On the monadicity over graphs of categories with limits
Max Kelly – 5 March 1997
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.