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.
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.
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.
Dear All,
Where does the Grothendieck construction come from? What is the original reference? Here is the construction.
A standard reference is (after Wikipedia, http://en.wikipedia.org/wiki/Grothendieck's_S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique): Grothendieck, Alexandre, Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Lecture notes in mathematics 224) (in French). Berlin; New York: Springer-Verlag, xxii+447. ISBN 3540056149. An updated version has been put in the arxiv: http://www.arxiv.org/abs/math.AG/0206203 The construction itself is defined in Section 8, as far as I remember. Artur
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.
Dear All,
Where does the Grothendieck construction come from? What is the origina= l reference? Here is the construction.
A standard reference is (after Wikipedia, http://en.wikipedia.org/wiki/Grothendieck's_S%C3%A9minaire_de_g%C3%A9om%C= 3%A9trie_alg%C3%A9brique): Grothendieck, Alexandre, S=E9minaire de G=E9om=E9trie Alg=E9brique du Boi= s Marie - 1960-61 - Revêtements =E9tales et groupe fondamental - (SGA 1) (Lect= ure notes in mathematics 224) (in French). Berlin; New York: Springer-Verlag, xxii+447. ISBN 3540056149. An updated version has been put in the arxiv: http://www.arxiv.org/abs/math.AG/0206203 The construction itself is defined in Section 8, as far as I remember. Artur
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.
Perhaps I should add that Saunders Mac Lane was always a bit unhappy with this terminology, and has told me repeatedly that "he knew it long before Grothendieck...". Ieke Moerdijk.
"he knew it long before Grothendieck..."
So maybe the construction itself is obvious, particularly if you know the semi-direct product or some other specialization (of the general construction). But the intrinic characterization of what the construction yields, that is, the definition of a fibration, seems less obvious. I'm sure everyone has a favorite example of that. For example, Carsten Fuhrmann gave an intrinsic description of the Kleisli category of a monad only in 1999. His home page is: http://www.cs.bath.ac.uk/~cf/ David
participants (5)
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Artur Zawlocki -
David Espinosa -
Gaucher Philippe -
I. Moerdijk -
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