Re: Enriched locally presentable categories

Re: Enriched locally presentable categories

[Max Kelly]

Mark Hovey's letter of 26 Dec - known as Boxing Day in the British-speaking
world - suggests that it might be a good idea to develop a theory of local
presentability and all that in the context of enriched categories. In fact
such a theory was developed in my paper [Structures defined by finite limits
in the enriched context I, Cahiers de Top. et Geom. Differentielles 23 (1982),
3 - 42]. Everything works very smoothly; but there are a few annoying
misprints, many of which seem to be my own fault. Further developments can be 
found in [Blackwell-Kelly-Power, Two-dimensional monad theory, J. Pure Appl. 
Algebra 59 (1989), 1 - 41] and in [Kelly-Power, Adjunctions whose counits are 
coequalizers and presentations of finitary enriched monads, J. Pure Appl. 
Algebra 89 (1993), 163 - 179], among other papers of myself and of others;
Brian Day, Steve Lack, John Power, and Ross Street have all written on related
matters.

Please accept, Mark, my best wishes for your future work in this direction. In 
any case, New Year's Eve is a fine time to send greetings more generally to
Bob and all on this Bulletin Board.

Warm regards - Max Kelly.
      

Enriched locally presentable categories

[Mark Hovey]

I am still trying to understand some enriched category theory.  Suppose
V is a closed symmetric monoidal category that is also locally
presentable.  Suppose C is a small V-category.  I am interested in the
category of V-functors from C to V, and, in particular, I want to know
that it is locally presentable.  Might need some hypotheses on C for
this, but I would prefer to avoid hypotheses on the actual functors.

This time I have actually looked in Kelly's book and I did not see it,
but I confess to finding this subject rough going so might have missed
it.  On the other hand, my library is closed for the holiday, so I have
not looked at Adamek and Rosicky's book on enriched category theory yet.

I guess the generators ought to be the representable functors.  I know
everything is a weighted colimit of representables, but I don't know
whether this colimit is filtered enough, nor do I know whether one can
get away with weighted colimits instead of ordinary ones.  

One direction this might go is to develop a theory of locally
presentable in an enriched sense, using weighted colimits instead of
colimits.  I would prefer to avoid that if possible.  

Happy holidays to all.
                 Mark Hovey