Sunequivalent

Sunequivalent is a term used in some mathematical contexts to describe a relation of equivalence among objects that is defined relative to a distinguished endofunctor or transformation called the sun operator. The concept is informal in many areas and can be specialized to fit the structures at hand, such as categories, modules, or dynamical systems. In general, sunequivalence identifies objects that become the same after applying the sun operation a suitable number of times.

Formally, let C be a category equipped with an endofunctor Sun: C → C. Two objects A and B in C are sunequivalent if there exist nonnegative integers m and n and an isomorphism Sun^m(A) ≅ Sun^n(B). This definition captures the idea that A and B are indistinguishable once they are evolved, reindexed, or transformed by Sun to the same or isomorphic form. If Sun is the identity functor, sunequivalence reduces to ordinary isomorphism. If Sun corresponds to a shift or suspension, sunequivalence mirrors a stable or periodized notion of equivalence.

Properties and implications vary with context, but, in typical settings, sunequivalence is an equivalence relation whenever isomorphism is compatible with Sun and composition preserves the relevant structure. It often serves to classify objects up to the action of Sun rather than up to strict isomorphism.

Contexts and examples include algebraic topology (Sun as a suspension functor), homological algebra (Sun as a shift functor on complexes), and abstract dynamics (Sun as an evolution operator). The term is not yet standardized and usage can differ by author or field.