# HNN-extension as a 2-colimit

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.

• 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. – David Roberts Feb 17 at 6:26
• David, I think this is clear from the last sentence "we're considering Grp as a full subcategory of Cat". – Martin Brandenburg Feb 17 at 9:18

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