I know that similar questions have been asked in the past and, even if some useful explanations/clarifications have been given (so now I know or, at least, I believe I know what results I should expect to be true) here I am asking mostly for a single reference (or series of references) that can allow me to use the following $2$-categorical results without having to prove them in my paper (in what follows I will use $2$-category/$2$-functor/$2$-colimit/etc. for the non-strict notions, that is, what was classically the corresponding "bi" notion. I will specify "strict" when needed):
- First of all it would be nice to have a proof for the various possible definitions of a $2$-adjoint pair between strict $2$-categories (I imagine that, as for the $1$-categorical notion, one should be able to introduce them through an equivalence of suitable $Hom$-categories, or via the $2$-commutativity of suitable diagrams involving the co/units, etc.);
- Show that, given a $2$-adjoint pair between strict $2$-categories, the left/right adjoint preserves $2$-co/limits. In particular, I would need to show that $2$-pushouts are preserved by the following notions of "completion" (that come as left adjoints in a $2$-adjoint pair), that I am listing here mostly as a motivation and to give some context:
- the "additivization of (small) preadditive categories" which is the $2$-left adjoint to the inclusion of the strict $2$-category of small additive categories, that is, small pointed categories with all binary biproducts, (additive functors and natural transformations) into the strict $2$-category of small pre-additive categories, that is, small $\mathrm{Ab}$-categories, (additive functors and natural transformations);
- the "idempotent completion of (small) additive categories" which is just the $2$-left adjoint to the inclusion of the strict $2$-category of small idempotent complete additive categories (additive functors and natural transformations) into the strict $2$-category of small additive categories;
- in particular, the composition of the above two adjunctions gives the Cauchy completion of a small pre-additive category (e.g., if we start with a ring $R$, we get $\mathrm{proj}(R)$, the category of finitely generated projectives).
Furthermore, I would also need to consider the following notions of completion, of which I am a bit more unsure (for size issues, but I do not actually think they should constitute any serious problem):
- "the coproduct completion of additive categories", the $2$-left adjoint to the inclusion of the strict $2$-category of coproduct complete additive categories (coproduct preserving functors, and natural transformations) into the strict $2$-category of additive categories;
- "the $\mathrm{Ind}$-completion of additive categories" which is the $2$-left adjoint to the inclusion of the strict $2$-category of $\mathrm{Ind}$-complete additive categories ($\mathrm{Ind}$-cocontinuous functors and natural transformations) into the strict $2$-category of additive categories.
(Continuing with the previous examples, the above notions of completions, possibly together with an additional idempotent completion, are meant to extend $\mathrm{proj}(R)$ first to $\mathrm{Proj}(R)$, the category of all projectives, and then to $\mathrm{Flat}(R)$, the category of flats).
