Examples
Suppose we have functors then . In this case, the category of sets is complete, so we need only form the equalizer and in this case
the natural transformations from to . Intuitively, a natural transformation from to is a morphism from to for every in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
Let be a simplicial set. That is, is a functor . The Discrete topology gives a functor, where is the category of topological spaces. Moreover, there is a map which sends the object of to the standard simplex inside . Finally there is a functor which takes the product of two topological spaces. Define to be the composition of this product functor with . The coend of is the geometric realization of .
Read more about this topic: End (category Theory)
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