Timeline for Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks
Current License: CC BY-SA 4.0
12 events
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| Dec 3, 2019 at 20:56 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Sep 1, 2019 at 11:15 | comment | added | Alexander Körschgen | Ok, I think I got it now. $V_C \to c$ is a cartesian natural transformation of functors on $Elt(X)$, and this is also true for its image under $F$. The descent argument in $\mathcal{T}$ can also be performed by applying Thm 6.1.0.6 in HTT. So, without checking every step in detail, your proof seems fine to me. | |
| Aug 28, 2019 at 17:07 | comment | added | Simon Henry | (I'm wirting $F$ for $\widehat{F}$ ) one shows that $F(V_c)$ is indeed the pullback $F(c) \times_{F(X)} F(V)$ by observing that the natural transformation (in c) $F(V_c) -> F(c)$ is cartesian (because it is cartesian in $Prsh(C)$ and $F$ preserve pulblack whose bottom object is representable), and hence by descent (more precisely, effectiveness of colimits in $\mathcal{T}$ this time), each $F(V_c) -> F(c)$ is the pullback of the colimitis, i.e. of $F(V)->F(X)$ (because this is the colimit in $Prsh(C)$ and $F$ preserves colimits). That 's essentially what I tried to explain in the answer. | |
| Aug 28, 2019 at 16:52 | comment | added | Alexander Körschgen | Regarding the proof: The crucial point is that $\hat{F}$ preserves pullbacks constructed from diagrams $c \to X \leftarrow V$ with $c$ representable. I don't see why this is true. In the beginning, you only get preservation of pullbacks of the form $X \to c \leftarrow Y$ (because $\hat{F}/c$ preserves finite limits). | |
| Aug 28, 2019 at 16:48 | comment | added | Alexander Körschgen | Thank you for the clarification of the term "descent". I found that Lurie also uses this terminology at least at one point, namely in the motivational part after Fact 6.1.1.6. | |
| Aug 27, 2019 at 16:24 | comment | added | Simon Henry | Ah found it: that use of the word "descent" is from C.Rezk preprint on higher toposes ( faculty.math.illinois.edu/~rezk/homotopy-topos-sketch.pdf ), (where he attribute it to Lurie actually) see 6.5 there. | |
| Aug 27, 2019 at 12:45 | comment | added | Simon Henry | [...] But I browse trough Lurie's 'Higher topos theory' to find a reference and I realized he never calls this descent in the book. The precise statement I'm refering to is for example condition (2) in theorem 6.1.0.6 of HTT. At the point you are referring I'm just using universality of colimits (so just the fact that as the category is locally Cartesian closed pullback preserve colimits, that part would actually still holds in set) but effectiveness is also used lated. (And thanks for looking at the proof ! ) | |
| Aug 27, 2019 at 12:41 | comment | added | Simon Henry | @AlexanderKörschgen : Regarding the counter-example in Set. When working in space taking homotopy orbits induce an equivalence of $\infty$-categories between space with a G-action and spaces over BG. An equivalence of category this preserves all limits, and so the homotopy orbit functors, which is just the composition of this equivalence with forgeting the map to BG reserve all contractible limits. Regarding Descent, I have to appologize I was refering to the fact that in an $\infty$-topos all colimits are effective and universal. I thought it was usual to call this "descent" [...] | |
| Aug 27, 2019 at 10:58 | comment | added | Alexander Körschgen | Reviewing your counterexample from the non ∞-case: If the proposition were true, we could take $C = G$ a group, considered as an ordinary category, considered as an ∞-category, $\mathcal{T}$ the ∞-topos of spaces, and $F: G \to \ast \to \mathcal{T}$ the functor which sends everything to the one-point space. The (∞-)left Kan extension $\widehat{F}\colon Prsh(G) = Fun(G^{op},\mathcal{T}) \to \mathcal{T}$ should be the homotopy orbit functor. Then we could conclude that homotopy orbits preserve (∞)-fiber products (i.e., homotopy pullbacks). Is that true? | |
| Aug 27, 2019 at 10:53 | comment | added | Alexander Körschgen | Why do you have $V = colim V_c$? This is a statement in $Prsh(C)$, so I'm wondering why descent is helpful here. | |
| Aug 21, 2019 at 19:52 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Aug 21, 2019 at 0:06 | history | answered | Simon Henry | CC BY-SA 4.0 |