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Alexander Gietelink Oldenziel (Nov 19 2020 at 09:35):A monoidal structure on the category of presentable categories and limit preserving functors
There is a canonical monoidal closed structure on the category of presentable categories and cocontinuous (=colimit preserving) functors.
I am wondering whether there is a monoidal structure on the category of presentable categories and continuous functors.
These categories are almost dual, except for the following snag: a colimit preserving functor between presentable categories is automatically has a right adjoint, but a limit preserving functor needs to be accesible to have a left adjoint.
GIven a monoidal structure on a category , there is a tautological monoidal structure on its opposite , so the subcategory of presentable categories and continuous accesible fucntor does have a monoidal structure.
Nathanael Arkor (Nov 19 2020 at 11:04):Is there a particular reason you're interested in continuous functors, without the accessibility requirement? It doesn't seem like such a natural thing to consider.