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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. $\endgroup$ Feb 28, 2019 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, 2019 at 8:41

1 Answer 1

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This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category Theory, and more explicitly as Proposition 11.4 and Theorem 11.5 of Fiore's Enrichment and Representation Theorems for Categories of Domains and Continuous Functions (in the case of small cocompletions, though the result is easily modified to work with a class of colimits instead). In summary, for classes $\Phi, \Psi$ of colimits for which the small category $\mathbf B$ is $\Phi$-cocomplete, there is a conservative $\Psi$-cocompletion $\widehat {\mathbf B}_\Phi$ of $\mathbf B$ preserving the $\Phi$-colimits. This means that the restriction of the (restricted) Yoneda embedding $\mathbf B \to \widehat {\mathbf B}_\Phi$ is $\Phi$-cocontinuous and exhibits a bijection between $\Phi$-cocontinuous functors $\mathbf B \to \mathbf C$ into cocomplete categories $\mathbf C$, and $\Phi$- and $\Psi$-cocontinuous functors $\widehat {\mathbf B}_\Phi \to \mathbf C$. Explicitly, $\widehat {\mathbf B}_\Phi$ is the subcategory of the category of presheaves on $\mathbf B$ which are $\Psi$-colimits of representables taking $\Phi$-cocones to limiting $\Phi$-cones.

In your setting, take $\Phi$ to be the class of colimits in the image of $\iota : \mathbf A \to \mathbf B$, and take $\Psi$ to be the class of finite colimits. Then $\widehat {\mathbf B}_\Phi$ is exactly the finite cocompletion of $\mathbf B$ relative to $\iota$.

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  • $\begingroup$ Thanks this is very useful ! Just to be clear: are you saying the description I sketched in my question is correct ? Also, I'm a bit unclear on whether your construction requires, say, $\iota$ to reflect finite colimits. E.g. I'm not, I think, making the assumption that the image in $B$ of a diagram in $A$ has a colimit in $B$, nor am I assming this colimit is the image of the one computed in $A$. $\endgroup$
    – Adrien
    May 16, 2021 at 12:08
  • $\begingroup$ @Adrien: if I'm not misunderstanding your description, I think it is not correct. For instance, take $\iota$ to be the identity on $B$. Then the finite colimits of representables of $B$ form the cocompletion under finite colimits, which ignores all existing finite colimits in $B$. In answer to your second question, you ought not to need any requirement on $\iota$, since $\iota$ itself is not crucial: the only important data is which colimits are in the image of $\iota$. $\endgroup$
    – varkor
    May 16, 2021 at 12:23
  • $\begingroup$ Yes, but then I suggest to take those that become representable upon restriction along $\iota$, so in the case $A=B$ and $\iota=id$ it will give back $B$ as it should, i think ? For your second point then could you clarify what you mean exactly by "the class of colimits in the image of $\iota$" ? Do you actually mean finite diagrams valued in the image of $\iota$ ? Sorry if I'm being daft. $\endgroup$
    – Adrien
    May 16, 2021 at 16:09
  • $\begingroup$ @Adrien: I think I misunderstood what you meant initially. But in this case, say that $B$ has only some finite colimits, and take $\iota$ to be the identity. Then $B_A = B$, but this doesn't have all finite colimits, so is not correct. By "the class of colimits in the image of $\iota$", I mean the diagrams with colimits in $B$ that arise by postcomposing diagrams with colimits in $A$ by $\iota$. $\endgroup$
    – varkor
    May 16, 2021 at 20:55
  • $\begingroup$ Sure, but $A$ is assumed to be finitely cocomplete, so if $\iota$ is the identity then so is $B$ :) Thanks for clarifying, that makes sense. $\endgroup$
    – Adrien
    May 16, 2021 at 21:10

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