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$\newcommand{\k}{\mathbf k}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\B}{\mathcal B}$ $\newcommand{\C}{\mathcal C}$ I'm wondering if anyone knows a reference for the following construction: let $\k$ be a field, say, and assume for convenience everything below is $\k$-linear, and that every category is essentially small. Hereafter "right exact" means "which commutes with finite colimits".

Let $\A$ be a finitely cocomplete category, and $\B$ be an arbitrary (still $\k$-linear and essentially small) category. Suppose we are given a functor $$ \iota :\A \longrightarrow \B$$ which I'm happy to assume essentially surjective if that helps. Define the relative $\A$-cocompletion of $\B$ to be a category $\B_\A$ equipped with a functor $\nu:\B\rightarrow \B_\A$, universal for the following properties:

  1. $\B_\A$ is finitely cocomplete
  2. for any finitely cocomplete $\C$, restriction along $\nu$ induces a natural equivalence between: (a) right exact functors $\B_\A \rightarrow \C$ and (b) just functors $\B \rightarrow \C $ such that the composition $\A \rightarrow \B \rightarrow \C$ is right exact.

In words, I want to complete $\B$ under finite colimit, without duplicating those already existing in the image of $\A$. If $\A$ is $Vect$ this should just be the free finite cocompletion.

Remember that the finite cocompletion of $\B$ is the full-subcategory of (linear) presheaves on $\B$ which are finite colimits of representables. I believe $\B_\A$ is then the full subcategory of that, of those presheaves having the property that their restriction along $\iota$ is representable. An example which is an inspiration for this definition is the construction of the category of modules over a monad, from the category of free modules, see e.g. https://ncatlab.org/nlab/show/Eilenberg-Moore+category#AsColimitCompletionOfKleisliCategory.

I'm in the process of checking that myself, and it's probably just formal, but I would much rather have a reference where it's done properly.

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    $\begingroup$ I don't have a reference for this particular construction, but I think it is an instance of a general notion: a (2-categorical) semifinal lift of the unary sink $U({\cal A}) \to \cal B$, where $U$ is the forgetful functor from finitely-cocomplete categories to all categories. $\endgroup$ – Mike Shulman Feb 28 at 0:44
  • $\begingroup$ Indeed, thanks, I wasn't aware of that notion. I should add that though I'd like to have an explicit construction, what I really care about apart from the mere exists of that thing is whether $\nu$ is fully faithful. $\endgroup$ – Adrien Feb 28 at 8:41

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