Homology turns shapes into sequences of abelian groups
Homology began in algebraic topology as a way to turn a shape's holes into algebra. At its core is a chain complex, a string of abelian groups linked by maps whose back-to-back composition is always zero, and the homology groups it yields are treated as fundamental invariants.
The word has three linked uses. There is the homology of a chain complex itself. There is the homology of any object to which a chain complex can be attached, with the different recipes for attaching them grouped into homology theories. And there is the homology of a topological space, which connects to popular ideas like the holes in a surface or the cycles in a graph. A mirror notion, cohomology, comes from cochain complexes.
In a chain complex the elements are called chains, and the maps between neighbouring groups are boundary homomorphisms. Cycles are chains the boundary map sends to zero, boundaries are chains that are themselves the boundary of something one level up, and the nth homology group is the quotient of cycles by boundaries. The theory is often named after its complex: singular, Morse, Khovanov and Hochschild homology come from singular, Morse, Khovanov and Hochschild complexes. In category-theory terms a homology theory is a functor into abelian groups, or a derived functor measuring how far another functor is from being exact.
For well-behaved spaces, any theory obeying the Eilenberg–Steenrod axioms gives the same groups as singular homology, so people simply speak of a space's homology. Graph homology suits one-dimensional spaces and is a special case of simplicial homology, which cuts a space into simplices, triangles generalised to any dimension. Singular homology allows more general maps of simplices, while cellular homology uses disks instead.
The inspiration was that holes tell shapes apart: a figure eight has more than a circle, and an inner-tube torus has different holes from a basketball sphere. The figure eight's two loops are examples of 1-cycles, and cycles can be added formally, as symbols, rather than joined geometrically.
Source: Homology (mathematics)
