structural:expansion, linguistic:voice_change, structural:scholar_edition
Finite calculus

For every abelian group G {\displaystyle G}, the set Hom Set (Z, U (G)) {\displaystyle {\text{Hom}}_{\textbf {Set}}(\mathbb {Z},U(G))} of functions from the integers to the underlying set of G {\displaystyle G} forms an abelian group V Z (G) {\displaystyle V_{\mathbb {Z} }(G)} under pointwise addition. (Here U {\displaystyle U} is the standard forgetful functor U: Ab → Set {\displaystyle U:{\textbf {Ab}}\to {\textbf {Set}}}.) Given an Ab {\displaystyle {\textbf {Ab}}} morphism φ: G → G ′ {\displaystyle \varphi:G\to G'}, the map V Z (φ): V Z (G) → V Z (G ′) {\displaystyle V_{\mathbb {Z} }(\varphi):V_{\mathbb {Z} }(G)\to V_{\mathbb {Z} }(G')} given by left composing φ {\displaystyle \varphi } with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor V Z: Ab → Ab {\displaystyle V_{\mathbb {Z} }:{\textbf {Ab}}\to {\textbf {Ab}}}. The finite difference operator Δ G {\displaystyle \Delta _{G}} taking each function f: Z → U (G) {\displaystyle f:\mathbb {Z} \to U(G)} to Δ (f): n ↦ f (n + 1) − f (n) {\displaystyle \Delta (f):n\mapsto f(n+1)-f(n)} is a map from V Z (G) {\displaystyle V_{\mathbb {Z} }(G)} to itself, and the collection Δ {\displaystyle \Delta } of such maps gives a natural transformation Δ: V Z → V Z {\displaystyle \Delta:V_{\mathbb {Z} }\to V_{\mathbb {Z} }}.