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Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that for each pair of 1-cells $f : A \to B$ and $g : A \to C$ in $\mathcal K$, the left extension $\mathrm{lan}_f g : B \to C$ exists in $\mathcal K'$? That is, is it possible to freely adjoin left extensions for every pair of 1-cells with common codomain?

If there is not an explicit construction (in the literature or elsewhere), I would be happy for a demonstration that a construction exists. I suspect one can prove that a syntactic construction works, but I would rather avoid trying this if there is a more elegant approach.

A related question regards adjoining adjoints to 1-cells in bicategories, for which a partial solution can be found in Dawson–Paré–Pronk's Adjoining adjoints.

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    $\begingroup$ Asking for the existence of a universal such functor $\mathcal{K} \to \mathcal{K}'$, and for that functor to be locally fully faithful, are two very different things. The former should be just an exercise in 2-dimensional universal algebra; the latter is more of a coherence / cut-elimination result that would probably require a more careful syntactic-like analysis of the particular situation. Which statement are you more interested in? $\endgroup$ Sep 21, 2021 at 20:30
  • $\begingroup$ For the problem I am interested in, it should suffice to find any faithful locally fully faithful pseudofunctor into a bicategory admitting all left extensions. It seemed neatest to do so universally, but this is not actually important. Perhaps there is an easier solution in this case. $\endgroup$
    – varkor
    Sep 21, 2021 at 20:47
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    $\begingroup$ Well, faithfulness (by which I assume you mean injectivity on 1-cells) is trivial; any pseudofunctor is equivalent to a faithful one. For full-faithfulness, perhaps some kind of Yoneda embedding? $\endgroup$ Sep 22, 2021 at 20:59
  • $\begingroup$ Ah, I hadn't realised that about pseudofunctors. It makes sense intuitively, but is there a simple proof of that fact? For a non-universal embedding, what I have in mind is that every monoidal category embeds fully faithfully in a left-closed monoidal category via Yoneda and Day convolution. Therefore a local bicompletion should embed a bicategory locally fully faithfully into a left-closed bicategory. I think one wants left-coclosure here, so probably one needs to take the local completion instead, but it seems like everything ought to work out. $\endgroup$
    – varkor
    Sep 22, 2021 at 21:46
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    $\begingroup$ Any functor factors as an injective-on-objects one followed by an equivalence. Do that to each action on hom-categories of your pseudofunctor. Then transport the bicategory structure of the codomain across the equivalences of hom-categories. $\endgroup$ Sep 23, 2021 at 3:54

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A partial answer is contained in Betti's Formal theory of internal categories (page 49), where he states that the bicategory $\mathbf{Dist}(\mathcal E)$ of $\mathcal E$-internal distributors is the free completion of $\mathbf{Cat}(\mathcal E)$ under right extensions, at least under the assumption that the underlying class of objects is preserved. It seems plausible this approach could be generalised, but likely not to arbitrary 2-categories, which would probably require a different approach.

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  • $\begingroup$ Great! I still have no more votes for today, till the system clock resets, but I certainly support returning to not leave old questions unanswered. $\endgroup$ Jan 23 at 16:27
  • $\begingroup$ I finally remembered to upvote this, rather than spending the 35 or so votes per day on random other questions. Good that you came back and added the answer. $\endgroup$ Jan 25 at 21:34

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