Re: Lax Indexed Functors?

[categories <cat-dist@mta.ca>]

Date: Fri, 31 Jan 1997 14:48:30 GMT
From: Sandro Fusco <sfusco@mathstat.yorku.ca>

On Jan 29,  4:05pm, categories wrote:
> Subject: Lax Indexed Functors?
> Date: Wed, 29 Jan 1997 18:58:44 +0100 (MET)
> From: Alfio Martini <alfio@cs.tu-berlin.de>
>
> To give  an impression about what we are doing,
> I will give the definition that turns out to be the
> adequate one for our purposes:
>
> A lax indexed functor F from an indexed category C:IND->CAT  to an
> indexed category D:IND->CAT is given by  functors F(i):C(i)->D(i) for
> each i in |IND| and by natural transformations  F(g):C(g);F(j)=>F(i);D(g)
> for each morphism g:i->j in IND such that the following compositionality
> condition is satisfied for any g:i->j and h:j->k in IND:
>
>    F(g;h) = (C(g);F(h))*(F(g);D(h)).            --------(1)
>
>
> To get the right feeling and insight we have developed all necessary
> results by ourselves. Especially we were interested in the generalization
> of the Grothendieck construction to "lax indexed functors".
>
> Thanks for any help.
>
> With all best wishes,
>
> Alfio Martini.
>
>
>-- End of excerpt from categories



Greetings!

Concerning Alfio Martini's message, I would like to point out that I have
been using a similar notion of what I also called "lax indexed functors".

The only difference is that instead of condition (1) I have the following:

 (\Phi_g,h ; F(k)) * F(g;h) = (C(g);F(h)) * (F(g);D(h)) * (F(i) ; \Phi'_g,h)

where \Phi_g,h:C(g);C(h)->C(g;h) is the natural isomorphism associated
with the indexed category C and \Phi'_g,h is the natural isomorphism
associated with D (the reason being that my indexed categories
C, D: IND->CAT are basically pseudofunctors).

As shown in my thesis abstract below, a generalized Grothendieck
construction is established.

I expect to have copies of my thesis available by the end of April 1997.


Yours truly,

Sandro Fusco


-------------

Doctoral Thesis

Title:
Stable Functors and the Grothendieck Construction.

Abstract:
In classical domain theory, Scott-continuous functions of partially ordered
sets (posets) are used to model approximation processes. When replacing
posets by (the more general) categories, the so called "stable functors" take
on the role of Scott-continuous functions. Different notions of stable functors
were studied intensively by various researchers during the past ten years
(notably by Paul Taylor of Imperial College, London, and Walter Tholen of
York University, Toronto). These notions are closely related to generalizations
of two fundamental notions of category theory, adjoint functors, and
fibrations, with no apparent link between the two approaches.
In this thesis, we
1. introduce and investigate a satisfactory notion and theory of stable
 functors based on "factorizations relative to a functor" (as given by
 both, generalized adjoints and fibrations);
2. establish the Grothendieck construction as a 2-functor \Gamma whose
 domain is the 2-category of \{cal X}-indexed categories and whose target
 is the suitably defined 2-category of stable functors with codomain \{cal X},
 in such a way that \Gamma is part of a higher-dimensional adjunction.
This latter part can be exploited at various levels of generality, yielding in
particular the well-known equivalence between \{cal X}-indexed categories and
cloven fibrations.



-- 

Sandro Fusco  <sfusco@mathstat.yorku.ca>
Dept. of Mathematics and Statistics
York University
North York, Ontario
Canada  M3J 1P3
 
Tel:  (416) 736-2100  Ext. 40617
Fax: (416) 736-5757