Geodesic
On the monadicity of categories with chosen colimits
Theory and applications of categories, Tome 7 (2000), pp. 148-170.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

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.
Classification : 18A35, 18C15, 18D20.
Keywords: monadicity, categories with limits, weighted limits, enriched categories. 
@article{TAC_2000_7_a6,
     author = {G. M. Kelly and Stephen Lack},
     title = {On the monadicity of categories with chosen colimits},
     journal = {Theory and applications of categories},
     pages = {148--170},
     publisher = {mathdoc},
     volume = {7},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2000_7_a6/}
}
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G. M. Kelly; Stephen Lack. On the monadicity of categories with chosen colimits. Theory and applications of categories, Tome 7 (2000), pp. 148-170. http://geodesic.mathdoc.fr/item/TAC_2000_7_a6/