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In Lack's A 2-categories companion, he states

There are general results asserting that any bicategory is biequivalent to a 2-category, but in fact naturally occurring bicategories tend to be biequivalent to naturally occurring 2-categories.

The examples given by Lack are (special cases of):

  • The bicategory of (enriched) profunctors is biequivalent to the 2-category of presheaf categories and (enriched) cocontinuous functors.
  • The bicategory of polynomials (resp. spans) in a locally cartesian closed category $\mathcal E$ is biequivalent to the 2-category of (linear) polynomial functors on slices of $\mathcal E$.

It would be nice to have more examples to justify this observation, or to have some (apparent) counterexamples to the claim. What are some some other examples of Lack's remark? Are there any "naturally occurring" bicategories that do not appear to have "naturally occurring" biequivalent 2-categories?

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    $\begingroup$ Very nice question indeed; I don’t know any answers, but look forward to seeing them! $\endgroup$
    – Peter LeFanu Lumsdaine
    May 10, 2022 at 9:08
  • $\begingroup$ I don't know enough to answer the question, but another example is (Joyal's ? or I forget who) the "Yoneda" proof of coherence for monoidal categories, namely that a monoidal category is equivalent to a strict one by mapping it to a certain category of endofunctors. Of course, by delooping, this gives an example of a bicategory being equivalent to a reasonably natural 2-category (namely "linear endofunctors") $\endgroup$
    – Maxime Ramzi
    May 10, 2022 at 9:55
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    $\begingroup$ @MaximeRamzi: this is (a special case of) the proof of the coherence theorem for bicategories via Yoneda. I don't think it is reasonable to consider these 2-categories "naturally occurring", because otherwise Lack's observation becomes trivial. Of course, the observation isn't well-defined in the first place, so determining what "naturally occurring" even means is subtle, but I think it's fair to say that we should not expect any general coherence theorem to produce examples. $\endgroup$
    – varkor
    May 10, 2022 at 10:17
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    $\begingroup$ I find this counter-intuitive. If the claim would be true, why working with bicategories? $\endgroup$
    – Konrad Waldorf
    May 11, 2022 at 6:08
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    $\begingroup$ E.g., what is the naturally occurring 2-category that is equivalent to the bicategory of algebras, bimodules, and intertwiners? $\endgroup$
    – Konrad Waldorf
    May 11, 2022 at 6:09

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