points × states → values
Chu spaces
A Chu space is a rectangular table of observations. Its rows are points, its columns arestates or tests, and each cell records a value in a fixed set K. This tiny format carries a surprising amount of structure: duality is matrix transposition, and maps are pairs of functions that preserve every observation.
separation diagnostics
orthogonality profile
morphism finder
A morphism (f,g): (A,r,X) → (B,s,Y) uses a point map f: A → B and a backwards state map g: Y → X. The adjointness condition iss(f(a), y) = r(a, g(y)) for every point a and state y.
definition
Fix a set K of truth values, colors, scores, or observations. A Chu space over Kis a triple (A, r, X) where A is a set of points, X is a set of states, and r: A × X → K is the evaluation map.
a ⊥ x = r(a,x)
dual
The dual space swaps rows and columns: (A,r,X)* = (X, rᵀ, A). In the dual, old states become points and old points become states. Applying duality twice gets back the original space.
separated and extensional
- Separated: no two points have the same row.
- Extensional: no two states have the same column.
- Biextensional collapse: identify equal rows and equal columns to remove observational duplicates.
why they matter
Chu spaces give a symmetric language for objects and predicates. They appear in categorical logic, linear logic, formal concept analysis, topology, automata, and models where systems are understood by the tests they pass and the observations they produce.