Subject: Lax Indexed Functors?

From: categories <[email-masked]>
Date: Wed, 29 Jan 1997 16:05:38 -0400 (AST)

Date: Wed, 29 Jan 1997 18:58:44 +0100 (MET)
From: Alfio Martini <[email-masked]>

At the moment we are investigating logical systems and their relations based
on the formal concepts of institution (Goguen/Burstall) and entailment
system/logic (Meseguer). For our analysis we found  appropriate to use
concepts like "lax indexed functors" thereby having in mind the
corresponding definitions for ordered categories in 
"Extending properties to categories of partial maps" from
Barry Jay [Jay90] (TR ECS-LFCS-90-107).

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):C(i)->D(j)
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)).

(We also need the other version where F(g) goes from F(i);D(g) to C(g);F(j).)

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".

Now, before fixing these things in a technical report, were are looking for
corresponding references of work already done in this direction. Especially we
don't want to introduce new names for already known concepts. We need
some advice here...

Our observation is that we have essentially used for many concepts and
results the 2-categorical structure of CAT. Thus we strongly believe that
somebody has already defined and investigated "lax functors" and "lax
natural transformations" for 2-categories.

Thanks for any help.

With all best wishes,

Alfio Martini.

---

From: categories <[email-masked]>
Date: Fri, 31 Jan 1997 14:14:32 -0400 (AST)

Date: Fri, 31 Jan 1997 14:48:30 GMT
From: Sandro Fusco <[email-masked]>

On Jan 29,  4:05pm, categories wrote:
> Subject: Lax Indexed Functors?
> 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)
>    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  <[email-masked]>
Dept. of Mathematics and Statistics
York University
North York, Ontario
Canada  M3J 1P3
 
Tel:  (416) 736-2100  Ext. 40617
Fax: (416) 736-5757