Dear Keith, This reply to the reply is actually more ambiguous... What do you mean with "extension"? Anyway, I find it interesting, in the groupoids case, the paper: "Categorical non abelian cohomology, and the Schreier theory of groupoids", V. Blanco, M. Bullejos, E. Faro, on the arXiv as math.CT/0410202. Best regards, Beppe Metere. Quoting "Dr. Keith G. Bowden" <k.bowden@physics.bbk.ac.uk>:
Dear Bill,
Your reply is slightly ambiguous.
Do you mean that you call it the semidirect product by extension of the semi-direct product in group theory?
Regards, Keith Bowden
----- Original Message ----- From: <wlawvere@buffalo.edu> To: <categories@mta.ca> Sent: Wednesday, January 17, 2007 1:23 AM
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.
Quoting "Dr. Keith G. Bowden" <k.bowden@physics.bbk.ac.uk>:
Dear Bill,
Your reply is slightly ambiguous.
Do you mean that you call it the semidirect product by extension of the semi-direct product in group theory?
Regards, Keith Bowden
----- Original Message ----- From: <wlawvere@buffalo.edu> To: <categories@mta.ca> Sent: Wednesday, January 17, 2007 1:23 AM
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.
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