* Grothendieck, Yoneda, Colimits
@ 2017-11-03 11:51 Uwe Egbert Wolter
2017-11-04 2:42 ` Ross Street
0 siblings, 1 reply; 2+ messages in thread
From: Uwe Egbert Wolter @ 2017-11-03 11:51 UTC (permalink / raw)
To: Categories list
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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* Re: Grothendieck, Yoneda, Colimits
2017-11-03 11:51 Grothendieck, Yoneda, Colimits Uwe Egbert Wolter
@ 2017-11-04 2:42 ` Ross Street
0 siblings, 0 replies; 2+ messages in thread
From: Ross Street @ 2017-11-04 2:42 UTC (permalink / raw)
To: Uwe Egbert Wolter; +Cc: categories@mta.ca list
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/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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