categories: weighted limits

[Steve Lack <stevel@maths.usyd.edu.au>]

Mark Hovey writes:
 > What are the standard references for weighted limits and colimits in
 > enriched categories?  I know about Borceux, volume 2, chapter 6, but
 > that does not go far enough. 

Have a look at Chapter 3 of Max Kelly's book ``The basic concepts
of enriched category theory'', LMS lecture note series 64.

 > 
 > More precisely, I want to know how functorial the weighted colimit is in
 > the weight.  Given a V-natural transformation F --> F', presumably I get
 > some kind of map from colim_F G to colim_F' G (or the other way
 > around).  I would like a reference for this fact and related functoriality
 > facts.  

Yes, it is functorial in F, in the way that you have written above.

 > 
 > Presumably the weighted colimit is a bifunctor in the weight and the
 > functor one is taking the colimit of, and presumably this bifunctor has
 > various good properties.  Has anybody ever written these down?

Once again, yes. There's quite a lot in the reference above. Kelly writes
F*G for what you have called colim_F G, and {F,G} for what you would 
presumably call lim_F G. He develops results based on the intuition that
* is a kind of tensor product, and {-,-} a kind of internal hom, and 
proves results like the associativity of * and {F*G,H}={F,{G,H}}. Of course
this sounds a bit odd, because the F and the G live in different categories,
but you can actually make sense of these things. There is even an isomorphism
F*G=G*F in the case of colimits in V itself.

There's another approach, which probably goes back to 

Street and Walters, Yoneda structures on 2-categories, J. Algebra 
50:350-379, 1978

and has been developed by Street and many others since. In this
approach you define a bicategory, often called V-Prof or V-Mod,
in which an object is a V-category, and a morphism from A to B,
often written A-|->B is a V-functor from A to [B^op,V], and a 
2-cell is a natural transformation. (This direction of the 1-cells
and 2-cells in this definition is not universally adopted.)
Then composition is given by colimit, in the sense that if f:A-|->B 
and g:B-|->C, then gf:A-|->C is defined by gf(a,c)=g(-,c)*f(a,-). 
Then associativity (up to isomorphism) of composition in this 
bicategory is the associaitivity of * referred to above. In fact 
this bicategory is closed, in the sense that composition has an adjoint, 
constructed using {-,-}.

Steve Lack.