Monadicity questions

[David Yetter <dyetter@ksu.edu>]

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We are all, of course, familiar with Beck's Theorem.  I'm rather hoping that there are results in the literature that will save me from having to prove that the underlying functor U creates coequalizers for U-split pairs in the context of two quite different projects I'm working on.  Thus, I have two questions for the community:


  1.
It seems obvious to me that for the same reason categories of models of finitary algebraic (equational) theories are monadic over Set, if one has (I think I'm using the term correctly here) a conservative extension of an equational theory (by which I mean, add operations and equations in such a way that no new equations are imposed on the operations of the original theory), then the category of models of the extension is monadic over the category of models of the original theory.  Surely this is either explicitly stated and proved somewhere, or follows easily from some result I simply have not encountered.   Citations?

  2.   What is the most general sort of theory whose models are monadic over Set? Or if that is not known, what sorts of theories have monadic categories of models over Set?   Are there multisorted generalizations of any results of that sort, ideally not just to Set^\alpha, where \alpha is the cardinality of a set of sorts, but to things like Graph?  Again some citations would be much appreciated.

Best Thoughts,
David Yetter
University Distinguished Professor
Department of Mathematics
Kansas State University

(That is the first and last time I'll use that signature block in writing to the list, but I thought I'd do it once since I thought the community would be gratified that a categorist was so honored.  After this it's back to David Y. or D.Y.)



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