Re: Grothendieck construction

[wlawvere@buffalo.edu]

Because Grothendieck made many constructions that 
became iconic, the terminology is ambiguous. 
I call this construction 
      "the Grothendieck semi-direct product"
because the formula for composition of these 
morphisms is exactly the same as in the very special 
case where I is a group.
Of course the result of the construction is a single 
category "fibered" over I and every fibred category 
so arises.
The original example for me (1959) was that from 
Cartan-Eilenberg where I is a category of rings and
H(i) is the category of modules over i.  Because 
J. L. Kelley had proposed "galactic" as the analogue
at the Cat level of the traditional "local" at the level 
of a space, I called such an H a "galactic cluster" .
The "fibration' terminology  and the accompanying
results and definitions for descent etc were presented 
by AG in Paris seminars in the very early 1960's and 
can probably be accessed elecronically now.

Best wishes
Bill

Quoting Gaucher Philippe <Philippe.Gaucher@pps.jussieu.fr>:

> Dear All,
> 
> Where does the Grothendieck construction come from? What is the
> original
> reference? Here is the construction.
> 
> Take a functor H:I-->Cat (the category of small categories)
> 
> The objects are the pairs (i,a) where a is an object of H(i).
> A morphism (i,a)-->(j,b) consists of a morphism f:i-->j of I and a
> morphism
> H(f)(a)-->b of H(j).
> 
> pg.
>