On the monadicity of categories with chosen colimits

G. M. Kelly, Stephen Lack

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    14 Citations (Scopus)


    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 languageEnglish
    Pages (from-to)148-170
    Number of pages23
    JournalTheory and Applications of Categories
    Publication statusPublished - 2000


    • Categories with limits
    • Enriched categories
    • Monadicity
    • Weighted limits


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