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On the monadicity of categories with chosen colimits

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G. M. Kelly and Stephen Lack

There is a 2-category {\cal J}{\bf-Colim} of small categories equipped
with a choice of colimit for each diagram whose domain $J$ lies in a given
small class {\cal J} of small categories, functors strictly preserving
such colimits, and natural transformations. The evident forgetful
2-functor from {\cal J}{\bf-Colim} to the 2-category {\bf 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 {\cal V}{\bf-Cat} of small {\cal V}-categories to {\cal
V}-categories with object-set in some larger universe. In each case, the
functors preserving the colimits in the usual ``up-to-isomorphism'' sense
are recovered as the {\em pseudomorphisms} between algebras for the
2-monad in question.

Keywords: monadicity, categories with limits, weighted limits, enriched categories.

2000 MSC: 18A35, 18C15, 18D20.

*Theory and Applications of Categories*, Vol. 7, 2000, No. 7, pp 148-170.

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