Bar, barcode, persistence diagram. A topological feature born at one filtration value and dead at another is a bar [birth, death). The barcode is the multiset of bars; the persistence diagram plots them as points. A bar that never dies is essential; its death is Infinity. Its length, death - birth, is its persistence.

Betti number. The number of independent features of a given degree: components (degree 0), loops (1), voids (2).

Boundary, coboundary. The boundary of a cell is the signed sum of its faces; the coboundary lists the cells that have it as a face. A boundary matrix holds the boundaries as columns; persistence is computed by reducing it.

Cell, simplex, cube. The building blocks of a complex. A simplex is a set of vertices; a cube is a cell of a grid, as in an image.

Chain, cycle, cocycle. A chain is a combination of cells with coefficients. A cycle is a chain with zero boundary, a cocycle a cochain with zero coboundary.

Complex, filtration. A complex is a set of cells closed under taking faces. A filtration gives each cell a value at which it enters, so that faces enter no later than the cells they bound.

Coefficients, field. The numbers chains are built from. tda4j works over a prime field (characteristic = p) or the reals (0). The default is the field with 17 elements.

Engine. An algorithm that computes persistence, for example Ripser's. They are interchangeable for the same input, with different speeds and abilities; see choosing an engine.

Representative. A cycle (or cocycle) that witnesses a bar: it is born when the bar is born and is a nonzero class until the bar dies.

Stream. A filtered complex given as its cells in filtration order.

Vietoris-Rips, Čech, alpha, witness, Dowker, cubical. Ways to build a filtered complex from data; see complexes.