## For which classes L of limits is the forgetful functor from the category of L-complete categories and L-continuous functors to the category of graphs and graph-morphisms a monadic functor?

### Max Kelly – 2 October 1996

m-limits, for each m in M, and whose morphisms are the functors preserving these limits on the nose. Writing Gph for the category of (small) graphs, we have a forgetful functor U from Cat_M to Gph sending a category with M-limits to its underlying graph. For certain sets M it is known that this functor U is monadic --- but the proofs use a different ``trick'' for each such M. We show that: U need not even be of descent type; it is so if each category m in M is freely generated by some graph; even then U need not be monadic; U is indeed monadic if each such graph is loop-free.

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