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In the spirit of this question, it would be interesting to give a characterization of HNN extensions as a 2-colimit. If $G$ is a group and $\alpha:H \xrightarrow{\cong} K$ is an isomorphism between two subgroups of $G$, then I think that the HNN extension $G_{\alpha}$ has the following universal property: if we let $i_1:H \hookrightarrow G,i_2:K \hookrightarrow G$ be the canonical inclusions, then the set of group homomorphisms $G_{\alpha} \to T$ are in natural bijection to pairs $(f,t)$ where $f:G \to T$ is a group homomorphism and $t:f \circ i_2 \circ \alpha \Rightarrow f \circ i_1$ is a 2-morphism. Here we're considering $\mathbf{Grp}$ as a full subcategory of $\mathbf{Cat}$ which carries a standard 2-category structure.

I'm just wondering if this universal property can be phrased as some kind of 2-colimit.

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  • $\begingroup$ I assume you are meaning to think of groups as one-object categories? It's not obvious to people who haven't thought about it this way before. $\endgroup$
    – David Roberts ♦
    Commented Feb 17, 2020 at 6:26
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    $\begingroup$ David, I think this is clear from the last sentence "we're considering Grp as a full subcategory of Cat". $\endgroup$
    – Martin Brandenburg
    Commented Feb 17, 2020 at 9:18

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Assuming your universal property is true, it exactly says that the HNN extension is the coinserter of $(i_2 \circ \alpha,i_1) : H \rightrightarrows G$.

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