Algebraic constructionfor spaceswithvariable dimension

An algebraic construction is presented that describes the variation of dimensionandrankoftensorspaces. Spaces 𝑅𝑛𝑚-𝑛-dimensional𝑚-rankrealvectorspaces-areintroduced,alongwithdimension-raisingandloweringfunctors 𝑁±andrank-raisingand lowering functors 𝛼±. Rules for direct sum and tensor product of objects withdifferent dimensions and ranks are formulated. An operator space 𝑋 is built that isclosedandcommutative. Dimensionalityemergesasavariable;sectionsofconstantdimension are Riemannian, while the full space is non-Riemannian. The functors𝑁±,𝛼±show a notable similarity with creation and annihilation operators of secondquantization. The resulting structure provides an algebraic foundation for theorieswithdynamical dimensionality.