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nLab > Latest Changes: vector G-space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2020

    Recording the result from Triantafillou 82, characterizing injective/projective objects in diagrams of vector spaces over (the opposite of) the orbit category.

    (The degreewise ingredients in the rational model for topological G-spaces)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2020

    I have spelled out, in the Examples-section here, the injective dual vector ℤ 2\mathbb{Z}_2-spaces

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2020

    I have added the statement (here) that tensor product preserves injectivity of finite dimensional vector GG-spaces, from:

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2020
    • (edited Sep 29th 2020)

    I still find it confusing, though, how Scull 01, Prop. 7.36 reviews this result and its previous incorrect statements. Because, that Prop. 7.36 is still missing the technical conditions of Golasinski 97b, Lemma 3.6, no?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 29th 2020

    Amusingly, the presheaf topos on the ℤ 2\mathbb{Z}_2 orbit category has appeared before in work of Fourman and Scedrov, showing the “world’s simplest” axiom of choice fails.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2020

    On a similar but maybe complementary note, I find it curious that in all the literature on (unstable) equivariant rational homotopy theory (as referenced here) I don’t find a single example discussed (specifically: an example of a minimal model for some non-trivial GG-space).

    I have just worked out myself the minimal ℤ 2\mathbb{Z}_2-equivariant model for twistor space equipped with its ℤ 2=AntiDiag(−1)⊂Sp(2)\mathbb{Z}_2 = AntiDiag(-1) \subset Sp(2)-action, and it’s fascinating. Now I am wondering if this is the first example ever in unstable equivariant RHT? If you see any other example in the literature, let me know.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2023

    I have spelled out (here) more details in the proof of Prop. 3.4 from Triantafillou 1982, characterizing the projective objects in Func(Orb G op,ℚVect)Func(Orb_G^{op}, \mathbb{Q}Vect) (highlighting the crucial use of a choice of splitting of some SES, which Triantafillou left notationally implicit.)

    In copying this over from the Sandbox, where I wrote it, I notice that my notational conventions in the new material now differ from when I last worked on this entry. This would deserve to be harmonized, but for the moment I just left some warning messages. But the section I added is self-contained.

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2023
    • (edited Nov 29th 2023)

    Below the construction of the comparison map (here) I added a TikZ-diagram making yet more explicit the claim that the map is indeed natural.

    (This is in conversation with somebody trying to follow Triantafillou 1982)

    diff, v11, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2023

    have now also spelled out the proof of this Lemma.

    diff, v11, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2023
    • (edited Nov 29th 2023)

    started now an analogous subsection (here) for the dual discussion of injective objects, which is left implicit in Triantafillou 1982

    (this doesn’t go far for the moment, but I am running out of steam now…)

    diff, v12, current

1 to 10 of 10

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