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Ugh, I couldn’t make heads or tails of the question until it occurred to me that ’xmod’ is perhaps textmessagespeak for “a crossed module”. Please let’s write in standard English.
@elif: Try it in some simple examples, and for small values of $n$? Are you meaning xmods of algebras or of groups? In the group case replacing square by commutator I think the answer is known using work by Baues and Conduché, and also it can be looked at from the point of view of the tensor square in that case. The only thing to do is to work it out!!!!!
Are you meaning xmods
Et tu, Tim? Or maybe everyone who deals with crossed modules uses this abbreviation, and I didn’t know it?
X is a cross so XMod is a suitable notation for the category of crossed modules, XComp, XSquare, etc, follow suit.
In my idiolect, a “cross” is a vertical line crossing a horizontal one, not two diagonal lines crossing each other. But in any case, even if the category of crossed modules is called XMod, that doesn’t justify writing “xmod” to mean “crossed module” in mathematical English.
In my idiolect, ‘cross’ means as Mike says, except in a mathematical context; compare the cross product ‘$\times$’. That said, when I saw the OP, I first thought that it was some TeX thing; compare \bmod
and \pmod
. Once explained, however, it seems fine.
Although the + is the usual cross shaped symbol, the Saltire cross of Scotland is such a diagonal cross. Diagonal cross shaped flags are also used by St Patrick (thus making up two of the parts of the Union Flag of the UK with the Saltire), and in various other heraldic devices, (e.g. the Neville family). (Non-mathematical point: there are three parts to the Union Flag, yet there are four countries in the United Kingdom.)
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