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Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and a $\lambda$-small 1-category $C$; we have a bicategory of monads $Mnd(C)$. Is there for all $\kappa$ a pair $\lambda$ and a $\lambda$-small 1-category $C$, such that $\kappa$-Cat $=$ $Mnd(C)$?

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  • $\begingroup$ Wait. How do you get a "bicategory of monads" on a 1-category $C$ ? I now how to get a "bicategory of monads in a bicategory" or a "1-category of monads on a 1-category $C$ " but I don't see what category you are talking about here. $\endgroup$
    – Simon Henry
    Commented Jan 5, 2023 at 2:56
  • $\begingroup$ Taking 2-cells as natural transformations between underlying morphisms of monad morphisms, that satisfy; commutativity of a diagram as defined here github.com/agda/agda-categories/blob/master/src/Categories/…, I think this should be the construction in ncatlab.org/nlab/show/monad#the_bicategory_of_monads, I am trying to arrange it into bicategory in agda right now, to check that my understanding is correct. $\endgroup$
    – Ilk
    Commented Jan 5, 2023 at 15:17
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    $\begingroup$ A monad morphism for two monads on a fixed category $C$ is already a natural transformation, so there no natural transformation between morphisms. I don't fully understand the Agda code you sent me, but it seems to refers to the notion of morphism between a monad on $C$ and a monad on another category $D$. In this case there is indeed a notion of 2-cell, but then you are not talking about the category of monads on $C$, you are talking about the bicategory of monads in the bicategory $Cat$ of categories. $\endgroup$
    – Simon Henry
    Commented Jan 5, 2023 at 15:24
  • $\begingroup$ I wrote the code. It indeed refers to the notion of monad map that allows for 2-cells, as in Street "formal theory". $\endgroup$
    – fosco
    Commented Jan 6, 2023 at 16:57
  • $\begingroup$ @fosco do do you mean 2-cells as in 2-cells, turning 1-category of monads on C into a 2 category? or 2-cells in the bicategory of monads in Cat sense? $\endgroup$
    – Ilk
    Commented Jan 7, 2023 at 16:30

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