# Relative cocompletion of a category

$$\newcommand{\k}{\mathbf k}$$ $$\newcommand{\A}{\mathcal A}$$ $$\newcommand{\B}{\mathcal B}$$ $$\newcommand{\C}{\mathcal C}$$ 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.

• 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. – Mike Shulman Feb 28 at 0:44
• 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. – Adrien Feb 28 at 8:41