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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?

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    $\begingroup$ 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. $\endgroup$ – Qiaochu Yuan Feb 26 '17 at 19:55
  • $\begingroup$ 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? $\endgroup$ – user321268 Feb 26 '17 at 20:13
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If you want to understand how the category of groups becomes a 2-category, and in general if you want to understand 2-categories, you ought to be familiar with the canonical example of a (strict) 2-category, namely Cat itself. This the the category of all (small) categories, and has as its objects categories, as its morphisms functors, and as its 2-morphisms natural transformations. Taking a group to be a category with one object, Grp becomes a full subcategory of Cat. Ultimately, I would recommend that you try to spend some more time understanding all these definitions, as once you have that set, the 2-category structure of Grp will become pretty straightforward.

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  • $\begingroup$ Thank you for your reply! Do you know if there is any kind of embedding of an arbitrary 2-category in Cat? $\endgroup$ – user321268 Feb 26 '17 at 20:47
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    $\begingroup$ I highly doubt it. Grp is a subcategory of Cat because it's defined that way. It's not like Cat is necessarily the end-all-be-all of 2-categories. $\endgroup$ – silvascientist Feb 26 '17 at 20:50
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    $\begingroup$ Regarding 2-categories, and more generally higher categories, I don't know of any good references; I've only picked them up off of resources scattered across the Internet. For ordinary category theory, I've picked up Topos Theory: The Categorial Analysis of Logic, by Robert Goldblatt, which starts out with a gentle introduction to Category Theory before it goes into the more foundational material. I think it does a good job of providing some decent motivation for the material, which can otherwise be difficult to get a grip on, what with all the abstract nonsense floating around. $\endgroup$ – silvascientist Feb 26 '17 at 20:55
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    $\begingroup$ Another good book seems to be Category Theory in Context, by Emily Riehl. I haven't read it thoroughly; it seems a bit more advanced in its strictly category-theoretic development, however it still aims to present category theory to the novice. Both books are very inexpensive - about $20 on Amazon. They are Dover books after all. $\endgroup$ – silvascientist Feb 26 '17 at 20:57
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    $\begingroup$ I don't think there are any textbook treatments of 2-categories, but Steve Lack's paper "A 2-Categories Companion", available at his website, is a great introduction. By the way, on your embedding question: the analogies question for categories is whether every category embeds into Set, which is false. $\endgroup$ – Kevin Arlin Feb 26 '17 at 22:42

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