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On The Monadicity Of Categories With Chosen Colimits
 THEORY APPL. CATEG
, 2000
"... There is a 2category JColim 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 2functor from JColim to the 2ca ..."
Abstract

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There is a 2category JColim 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 2functor from JColim to the 2category 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 2category VCat of small Vcategories to Vcategories with objectset in some larger universe. In each case, the functors preserving the colimits in the usual "uptoisomorphism" sense are recovered as the pseudomorphisms between algebras for the 2monad in question.