GaKmorphism

GaKmorphism is a concept in abstract mathematics describing a class of structure-preserving maps between GaK-structured objects within a category. The GaK-structure assigns to each object X a family of endomorphisms Ga_k: X → X indexed by elements k of a monoid K, such that the family interacts coherently with the category’s composition. In many formulations, Ga_1 is the identity on X and Ga_k ∘ Ga_l equals Ga_{kl} (or satisfies a related monoid action condition), making each X into a GaK-object with an action of K.

A GaKmorphism f: X → Y between GaK-structured objects is a morphism in the category that respects the GaK-actions. Concretely, for every k in K, f ∘ Ga_k^X = Ga_k^Y ∘ f. This commuting requirement ensures that the endomorphisms encoding the GaK-structure are preserved by f, so the symmetry or dynamical aspect encoded by Ga_k is carried along by the map.

Key properties include closure under composition: the composition of GaKmorphisms is again a GaKmorphism, and the identity morphism on any GaK-object is a GaKmorphism. Subobjects and quotients can inherit compatible GaK-structures when the endomorphisms restrict or descend appropriately, allowing broader categorical constructions to respect the GaK-data.

Examples help clarify the idea. If K is the trivial monoid, every morphism is a GaKmorphism. If Ga_k equals the k-th power of a fixed endomorphism α, then GaKmorphisms are precisely the morphisms that commute with α. Related notions include equivariant maps in group actions and monoid actions on objects in a category.

GaKmorphism is used to study categories endowed with symmetric, periodic, or dynamical structures encoded by a monoid action, enabling analysis of structure-preserving maps in such contexts.