Morphisms of diagrams

Morphisms of diagrams

[categories]

Date: Tue, 18 Mar 1997 10:20:19 -0500
From: Charles Wells <charles@freude.com>

Let C be a category and I and I' graphs (or categories if
you prefer).  Define a morphism of diagrams
psi:(delta:I-->C)-->(delta':I'-->C) to be a graph morphism (or
functor if you prefer) psi:I-->I' together with a natural
transformation alpha:delta' o psi-->delta.  This definition
turns Lim into a contravariant functor from the category of
diagrams to C (when C is complete, anyway).

I believe this construction has been familiar since the early
days of category theory, but I don't know a reference and would
be glad to learn of any.

By the way, Barr in SLN 236 (page 52) defines an entirely
different notion of morphism of diagrams which Tholen and Tozzi
develop extensively in "Completions of Categories and Initial
Completions", Cahiers 1989, pages 127-156.  This makes Lim a
covariant functor.



Charles Wells, 105 South Cedar Street, Oberlin, Ohio 44074, USA.
(I am on sabbatical until 20 August 1997 and cannot easily be reached
at Case Western Reserve University.) EMAIL: cfw2@po.cwru.edu.
HOME PHONE: 216 774 1926.  FAX: Same as home phone.
HOME PAGE: URL http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/home.html

"Some have said that I can't sing.  But no one will say that I _didn't_ sing."
                                                   --Florence Foster Jenkins

Re: Morphisms of diagrams

[categories]

Date: Fri, 21 Mar 97 15:15:56 +1100
From: Max Kelly <kelly_m@maths.su.oz.au>

Charles Wells asked the following:
__________
Let C be a category and I and I' graphs (or categories if
you prefer).  Define a morphism of diagrams
psi:(delta:I-->C)-->(delta':I'-->C) to be a graph morphism (or
functor if you prefer) psi:I-->I' together with a natural
transformation alpha:delta' o psi-->delta.  This definition
turns Lim into a contravariant functor from the category of
diagrams to C (when C is complete, anyway).

I believe this construction has been familiar since the early
days of category theory, but I don't know a reference and would
be glad to learn of any.
______________   

Steve Lack replied with the folowing information:
____________
The dual construction (i.e. for colimits) appears in
	Rene Guitart, ``Remarques sur les machines et les
	structures'', Cahiers XV-2 (1974);
and its sequel
	Rene Guitart and Luc Van den Bril, ``Decompositions
	et lax-completions'', Cahiers XVIII-4 (1977);
where further references are also given.
_____________

I am writing at the university, with my files at home; but my
memory is that the construction was introduced by Eilenberg
and Mac Lane in 1945, in a paper called something like "On a
general theory of natural equivalences".

Max Kelly. 

Re: Morphisms of diagrams

[categories]

Date: Thu, 20 Mar 1997 14:53:31 +1100 (EST)
From: Steve Lack <stevel@maths.su.oz.au>

> Date: Tue, 18 Mar 1997 11:24:35 -0400 (AST)
> From: categories <cat-dist@mta.ca>
> 
> Date: Tue, 18 Mar 1997 10:20:19 -0500
> From: Charles Wells <charles@freude.com>
> 
> Let C be a category and I and I' graphs (or categories if
> you prefer).  Define a morphism of diagrams
> psi:(delta:I-->C)-->(delta':I'-->C) to be a graph morphism (or
> functor if you prefer) psi:I-->I' together with a natural
> transformation alpha:delta' o psi-->delta.  This definition
> turns Lim into a contravariant functor from the category of
> diagrams to C (when C is complete, anyway).
> 
> I believe this construction has been familiar since the early
> days of category theory, but I don't know a reference and would
> be glad to learn of any.

The dual construction (i.e. for colimits) appears in
	Rene Guitart, ``Remarques sur les machines et les
	structures'', Cahiers XV-2 (1974);
and its sequel
	Rene Guitart and Luc Van den Bril, ``Decompositions
	et lax-completions'', Cahiers XVIII-4 (1977);
where further references are also given.

Steve Lack.