Grothendieck, Yoneda, Colimits

Grothendieck, Yoneda, Colimits

[Uwe Egbert Wolter]

Dear all,

I'm sure the following observation has been made before. Hopefully,
somebody can provide a reference.

Many thanks in advance

Uwe Wolter

***************************************************

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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Re: Grothendieck, Yoneda, Colimits

[Ross Street]

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
> 
> ***************************************************
> 
> 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.
> 
> 
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



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