See Section 1 on Functor Categories of Chapter Two of the Springer book by Gabriel-Zisman 1967 ``Calculus of Fractions and Homotopy Theory''. That is where I first learnt it. In other terminology, it is about the Yoneda embedding being dense. ==Ross
On 3 Nov 2017, at 10:51 PM, Uwe Egbert Wolter <Uwe.Wolter@uib.no> wrote:
Dear all,
I'm sure the following observation has been made before. Hopefully, somebody can provide a reference.
Many thanks in advance
Uwe Wolter
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We consider a small category B and an element M:B --> Set of the functor category Set^B. By G(B,M) we denote the category obtained by the Grothendieck construction with objects (x,b) where b is an object in B and x an element in the set M(b). G(M):G(B,M) --> B is the corresponding split discrete obfibration. Composing G(M)^op with the Yoneda embedding Y:B^op --> Set^B gives us a diagram in Set^B with shape G(B,M)^op.
The observation is that M is the colimit of this diagram where for each (x,b) in G(B,M) the projection p(x,b):B(b,_) ==> M is the natural transformation determined, according to the Yoneda lemma, by the element x in the set b.
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