Re: semi direct product

[F W Lawvere <wlawvere@buffalo.edu>]

Dear colleagues,

The terminology
			"Grothendieck semi-direct product"
is just to reduce terminological ambiguity. For example, if a colloquium
talk is advertised by a title that includes the term "Grothendieck
construction", should we expect that it will involve the process of
passing from a galactic cluster to the associated fibration? No, not
necessarily, because that term is also routinely applied in other ways,
for example to his construction in K-theory.

	That Grothendieck construction within K-theory is also
tautological (the reflection of rigs into commutative rings), but
Grothendieck realized that it could have profound content: unlike the
cases taught in junior high school, the adjunction map can degrade
information, with the useful result that the ring becomes more calculable,
while still determining the rank of a module  at each point of a parameter
space; the successes of his approach led to further dissemination of the
philosophy that for measuring the objects in some category, the rigs that
are appropriate depend on the category.

	The principle that seemingly simple constructions can have
profound content was demonstrated many times by Grothendieck, not only by
the idea of K-theory, but by the fibered category concept of the present
discussion.

	The universal semi-direct product formula, (defining the
composition of pairs of the kind <actor, element acted upon>) was long
known in group theory. That it describes the total category of a galactic
cluster may well have been known to some before 1960, but the realization
and popularization of new geometrical applications justify attaching
Grothendieck's name to this kind of semi-direct product.

	Certain fibered categories, under names like "covariance system",
are a key ingredient giving operator theory a content that transcends the
study of linear operators as such. For example, on a base category of
smooth spaces we can consider for each X the category (with one object)
A(X) of smooth functions under multiplication. Then in the total category
one can recognize the "Canonical Commutation Relations" between q in a
fiber and m in the base (special operators p arise as limits of difference
quotients of families of such m). Despite the continuing restrictive
influence of Klein's "Erlanger Programm", one can note that m need NOT be
invertible; paths, inclusion maps, projections, etc. are typically maps m
in the base that can also operate on the q's. In this sense, the fibered
category is an extension of the group case.

Bill Lawvere


On Wed, 17 Jan 2007, Dr. Keith G. Bowden wrote:

> 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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>