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There is some discussion on the nLab on seeing the free cocompletion $\mathbf{Psh}(\mathbf{A}) = [\mathbf{A}^{op}, \mathbf{Sets}]$, as a pseudomonad. The Yoneda embedding $よ \colon \mathbf{A} \to \mathbf{Psh}(\mathbf{A})$ gives the unit and that the multiplication would be the unique functor $\mu \colon \mathbf{Psh}(\mathbf{Psh}(\mathbf{A})) \to \mathbf{Psh}(\mathbf{A})$ such that $\mu \circ よ = \mathrm{id}$. However, this creates some size issues.

On the other hand, composition of profunctors $p \colon \mathbf{B} \to \mathbf{Psh}(\mathbf{A})$ and $q \colon \mathbf{C} \to \mathbf{Psh}(\mathbf{B})$ can be defined (following Remark 5.6 in This is the (co)end...) as $\mathsf{Lan}_よ(p) \circ q$, which looks like composition in a Kleisli category for the (pseudo)monad.

Can we get the bicategory of profunctors as a Kleisli construction over the free-completion pseudomonad? Can we deal somehow with the size issues? Is there any reference where this construction is discussed?

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  • $\begingroup$ out of curiosity, what goes wrong if you fix a cardinal kappa and try to execute this plan for kappa-accessible categories, kappa-accessible presheaves, etc.? $\endgroup$
    – pupshaw
    Commented Jun 26, 2019 at 23:10

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This beautiful story take place in $\mathsf{Cat}$, the bicategory of locally small categories. There, the construction of small presheaves induces a pseudomonad whose pseudoalgebrabas are cocomplete categories and whose Kleisli category is $\mathsf{Prof}$. A part of this story is folklore and is partially discussed in Lex Colimits, by Garner and Lack, Prop 2.2.

Another part of this story is explained in Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures, with the relative approach, that tends to complicate a bit the things but shows a very nice direction.

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  • $\begingroup$ Prof generally refers to the bicategory whose objects are small categories and whose morphisms are profunctors, so it is only a full sub-bicategory of the Kleisli bicategory for the "small presheaves" pseudomonad, which has all locally small categories as objects and "small profunctors" as morphisms. The "relative" version is a good answer to the question on its own, but I don't think it uses the small-presheaf technology. $\endgroup$
    – Mike Shulman
    Commented Jun 27, 2019 at 8:03
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A nice account of this is given in Martin Hyland's Elements of a theory of algebraic theories.

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