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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2506.06885 |
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| _version_ | 1866913105254023168 |
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| author | Santacana, Andreu Ballus |
| author_facet | Santacana, Andreu Ballus |
| contents | 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 $ dμ_x(u)=\frac{π^{x/2}}{Γ(x/2)}u^{x/2-1}\,du. $ Its morphism component yields the radial-integration transport $ R(x,r)=\frac{π^rΓ(x/2)}{Γ(x/2+r)}, $ while the unit-interval observable recovers the Euclidean ball-volume formula $ V(x)=\frac{π^{x/2}}{Γ(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 $\mathrm{Rep}(O_t)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Dimension-Shift Category and Its Mellin-Gamma Representation Santacana, Andreu Ballus Representation Theory Classical Analysis and ODEs Category Theory 18M05, 28A33, 43A05, 33B15 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 $ dμ_x(u)=\frac{π^{x/2}}{Γ(x/2)}u^{x/2-1}\,du. $ Its morphism component yields the radial-integration transport $ R(x,r)=\frac{π^rΓ(x/2)}{Γ(x/2+r)}, $ while the unit-interval observable recovers the Euclidean ball-volume formula $ V(x)=\frac{π^{x/2}}{Γ(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 $\mathrm{Rep}(O_t)$. |
| title | The Dimension-Shift Category and Its Mellin-Gamma Representation |
| topic | Representation Theory Classical Analysis and ODEs Category Theory 18M05, 28A33, 43A05, 33B15 |
| url | https://arxiv.org/abs/2506.06885 |