## Abstract

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.

Original language | English |
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Pages (from-to) | 148-170 |

Number of pages | 23 |

Journal | Theory and Applications of Categories |

Volume | 7 |

Publication status | Published - 2000 |

## Keywords

- Categories with limits
- Enriched categories
- Monadicity
- Weighted limits