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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 11th 2015
• (edited Dec 11th 2015)

I have edited and rearranged just a little at normalizer, in order to clarify a little more.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeDec 12th 2015

Is “Weyl group” really standard terminology in that generality? I’d never seen that.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 14th 2015

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.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 14th 2015
• (edited Dec 14th 2015)

Thanks! (The link in your comment produced a Bad Gateway message, but I can probably find it.)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeDec 14th 2015
• (edited Dec 14th 2015)

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.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeDec 14th 2015

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.

• CommentRowNumber7.
• CommentAuthoreperzhand
• CommentTimeDec 16th 2015
may I ask a stupid question (if this is an offtopic please correct me)
I remember that normalizer was defined in purely cathegorical-theory terms - i.e. not as the subset of a group, but as a category on objects.
Whats the reason for defining it this way?
• CommentRowNumber8.
• CommentAuthoreperzhand
• CommentTimeDec 16th 2015
like here: http://arxiv.org/abs/1307.4845
• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeDec 17th 2015

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.

• CommentRowNumber10.
• CommentAuthoreperzhand
• CommentTimeDec 20th 2015
• (edited Dec 20th 2015)
Re #9:
Shall it be defined the same way in the ncatlab wiki? Or add it as an alternative definition? I.e. more general = bettter?
• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeDec 21st 2015

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJan 12th 2016

re #3: I have made the terminology issue more explicit at normalizer and in particular at the beginning of Weyl group.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeApr 8th 2021

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