Creation of weighted limits

Creation of weighted limits

[Steve Vickers]

It is well known that if a right adjoint with codomain Set is monadic
then it creates limits.

Is there an analogous result for a right adjoint with codomain Cat,
sufficing for it to create weighted limits?

Steve.


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Re: Creation of weighted limits

[Steve Lack]

Dear Steve,

For any monad T and any category C, the forgetful functor U^T:C^T->C 
from the category of algebras creates limits, both weighted and conical
(i.e. unweighted). In particular you could take C=Set.

This remains true if T is a V-enriched monad and C a V-enriched category,
In particular you could take C=V=Cat.

The special thing about the case of Cat is that you might want to replace 
the 2-category of algebras C^T by a variant involving morphisms which 
do not preserve algebra structure strictly, but only in some weaker sense.
In this case, the forgetful functor will create only some limits; exactly which
ones depend on exactly how you vary C^T. 

For the case where morphisms preserve algebra structure up to isomorphism, see
the papers,

Blackwell-Kelly-Power, 2-dimensional monad theory
Power-Robinson, PIE-limits

For the case where morphisms preserve algebra only up to a coherent comparison
map, not necessarily invertible, see 

Lack, Limits for lax morphisms
Lack-Shulman, Enhanced 2-categories and limits for lax morphisms.

The latter paper, which was also the basis for my CT2008 talk at Calais, proposes
moving beyond the framework of 2-categories, to "enhanced 2-categories", in which
you keep track both of the strict and the non-strict morphisms. This allows a much 
finer treatment of which limits lift in the lax case. (The theory there includes the 
pseudo case of preservation up to isomorphism, but the difference is much less
stark there.) 

Best wishes,

Steve Lack.


On 18/11/2011, at 9:31 PM, Steve Vickers wrote:

> It is well known that if a right adjoint with codomain Set is monadic
> then it creates limits.
> 
> Is there an analogous result for a right adjoint with codomain Cat,
> sufficing for it to create weighted limits?
> 
> Steve.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]