Parametrized equivalence relation on the global class of morphisms of a category and frame-theoretic examples

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.