On the Monadicity of Categories with Chosen Colimits

There is a 2-category J -Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from J -Colim to the 2-category Cat of small categories is known to be monadic. We extend this result by considering not just conical colimits, but general weighted colimits; not just ordinary categories but enriched ones; and not just small classes of colimits but large ones; in this last case we are forced to move from the 2-category V-Cat of small V-categories to V-categories with object-set in some larger universe. In each case, the functors preserving the colimits in the usual “upto-isomorphism” sense are recovered as the pseudomorphisms between algebras for the 2-monad in question.