Ivancevic_Applied-Diff-Geom

110 Applied Differential Geometry: A Modern Introductionh : x → e of B with β c = ω c h for all a ∈ C. We call ω the ending wedgewith components ω c , while the object e itself, by abuse of language, is calledthe end of S and written with integral notation as ∫ S(c, c); thusc∫S(c, c) →ωcS(c, c) = e.cNote that the ‘variable of integration’ c appears twice under the integralsign (once contravariant, once covariant) and is ‘bound’ by the integralsign, in that the result no longer depends on c and so is unchanged if‘c’ is replaced by any other letter standing for an object of the categoryC. These properties are like those of the letter x under the usual integralsymbol ∫ f(x) dx of calculus.Every end is manifestly a limit (see below) – specifically, a limit of asuitable diagram in X made up of pieces like S(b, b) → S(b, c) → S(c, c).For each functor T : C → X there is an isomorphism∫ ∫S(c, c) = T c ∼ = Lim T,ccvalid when either the end of the limit exists, carrying the ending wedge tothe limiting cone; the indicated notation thus allows us to write any limitas an integral (an end) without explicitly mentioning the dummy variable(the first variable c of S).A functor H : X → Y is said to preserve the end of a functor S :C op ×C → X when ω : e → .. S an end of S in X implies that Hω : He → .. HSis an and for HS; in symbols∫ ∫H S(c, c) = HS(c, c).cSimilarly, H creates the end of S when to each end v : y .. → HS in Y thereis a unique wedge ω : e .. → S with Hω = v, and this wedge ω is an end of S.The definition of the coend of a functor S : C op ×C → X is dual to thatof an end. A coend of S is a pair 〈d, ζ〉, consisting of an object d ∈ X and awedge ζ : S .. → d. The object d (when it exists, unique up to isomorphism)will usually be written with an integral sign and with the bound variable cas superscript; thus∫ cS(c, c) ζ c→ S(c, c) = d.c