Smooth and proper functors

Smooth and proper functors

[Hasse Riemann]

 

Hi category gurus and categorists

 

I have many questions about category theory but i start with one.

 

1>

What are smooth functors and proper functors, originating in pursuing stacks?

Both nontechnically and technicaly.

 

I know they are dual to each other and that they are characterized by cohomological properties

inspired by the proper or smooth base change theorem in algebraic geometry, but what is the relation?

(I don't know the statement of the theorems)

 

Finally, what are smooth and proper functors good for?

Are smooth and proper functors fibrations and cofibrations or Grothendieck fibrations and

Grothendieck op-fibrations in some model categories or derivators?

 

The only thing i could find about smooth and proper functors on internet is the last entrance in
http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_18.html

 

Best regards

Rafael Borowiecki

Re: Smooth and proper functors

[Jonathan CHICHE 齐正航]

Hi,

The following paper is very clear, I'm currently learning the basics  
of the subject with it: http://people.math.jussieu.fr/~maltsin/ps/ 
asphbl.ps. It's written in French. Another member of this mailing- 
list has asked me to translate it in English, I may be able to send  
you a rough translation in a few weeks.

Best,

Jonathan

Le 15 avr. 09 à 15:45, Hasse Riemann a écrit :

> Hi category gurus and categorists
>
>
>
> I have many questions about category theory but i start with one.
>
>
>
> 1>
>
> What are smooth functors and proper functors, originating in  
> pursuing stacks?
>
> Both nontechnically and technicaly.
>
>
>
> I know they are dual to each other and that they are characterized  
> by cohomological properties
>
> inspired by the proper or smooth base change theorem in algebraic  
> geometry, but what is the relation?
>
> (I don't know the statement of the theorems)
>
>
>
> Finally, what are smooth and proper functors good for?
>
> Are smooth and proper functors fibrations and cofibrations or  
> Grothendieck fibrations and
>
> Grothendieck op-fibrations in some model categories or derivators?
>
>
>
> The only thing i could find about smooth and proper functors on  
> internet is the last entrance in
> http://golem.ph.utexas.edu/category/2008/01/ 
> geometric_representation_theor_18.html
>
>
>
> Best regards
>
> Rafael Borowiecki

Re: Smooth and proper functors

[Andreas Holmstrom]

Hi Rafael,

I don't know much about this, but I listened to an excellent talk of
Maltsiniotis a few months ago at IHES and posted the scanned notes in
a blog post here:

http://homotopical.wordpress.com/2009/01/26/maltsinotis-grothendieck-and-homotopical-algebra/

These notes (on page 11-12) contain at least the definition of proper
and smooth functors, and the duality statement, so maybe they can be
of some limited use. Hopefully other people on this list can provide
some more substantial information.

Best regards,
Andreas Holmstrom



2009/4/15 Hasse Riemann <rafaelb77@hotmail.com>:
>
>
>
> Hi category gurus and categorists
>
>
>
> I have many questions about category theory but i start with one.
>
>
>
> 1>
>
> What are smooth functors and proper functors, originating in pursuing stacks?
>
> Both nontechnically and technicaly.
>
>
>
> I know they are dual to each other and that they are characterized by cohomological properties
>
> inspired by the proper or smooth base change theorem in algebraic geometry, but what is the relation?
>
> (I don't know the statement of the theorems)
>
>
>
> Finally, what are smooth and proper functors good for?
>
> Are smooth and proper functors fibrations and cofibrations or Grothendieck fibrations and
>
> Grothendieck op-fibrations in some model categories or derivators?
>
>
>
> The only thing i could find about smooth and proper functors on internet is the last entrance in
> http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_18.html
>
>
>
> Best regards
>
> Rafael Borowiecki
>
>
>