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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)
I have added the statement (here) that tensor product preserves injectivity of finite dimensional vector $G$-spaces, from:
Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, GolasinskiInjectiveModels.pdf:file)
Marek Golasiński, Injective models of $G$-disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)
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?
Amusingly, the presheaf topos on the $\mathbb{Z}_2$ orbit category has appeared before in work of Fourman and Scedrov, showing the “world’s simplest” axiom of choice fails.
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 $G$-space).
I have just worked out myself the minimal $\mathbb{Z}_2$-equivariant model for twistor space equipped with its $\mathbb{Z}_2 = AntiDiag(-1) \subset Sp(2)$-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.
I have spelled out (here) more details in the proof of Prop. 3.4 from Triantafillou 1982, characterizing the projective objects in $Func(Orb_G^{op}, \mathbb{Q}Vect)$ (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.
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)
have now also spelled out the proof of this Lemma.
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…)
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