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
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.
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.
participants (3)
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Charles Wells -
Max Kelly -
Steve Lack