Timeline for Understanding equivariance of the Tate construction $(-)^{tC_P}$
Current License: CC BY-SA 4.0
14 events
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| Nov 25, 2020 at 0:54 | comment | added | Dylan Wilson | (I think that should say `$\mathsf{Fun}^G(C, D^{triv})$'.) I don't think 'enriched' is so important here (or elsewhere in the paper). The functor $\mathsf{Fun}(-, \mathcal{D}): \mathsf{Cat}^{op} \to \mathsf{Cat}$ is a right adjoint so it sends colimits to limits; in particular colimits over $BG$-shaped diagrams get sent to limits. | |
| Nov 24, 2020 at 23:34 | comment | added | Bryan Shih | Ah! Thanks a lot Dylan. Ok, Denis! It seems to me that the whole paper is based a lot on enriched category theory (enriched in $Cat$) of the category $Cat^{BG}$ where $G$ is a simplicial group. For example: we have the adjunction $Fun^{BG}(C,D^{triv}) = Fun(C_{hG}, D^{triv})$ (Hopefully i got this correct). Are there any articles that systemically treats these "enriched" adjunctions? | |
| Nov 24, 2020 at 19:00 | comment | added | Dylan Wilson | @BryanShih if $F: \mathcal{C} \to \mathcal{D}$ is a functor, then it induces a functor $\mathsf{Fun}(BG, \mathcal{C}) \to \mathsf{Fun}(BG, \mathcal{D})$ by composition. Now take $\mathcal{C}=\mathcal{D} = \mathsf{Fun}(\mathsf{Sp}^S, \mathsf{Sp})$, $G=\mathrm{Aut}(S)$, and $F$ the 'Verdier localization' (again, modulo set theory- really one should work with the category of $\kappa$-accessible functors for some $\kappa$). | |
| Nov 24, 2020 at 15:37 | comment | added | Denis Nardin | It might be helpful to work out precisely what is a $BC_p$-action on a category. It is a fun exercise, and I won't spoil the answer here :). | |
| Nov 24, 2020 at 15:19 | history | edited | Bryan Shih | CC BY-SA 4.0 |
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| Nov 24, 2020 at 15:18 | comment | added | Bryan Shih | I'm still lost by "so this functor then induces a functor on Aut(S)-equivariant objects. " - but I should think about this for some more time then come back. And yes! I had a typo on Q1. | |
| Nov 24, 2020 at 13:16 | comment | added | Dylan Wilson | for Q2: Let S be a Kan complex. Then the diagonal p^*: Sp-->Sp^S is certainly Aut(S)-equivariant for the trivial action on the source. It follows that the right adjoint p_* inherits an Aut(S)-action. Now, there is a localization functor Fun(Sp^S, Sp)-->Fun(Sp^S,S) (modulo set theory) which approximates a functor by one which annihilates the 'induced' objects; so this functor then induces a functor on Aut(S)-equivariant objects. The value of the functor on p_* is p_*^T. Now apply the discussion to the case S=BC_p and restrict the Aut(BC_p) action to BC_p, which acts on itself by translation | |
| Nov 24, 2020 at 13:10 | comment | added | Dylan Wilson | for your edit: a BC_p-equivariant object in Cat would be a functor from B(BC_p) to Cat, not from BC_p to Cat. | |
| Nov 24, 2020 at 1:59 | history | edited | Bryan Shih | CC BY-SA 4.0 |
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| Nov 24, 2020 at 0:55 | history | edited | LSpice | CC BY-SA 4.0 |
More links and slight proofreading
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| Nov 24, 2020 at 0:45 | comment | added | Bryan Shih | Thanks Dylan, for Q1, i edited slightly - is what I comment what we should expect? For Q2, Im confused with your comment, I hope you don't mind elaborating. | |
| Nov 24, 2020 at 0:41 | history | edited | Bryan Shih | CC BY-SA 4.0 |
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| Nov 23, 2020 at 23:43 | comment | added | Dylan Wilson | presumably $BC_p$ acts on itself by translation and hence on the functor category $\mathsf{Fun}(\mathrm{B}C_p, \mathsf{Sp})$, and it acts trivially on $\mathsf{Sp}$. (Be careful not to confuse a BC_p action for a C_p action!) for Q2, I'd guess they're pointing to I.4.1 since it gives a statement that is manifestly functorial in the Kan complex S. | |
| Nov 23, 2020 at 23:17 | history | asked | Bryan Shih | CC BY-SA 4.0 |