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