# ct.category theory – Relative cocompletion of a category

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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 $$nu:Brightarrow B_A$$, 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.