homologies

Noun

homology (countable and uncountable, plural homologies)

1. (projective geometry) specifically, such relationship in the context of the geometry of perspective.
   • Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology: prove that the lines joining the corresponding vertices meet in a point [called the centre of homology].1863, George Salmon, A Treatise on Conic Sections, Longman, Brown, Green, Longman, and Roberts, 4th Edition, page 61
2. (projective geometry) An automorphism of the projective plane (representing a perspective projection) that leaves all the points of some straight line (the homology axis) fixed and maps all the lines through some single point (the homology centre) onto themselves.
   • If the homology centre lies on the homology axis, the homology is said to be singular or parabolic; otherwise, it is called non-singular or hyperbolic.
3. A general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.
   • One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by...2000, Sibe Mardešić, Strong Shape and Homology, Springer, page v
4. Given a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of Gₙ₊₁.
   • Because of their connection with both homology and cohomology, chain complexes are an important topic of study in homological algebra.
5. The relationship, between elements, of being in the same group of the periodic table.
6. The relationship, between organic compounds, of being in the same homologous series.

See all 10 meanings of homology