The Dimension-Shift Category and Its Mellin-Gamma Representation

TL;DR

This paper introduces a category of dimension shifts and classifies scaling-covariant functors to Radon measures, revealing connections with Mellin-Gamma representations, Euclidean volume, and Deligne's interpolation category.

Contribution

It defines a new category of dimension shifts and classifies functors to Radon measures, linking Mellin-Gamma representations with geometric and categorical structures.

Findings

Gaussian normalization yields a unique functor with explicit density form.

Radial-integration transport formula derived from the functor.

Unit-interval observable recovers Euclidean ball-volume formula.

Abstract

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