H_n(f): H_n(X) → H_n(Y). Basic computations include the homology of spheres, which is Z in dimensions 0 and n and trivial otherwise.

In homological algebra, the term describes an abstract framework for studying chain complexes in abelian categories and their homology. This includes exact sequences, long exact sequences, and derived functors. Derived functors such as Ext and Tor measure the failure of exactness of Hom and tensor functors and are computed via projective or injective resolutions. Homological methods provide tools across mathematics, including algebraic geometry, representation theory, and algebraic topology, enabling comparisons of spaces and modules through general, categorical constructions.

Related concepts include cohomology, which is a dual theory, and persistent homology, used in data analysis to study features across scales.