Re: Morphisms of diagrams

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

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.