Grothendieck Construction

Grothendieck Construction

[Joseph Moeller]

I'm looking for a reference where a theorem
of this form was first proven:

> If (C,⊗_C) is (symmetric) monoidal, F: C -> Cat
a lax (symmetric) monoidal functor, and μ the associated
natural transformation, then the Grothendieck category
of F is (symmetric) monoidal with ⊗ defined by
(c,x)⊗(d,y)=(c⊗d,μ_c,d(x,y)).


The proof is a straightforward verification,
so I expect it has been done before.

thanks,
Joe Moeller


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

Re: Grothendieck construction

[David Espinosa]

> "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

Grothendieck construction

[I. Moerdijk]

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.

Re: Grothendieck construction

[Artur Zawlocki]

> 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.
>
>
>

Re: Grothendieck construction

[wlawvere]

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.
> 

Grothendieck construction

[Gaucher Philippe]

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.