In category theory, an end of a functor S : C o p × C → X is a universal extranatural transformation from an object e of X to S.
More explicitly, this is a pair ( e , ω ) , where e is an object of X and
ω : e → ¨ S is an extranatural transformation such that for every extranatural transformation
β : x → ¨ S there exists a unique morphism
h : x → e of X with
β a = ω a ∘ h for every object a of C.
By abuse of language the object e is often called the end of the functor S (forgetting ω ) and is written
e = ∫ c S ( c , c ) or just ∫ C S . Characterization as limit: If X is complete, the end can be described as the equaliser in the diagram
∫ c S ( c , c ) → ∏ c ∈ C S ( c , c ) ⇉ ∏ c → c ′ S ( c , c ′ ) , where the first morphism is induced by S ( c , c ) → S ( c , c ′ ) and the second morphism is induced by S ( c ′ , c ′ ) → S ( c , c ′ ) .
The definition of the coend of a functor S : C o p × C → X is the dual of the definition of an end.
Thus, a coend of S consists of a pair ( d , ζ ) , where d is an object of X and
ζ : S → ¨ d is an extranatural transformation, such that for every extranatural transformation
γ : S → ¨ x there exists a unique morphism
g : d → x of X with
γ a = g ∘ ζ a for every object a of C.
The coend d of the functor S is written
d = ∫ c S ( c , c ) or ∫ C S . Characterization as colimit: Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram
∫ c S ( c , c ) ← ∐ c ∈ C S ( c , c ) ⇇ ∐ c → c ′ S ( c ′ , c ) . Natural transformations:Suppose we have functors F , G : C → X then
H o m X ( F ( − ) , G ( − ) ) : C o p × C → S e t .
In this case, the category of sets is complete, so we need only form the equalizer and in this case
∫ c H o m X ( F ( c ) , G ( c ) ) = N a t ( F , G ) the natural transformations from F to G . Intuitively, a natural transformation from F to G is a morphism from F ( c ) to G ( c ) for every c in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
Geometric realizations:Let T be a simplicial set. That is, T is a functor Δ o p → S e t . The discrete topology gives a functor S e t → T o p , where T o p is the category of topological spaces. Moreover, there is a map γ : Δ → T o p which sends the object [ n ] of Δ to the standard n simplex inside R n + 1 . Finally there is a functor T o p × T o p → T o p which takes the product of two topological spaces.
Define S to be the composition of this product functor with T × γ . The coend of S is the geometric realization of T .
