Polyad

In mathematics, polyad is a concept of category theory introduced by Jean Bénabou in generalising monads.^[1] A polyad in a bicategory D is a bicategory morphism Φ from a locally punctual bicategory C to D, Φ : C → D. (A bicategory C is called locally punctual if all hom-categories C(X,Y) consist of one object and one morphism only.) Monads are polyads Φ : C → D where C has only one object.

In infrared spectroscopy, polyads are groups of vibrational modes (levels). As an example, consider the case of the methane molecule, ¹²CH₄, a spherical top. The molecular vibration spectroscopic wavenumbers verify approximate degeneracy properties. As a consequence, the vibrational levels are grouped into successive polyads. In order of increasing wavenumbers, the polyads are the ground state (origin), the dyad ({ν₂, ν₄}, around 1300-1500 cm^-1), the pentad ({2ν₂, ν₂ + ν₄, 2ν₄, ν₁, ν₃}, around 2600-3000 cm^-1), the octad ({3ν₂, 2ν₂ + ν₄, ν₂ + 2ν₄, 3ν₄, ν₁ + ν₂, ν₁ + ν₄, ν₃ + ν₂, ν₃ + ν₄}, around 3800-4600 cm^-1), the tetradecad ({4ν₂, ..., ν₃ + 2ν₄}, around 5100-6100 cm^-1), the icosad ({5ν₂, ..., ν₃ + 3ν₄}, around 6400-7600 cm^-1), the triacontad ({6ν₂, ..., ν₃ + 4ν₄}, around 7600-9200 cm^-1), and so on. More details can be found at https://doi.org/10.1002/9780470749593.hrs021.