Evaluationsfunktoren

Evaluationsfunktoren, or evaluation functors, are a standard construction in category theory. Given categories C and D and the functor category [D, C] (the category of all functors D → C with natural transformations), for each object d in D there is a functor ev_d: [D, C] → C defined by ev_d(F) = F(d) on objects and ev_d(α: F ⇒ G) = α_d: F(d) ⇒ G(d) on morphisms. In essence, ev_d reads off the value of a diagram at the fixed point d.

Properties of evaluation functors include that they preserve limits and colimits that exist pointwise in [D, C]. If C is complete, ev_d preserves limits; if C is cocomplete, ev_d preserves colimits. They behave well with respect to the varying object d, forming a natural family of functors indexed by D.

Adjoints and Kan extensions: When C has the appropriate (co)limits, ev_d has both left and right adjoints. The left adjoint L_d: C → [D, C] is given by (L_d c)(d') = ⨿_{D(d, d')} c, with the action on morphisms induced by composition. The right adjoint R_d: C → [D, C] is given by (R_d c)(d') = ∏_{D(d', d)} c, with the corresponding functorial structure. These adjoints arise as left and right Kan extensions along the inclusion of the representable at d.

Applications of evaluation functors include the pointwise construction of limits and colimits in functor categories, analysis of representable diagrams, and foundational aspects of ends, coends, and the Yoneda calculus. They provide a concrete way to extract the value of a diagram at a particular object and to study how diagrams vary across D.