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Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the category of commutative pseudomonoids $\mathrm{CMon}(\mathcal{C})$ has bicategorical pushouts.

This is true for symmetric monoidal categories $(\mathcal{C},\otimes)$. Here one constructs the pushout of monoid morphisms $(A,\mu_A,\eta_A) \leftarrow (C,\mu_C,\eta_C) \rightarrow (B,\mu_B,\eta_B)$ as the coequalizer $A \otimes_C B$ of the two evident morphisms $A \otimes C \otimes B \rightrightarrows A \otimes B$ in $\mathcal{C}$ and endows it with a monoid structure. The most familiar case for this is $\mathcal{C}=\mathsf{Ab}$, which yields the construction of pushouts of commutative rings.

I don't see why this should not work for symmetric monoidal bicategories. But I was told that one has to use codescent objects; I wonder why this is necessary. A reference for coproducts in $\mathrm{CMon}(\mathcal{C})$ (which is the special case $C=\mathbf{1}_{\mathcal{C}}$) is Theorem 5.2 in Schäppi's paper on ind-abelian categories.

Edit: I think I now understand why coequalizers are not enough. So assume that $\mathcal{C}$ has bicategorical codescent objects which are preserved by $\otimes$ in each variable (add additional assumptions if necessary). Does then $\mathrm{CMon}(\mathcal{C})$ have bicategorical pushouts? It seems that Schäppi uses this (in the special case $\mathcal{C}=\mathsf{cat}_{\mathrm{fc}/k}$, the category of essentially small finitely cocomplete $k$-linear categories) without proof here in Section 4.1. I assume that the codescent object is constructed in $\mathcal{C}$ (since we cannot just assume the existence of codescent objects in $\mathrm{CMon}(\mathcal{C}$), right?), but the proof that it has the structure of a commutative pseudomonoid, and that it is actually a codescent object in $\mathrm{CMon}(\mathcal{C})$, is missing.

I would already be happy for a detailed explanation or reference for the case $\mathcal{C}=\mathsf{cat}$, i.e. how to construct bicategorical pushouts of small symmetric monoidal categories.

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    $\begingroup$ Have you tried to endow this coequalizer with a pseudomonoid structure and prove its universal property? I expect that if you try to do that you'll see why you need codescent objects. $\endgroup$ – Mike Shulman Jul 14 '17 at 23:34
  • $\begingroup$ Thanks! I would be grateful for a more detailed explanation. @MikeShulman $\endgroup$ – Martin Brandenburg Dec 25 '19 at 21:57
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    $\begingroup$ The intuitive reason why you shouldn't expect coequalizers to work is that when working with higher categories it almost never works to talk about isomorphisms that don't satisfy coherence conditions, and a bicategorical coequalizer is adding an isomorphism without a coherence condition. $\endgroup$ – Mike Shulman Jan 3 at 15:20
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    $\begingroup$ It's a codescent object in CMon(C) because it's a reflexive codescent object, and reflexive codescent objects, like reflexive coequalizers and geometric realizations of simplicial objects, are automatically preserved in both variables together by any two-variable functor that preserves them in each variable separately. Unfortunately I can't think of a reference at the moment. $\endgroup$ – Mike Shulman Jan 3 at 15:37
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    $\begingroup$ Ah, thanks. Probably essentially recapitulating the construction of codescent objects in terms of coequalizers and coequifiers, then. $\endgroup$ – Mike Shulman Jan 13 at 16:56
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To summarize some of the comments:

I don't know a short answer for why a bicategorical coequalizer doesn't work. If you try to give the bicategorical coequalizer the structure and universal property, you'll find that it just doesn't work somewhere. The intuition is that in higher categories, when you have more coherence information, you generally have to use it rather than ignore it.

The fact that the codescent object is also a codescent object in commutative monoids follows from the fact that it is a reflexive codescent object, and that a two-variable functor preserving reflexive codescent objects in each variable separately also preserves them in both variables jointly. This categorifies the corresponding fact for reflexive coequalizers in 1-categories, and decategorifies a corresponding statement for geometric realizations of simplicial objects in $\infty$-categories; your question here was answered with a proof.

Finally, in $\rm Cat$ (or other locally presentable 2-categories) one can alternatively use the technology of Blackwell-Kelly-Power "Two-dimensional monad theory" to construct colimits in categories of (commutative) monoids, since they are of the form $T\rm Alg$ for an accessible 2-monad $T$.

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For a more elementary approach, see my preprint Bicategorical colimits of tensor categories.

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