Parametrized equivalence relation on the global class of morphisms of a category and frame-theoretic examples
Nizar El Idrissi
TL;DR
The paper addresses the problem of defining morphism equivalence beyond symmetry-based group actions by proposing a parametrized, higher-categorical notion of equivalence on the global class of morphisms. This is implemented by a construction that uses data $(\mathcal{C},\mathcal{D},\sigma,\tau_1,\tau_2)$ and 1- and 2-morphisms in a $2$-category to relate $m:A\to B$ and $\tilde{m}:\tilde{A}\to \tilde{B}$, with reflexivity, symmetry, and transitivity established. Two concrete instantiations are shown to fit into the framework: (i) group-action equivalence, recovering classical notions, and (ii) a coarse invariant-based equivalence for Bessel functions based on the pointwise norm of the analysis operator $\rho(f)=\|T_f(\cdot)\|_{L^2}$; these illustrate how invariants can supplement, or replace, symmetry in classification. The contributions provide a unifying language for comparing frames and morphisms via higher-categorical observables, potentially guiding a categorical invariant theory for frames and continuous frames. Future directions include characterizing equivalence classes under the coarse invariant and extending the framework to Banach-space settings.
Abstract
We introduce an equivalence relation on the global class of morphisms of a category that extends several classical notions of equivalence in mathematics. We show that the standard group-action equivalence is a special case of our framework. A more interesting example is provided by an equivalence relation defined on the class of Bessel families and based on a coarse invariant that is the pointwise norm of the analysis operator of the Bessel family -- this is also a special case of our framework, and the formalism developed here can be seen as a first step toward a categorical invariant theory for Bessel families and continuous frames.