endiagrammen

Endiagrammen are a concept in category theory used to describe ends and coends through universal diagrammatic constructions. For a functor F: C^op × C → D, an end ∫_C F(c, c) is an object E equipped with a family of arrows e_c: E → F(c, c) that satisfy dinaturality: for every morphism f: c → d in C, F(f, id_d) ∘ e_d = F(id_c, f) ∘ e_c. The end is universal with respect to this property: for any object X with a family of arrows x_c: X → F(c, c) satisfying the same relation, there is a unique u: X → E with e_c ∘ u = x_c for all c. The dual construction, the coend ∮_C F(c, c), comes with arrows F(c, c) → E that satisfy the dual dinaturality condition and have a corresponding universal property.

Ends are limits and coends are colimits in the appropriate sense. In many concrete categories ends and coends can be computed as equalizers, coequalizers, or via explicit diagrammatic constructions. Endiagrammen are the diagrams that encode these coherence conditions, often used to reason about natural transformations, invariants in representation theory, or tensor–Hom relations in enriched category theory.

Because ends and coends unify limits and colimits across a bifunctor, they appear in a variety of areas, including algebra, topology, and geometry. The terminology and notation vary by author, but the core idea is to capture a universal object that coherently summarizes a family of morphisms indexed by objects of C.