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It is known that fully faithful functors are closed under pushouts in Cat (e.g. Lemma 4.9 of this paper). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat of 2-categories, (strict) 2-functors, and 2-natural transformations? I expect this to be true, but giving an explicit description of a 2-pushout is daunting. Is there a simpler way to reason to prove this by reasoning entirely locally (i.e. in the hom-categories)?

I expect the fully weak setting to be more difficult, but if it is known that locally fully faithful pseudofunctors are closed under pseudopushouts in a bicategory of bicategories, this would also answer my question.

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  • $\begingroup$ To clarify: what precisely is your 2-category 2-Cat? Also, that ff functors are closed under pullback seems like an instance of limits commuting with limits (since a ff functor can be characterised by a limit diagram, IIRC). $\endgroup$
    – David Roberts ♦
    Commented Nov 6, 2021 at 2:08
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    $\begingroup$ @DavidRoberts: I've clarified my question and corrected a typo: the relevant fact is that fully faithful functors are closed under pushout, not pullback. $\endgroup$
    – varkor
    Commented Nov 6, 2021 at 2:40
  • $\begingroup$ OK, thanks! (I knew offhand that ff+**eso injective on objects** functors are closed under pushout, but I confess I didn't check your link) $\endgroup$
    – David Roberts ♦
    Commented Nov 6, 2021 at 3:13
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    $\begingroup$ @varkor But I don't know what a 2-pushout is. Is it a cocomma object? Is it a lax pushout? Or is it simply a pushout? Oh... does it mean "Cat-enriched pushout"? $\endgroup$
    – Tim Campion
    Commented Dec 14, 2021 at 18:16
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    $\begingroup$ Ok -- so that's the same thing as a $Cat$-enriched pushout. Because $Cat$ has cotensors, this is the same thing as a pushout in the underlying 1-category $Cat$. $\endgroup$
    – Tim Campion
    Commented Dec 14, 2021 at 22:58

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No. I'm not sure whether you're asking about pushout along an arbitrary 2-functor, or just about pushing out a locally fully faithful functor along another locally fully faithful functor. Either way the answer is no.

Consider the inclusion of 1-categories into 2-categories as the locally discrete ones. This functor is fully faithful. It has a right adjoint, so it preserves pushouts. And it identifies the faithful functors between 1-categories with the locally fully faithful functors between essentially discrete 2-categories. So it suffices to show that faithful functors of 1-categories are not closed under pushout in $Cat$.

So it suffices to show that injective homomorphisms of monoids are not closed under pushout in $Mon$.

For this, if you're asking about pushout along an arbitrary map, then it suffices to consider the pushout of the injection $\mathbb N \to \mathbb Z$ along the map $\mathbb N \to \mathbb N / (2=1)$, which is the non-injection $\mathbb N / (2=1) \to \ast$.

Otherwise, we're pushing out an injective homomorphism along another injective homomorphism. The result need not be an injective homomorphism -- see here (the first-linked paper at Benjamin Steinberg's answer there gives an example with $\leq 4$-element semigroups; adding disjoint unit elements these are $\leq 5$-element monoids).

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  • $\begingroup$ I'm probably missing something obvious, but why does it suffice to show that faithful functors are not closed under pushout? Why faithful and not fully faithful (which are closed under pushout)? $\endgroup$
    – varkor
    Commented Feb 17, 2022 at 3:32
  • $\begingroup$ I'm including 1-categories into 2-categories as the locally discrete ones. This functor has a right adjoint, so it preserves pushouts. $\endgroup$
    – Tim Campion
    Commented Feb 17, 2022 at 4:12
  • $\begingroup$ It's still not clear to me why it suffices to consider only the faithful functors. I'm interested in locally fully faithful functors: why is it still sufficient if we drop fullness? $\endgroup$
    – varkor
    Commented Feb 19, 2022 at 19:50
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    $\begingroup$ Let F: A —> B be a functor between 1-categories. Let i(F): i(A) —> i(B) be the corresponding 2-functor between locally discrete 2-categories. Then F is faithful iff i(F) is locally fully faithful. $\endgroup$
    – Tim Campion
    Commented Feb 19, 2022 at 20:00
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    $\begingroup$ If I were trying to prove your conjecture to be true, this should not suffice. But I’m proving it false, ie coming up with a counterexample. so it does suffice. $\endgroup$
    – Tim Campion
    Commented Feb 19, 2022 at 20:04

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