The Dimension-Shift Category and Its Mellin-Gamma Representation

2025/06/07 by Andreu Ballus Santacana, Santacana, Andreu Ballus
Mathematics · #18M05 #22E45 #28C10 #30B40 #33B15 #Category Theory (math.CT) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Point processes and geometric inequalities #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.06885

openalex publication_date 2025/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a thin category Dim⁺ of dimension shifts and a category RadMeas of positive Radon measures with Radon--Nikodym density morphisms. We classify scaling-covariant functors Dim⁺\toRadMeas whose morphisms are given by homogeneous densities. Gaussian normalization selects a unique functor with values dμ_x(u)=\fracπ^x/2Γ(x/2)u^x/2-1 du. Its morphism component yields the radial-integration transport R(x,r)=(π^rΓ(x/2))/(Γ(x/2+r)), while the unit-interval observable recovers the Euclidean ball-volume formula V(x)=\fracπ^x/2Γ(x/2+1). The two transports differ by the multiplicative coboundary of β(x)=x, identified with the categorical dimension of the standard object in Deligne's interpolation category Rep(O_t).

Citations

• A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on ℤ^d

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