# Why a group $G$ can form a 2-category?

I know (and understand) how a single group can form a category (of one object, morphisms are the action of the group elements as endomorphisms on it). However here Describing the Wreath product categorically., Qiaochu Yuan states that moreover a group $G$ forms a 2-category, which seems to be quite interesting but lacks of an explanation on the internet unfortunately. Can you please explain which the 2-morphisms are in that case? Also if someone knows, can you write some of the applications of this 2-category?

• What I mean is that the whole collection of groups forms a 2-category, not an individual group. This 2-category shows up in algebraic topology and is a conceptual way of understanding the classification of group extensions. Feb 26, 2017 at 19:55
• Firstly thank you for your reply! Oh, I see! So I totally misunderstood what you wrote out. So I rephrase, how the category of groups becomes a 2-category?
– user321268
Feb 26, 2017 at 20:13