## The fformal theory of monads

### Richard Garner – 2 May 2012

We explain an observation of Mike Shulman's; it shows how one can obtain the equipment Cat --> Prof as a free cocompletion in the world of bicategories enriched over a particular monoidal bicategory F. This F has as objects, arrows A -->
C from a discrete category to a cocomplete one, and as arrows, pseudocommutative squares. An F-bicategory is a category A with a pseudofunctor A --> C into a locally cocomplete bicategory (and
composition preserving colimits in each variable). The free cocompletion yielding Cat --> Prof is that of 1 --> Sigma(Set) with respect to tight coproducts and tight Kleisli objects. Various generalisations are discussed.

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