Using Mathematics to Simplify Q -Analysis

Abstract

Q-analysis involves modelling a relation Λ between two sets, A and B, by a pair of ‘conjugate’ simplicial complexes Λ, Λ′. To ‘understand’ Λ, we study the geometry of Λ and Λ′, hoping that the features we recognise can be carried back to the data that gives rise to Λ. There is a ‘Galois connection’ between Λ and Λ′ which yields a decreasing sequence of subcomplexes Λ ^r of Λ: Λ = Λ⁰ ⊇ Λ¹ ⊇ … ⊇ Λ^r ⊇ Λ^r+1 ⊇ … (and similarly for Λ′), Q-analysis conventionally calculates the numbers, Q_r, the number of components of Λ ^r , but this is merely zero-dimensional information. By using homology theory, we can calculate ‘Betti numbers’ R₀(Λ ^r ), R₁(Λ ^r ), …, R_q(Λ ^r ), … (where R₀ = Q_r) to obtain much more information about the original relation Λ. The purpose of this paper is to draw attention to the existence of these methods, and to give geometrical illustrations. Remarks will be made about the conventional exposition of Q-analysis, to cover the word ‘simplify’ in the title.