Kan Extensions are Partial Colimits

@article{Perrone2022KanEA,
  title={Kan Extensions are Partial Colimits},
  author={Paolo Perrone and Walter Tholen},
  journal={Applied Categorical Structures},
  year={2022}
}
One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues… 
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