Re: It it a good idea to use the term 2-group outside of its use in group thoery?

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Ronnie Brown wrote:

>I would  like to raise an objection  to using the term `2-group' as on nlab and elsehere since for the group theorists this has a specialised meaning: See the following wiki entry, especially the first 2 words:

>"In mathematics, given a prime number p, a p-group is [...]"

That the first 2 words are "In mathematics" rather than
"In group theory, a branch of mathematics," means nothing.
It's not like the Wikipedians had a discussion about it
and determined that p-groups appear throughout mathematics.

You do raise a good point, though.  The term '2-group' is a special case
of both 'p-group' and 'n-group', and these mean very different things.

I wouldn't want to give up 'n-group', so I find '2-group' appropriate
when (as on the n-Category Lab) one is discussing n-groups as well.
But in your example about the structure of finite crossed modules,
one can simply say 'crossed module', making a note that some literature
calls a crossed module a '2-group' (or even 'strict 2-group').

>"[...] Such groups are also called primary."
>there are claims that crossed modules, for example, can be thought of as `2-dimensional groups'

In extreme cases, these show the way: both 'p-group' and 'n-group'
are abbreviations, for 'p-primary group' and 'n-dimensional higher group'.
So one can always use the full name or specify which usage one's paper follows.


--Toby