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I would like to know if there is some uniform construction out of a given category $\mathcal C$ that freely throws in all quotients,to form a new category $\mathcal C'$. Preferably $\mathcal C'$ has all (small?) quotients, but if it only contains the quotients out of $\mathcal C$ I can also live with it.

I know that the presheaf category $[\mathcal C^{\mathrm {op}} , \mathrm{Set}]$ is the free cocompletion of $\mathcal C$, which means that I get all the quotients, but I also get a whole bunch of other stuff. Maybe it has to do with $\mathrm{Set}$ contains all the colimits. In fact $\mathrm{Set} = [\mathbb 1^{\mathrm {op}}, \mathrm{Set}]$, which is the free cocompletion of the category $\mathbb 1$. so maybe we should consider $[\mathcal C^{\mathrm {op}}, \mathbb Q]$ where $\mathbb Q$ is something like the category that somehow contains all quotients generated from $\mathbb 1$? But then $\mathbb Q = \mathbb 1$ since there are no new quotients to add at all...

On the other hand, maybe this is related to some sort of categorified process of "setoid-ification".

Is there existing results concerning this question? Any help is appreciated.

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    $\begingroup$ You should be more specific about what you mean by "quotient", but perhaps the ex/lex completion is what you are looking for. It can be constructed as a full subcategory of the category of presheaves. $\endgroup$
    – Zhen Lin
    Commented Dec 1, 2021 at 13:44
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    $\begingroup$ ncatlab.org/nlab/show/regular+and+exact+completions $\endgroup$
    – Mike Shulman
    Commented Dec 1, 2021 at 14:04
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    $\begingroup$ The ex/lex completion, and others like it, require the original category to have at least (weak) finite limits in order to define a notion of (pseudo) equivalence relation that is what you take "quotients" of. OTOH if by "quotient" you just mean "coequalizer" then you can just take the subcategory of the presheaf category determined by coequalizers of maps between representables (to just get coequalizers from $C$) or the closure of the representables under coequalizers inside the presheaf category (to get all coequalizers). $\endgroup$
    – Mike Shulman
    Commented Dec 1, 2021 at 14:06
  • $\begingroup$ (If you clarify what you mean by "quotient" then one of these remarks could be made into an answer.) $\endgroup$
    – Mike Shulman
    Commented Dec 4, 2021 at 4:25
  • $\begingroup$ @MikeShulman The ex/lex completion is closer to what I (vaguely) had in mind, but your comment is also worth exploring. If any one of you turn the comments into an answer I'll accept that. $\endgroup$
    – Trebor
    Commented Dec 4, 2021 at 12:46

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In general, the way to construct a free completion of a category under only some colimits is to take a full subcategory of the presheaf category $[C^{\rm op},\rm Set]$ that's the closure of the representables under the colimits in question. In particular we can do this for quotients, although there are different meanings we might pick for "quotient".

  1. If by "quotient" we mean simply a coequalizer, then the closure of the representables under coequalizers is the free cocompletion under coequalizers, with a universal property relative to mapping into other categories with coequalizers.

  2. If by "quotient" we mean the quotient of an internal equivalence relation, then the closure of the representables under such quotients is called the ex/lex completion. As long as the category $C$ already has finite limits, so that it makes sense to talk about internal equivalence relations therein, this has a universal property relative to finite-limit-preserving functors into exact categories. This can be generalized to other kinds of exact completions.

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