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nLab > Latest Changes: category of G sets

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    • CommentRowNumber1.
    • CommentAuthordec41
    • CommentTimeJul 14th 2016

    I’ve started a page category of G sets. I will continue to fill in more details as I have time. Some of the information from ZFA can probably be moved here.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2016

    Great, but it does need to be linked to existing pages such as G-set, topological G-space and permutation representation. You may find that this suggests some reorganisation of existing material.

    • CommentRowNumber3.
    • CommentAuthorDexter Chua
    • CommentTimeJul 14th 2016
    Yes. I think the G-set page can talk about topological groups as well. (I'm the same person as OP, but thought I should use my real name instead - didn't realize username was the same as the displayed name when signing up)
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2016

    One thing. We have tended to use a capitalised form ’X’ for ’the category of xs’, sometimes with abbreviations, so Set, Cat, Grpd, Infinity-Grpd, FinSet, etc. That policy would be awkward here since the ’G’ of ’G-set’ is already a capital. ’G-Set’ would be natural, but perhaps too subtle.

    • CommentRowNumber5.
    • CommentAuthorDexter Chua
    • CommentTimeJul 14th 2016
    The Elephant uses Cont(G) to denote the category, while MacLane and Moerdijk uses BG. I thought BG might be confusing here, since "B" is often taken to mean "delooping" in the nLab, so I opted for something that resembles the Elephant's usage.

    I tend to imagine Cont (or Cts) as something like a functor TopGrp -> Cat that sends each group to the corresponding category, similar to how we write Sh(C, J) for the category of sheaves on a site.

    Another possible notation is Cts-G-Set.
    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2016

    We also have MSet and Understanding M-Set.

    • CommentRowNumber7.
    • CommentAuthorDexter Chua
    • CommentTimeJul 14th 2016
    I'll go for GSet, then.
    • CommentRowNumber8.
    • CommentAuthorDexter Chua
    • CommentTimeJul 14th 2016
    Added a few more basic properties about the topos.

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