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I have edited and rearranged just a little at normalizer, in order to clarify a little more.
Is “Weyl group” really standard terminology in that generality? I’d never seen that.
It’s standard to use it in this generality at least in the field of equivariant homotopy theory. See e.g. page 16 of Lectures on Equivariant Stable Homotopy Theory, but the usage goes way back.
Thanks! (The link in your comment produced a Bad Gateway message, but I can probably find it.)
Ah, the link was to an nLab page, but presently Adeel is reconfiguring something and for the moment the whole nLab produces Bad Gateway errors. A direct link is, as you will have found out already, http://www.math.uni-bonn.de/people/schwede/equivariant.pdf. For earlier references: about every article on equivariant stable homotopy theory will do, notably by Greenlees-May.
I’ve run into this as well. I think searching the Internet for “Weyl group” doesn’t really turn up this meaning, so I guess it isn’t really a thing in (say) group theory; but it does seem to be standard in equivariant homotopy theory.
Re #7, presumably to find a general construction that works in different categories.
Examples. It is easy to check that the pointed categories Gp of groups, Rg of non commutative non unitary rings and R-Lie of Lie algebras on a ring R have normalizers in this sense.
There’s quite an industry of finding commonalities between categories, e.g., exactness properties. Here, this seems to be an extension of work on protomodular categories.
I should think it could appear latter in the page since the generalization is not so widely adopted yet. I’ve added a brief section along those lines, which could be expanded.
re #3: I have made the terminology issue more explicit at normalizer and in particular at the beginning of Weyl group.
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