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.
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