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Dylan Wilson
Dylan Wilson
12:32 AM
@AndreaMarino HA.5.4.5.2 followed by HA.5.4.5.3
 
 
8 hours later…
Yai0Phah
Yai0Phah
8:53 AM
I am confused about various structures on THH(F_p). By considerations of TC(F_p), we know that there is a S^1-equivariant map of algebras Z_p->THH(F_p). Does it factor non-equivariantly as Z_p->F_p->THH(F_p)?
 
 
1 hour later…
Andrea Marino
Andrea Marino
10:21 AM
@JonathanBeardsley: ah, didn't know that could be rendered! Next time I'll put dollars :)
 
 
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Bruno Stonek
Bruno Stonek
11:23 AM
Lurie's (HTT 4.3.2.2) definition of Kan extensions along inclusions is the usual colimit formula for (pointwise) Kan extensions. But his definition in the general case (HTT 4.3.3.2) is harder for me to decypher. Is it true that if $F:C\to D$ and $G:C\to E$ are any functors, then when $Lan_F G$ exists one can compute $Lan_F G(d)$ as $colim((F\downarrow d) \to C \to E)$?
 
Denis Nardin
Denis Nardin
@BrunoStonek Yes, if those colimits exist. I recommend you to check arxiv.org/abs/1809.05892 section 10 for an alternative approach (just take S=*)
 
Bruno Stonek
Bruno Stonek
11:42 AM
ok, that's reassuring. will have a look at the reference, thanks!
 
 
1 hour later…
Andrea Marino
Andrea Marino
1:10 PM
I looked at that stuff, @DylanWilson. If you follow the flow of remarks and theorems, there is always something dropped. So I have a pair of approaches I understood I could follow but incomplete.

1. For theorem 2.3.4.4, the unital infinty operad $ E_M, as B_M = E_M_{<1>}$ is a Kan complex, can be written as an assembly of a B_M family of reduced operads. Here it is not clear how to compute that the operads in the family are all equivalent to E_k. Secondly, in 2.3.3.4 it is stated: Given an assembly O \to O', where O' is the assembled infinity operad and O is C family of operads, if O is a …
2. In remark 5.4.2.13, he states the following. If you take BTop(k)^{otimes}, this is unital and BTop(k) is a Kan complex, so there exist an assembled BTop(k)-family of operads O \to BTop(k)^{otimes}. If B to BTop(k) is Kan fibration, we can consider:
A) The base changed family $O_B = O \times_{Btop(k)} B$;
B) The infty operad $E_B = BTop(k)^{otimes} \times_{BTop(k)^U} B^U$

He asserts that O_B \to E_B (presumably the map induced by O to Btop(k)^{\otimes}, BTop(k) \to BTop(k)^U, B \to B^U) exhibits E_B as an assembly of O_B. But.. Why? There is some stability of pullback wrt preoperadic equ…
Sorry for the long post..
 
 
2 hours later…
Dylan Wilson
Dylan Wilson
3:14 PM
@AndreaMarino I don't think anything's actually unjustified. For (1), the claim that E_{B} is an assembly of copies of E_k is HA.5.4.2.8 (see also the remark HA.5.4.2.9 that follows it). I'm not sure I understand what your second concern about 2.3.3.4 is though, could you elaborate?
regarding (2): $\mathbb{E}_B \to \mathbb{E}_{BTop(k)}$ is an approximation and $\mathbb{E}_k \to \mathbb{E}_{BTop(k)}$ is an approximation so $\mathbb{E}_k \to \mathbb{E}_B$ is also an approximation (see Corollary HA.2.3.3.18)
 
Andrea Marino
Andrea Marino
Yes, you are right, fibers are E_k by 5.4.2.8 /9.
 
Andrea Marino
Andrea Marino
3:28 PM
My second concern is that I can't see what is the colimit diagram, and why maps from O to O'' (where O is a C family and O'' an infinity operad) are determined by maps from the fibers O_c \to O''.

Indeed, I see that $ Hom_{Op_{gn}}(O,O'') \simeq Hom_{Op_{\infty}}(O',O'' )$ almost by definition of assembly, but I don't see $Hom_{Op_{gn}}(O,O'') = Hom_{Op_{\finfty}}(some colimit diagram in O_c, O'')$
 
Dylan Wilson
Dylan Wilson
4:02 PM
Lurie is not claiming that maps O to O'' are determined by giving unrelated maps on each fiber, he is saying that you can informally view the map as a family of maps, one for each fiber. The statement about being a colimit is again supposed to be informal (the formal version is the whole theory of assembly and "C-families"), but the idea is that, just like, say, a cocartesian fibration over C determines a functor C--->Cat_{infty}, we should think of a C-family of operads as trying to
encode a functor C--->Op_{infty}
but again, Lurie does not actually prove any type of straightening/unstraightening, he just works with the fibrations directly.
 
Andrea Marino
Andrea Marino
4:15 PM
Aaaah. Ok. I was deceived by the statement "This description is literally true when C is a Kan complex". So on balance: you suggest me to show the operadic equivalence via approximation, i.e. $\mathbb{E}_B \to \mathbb{E}_{BTop(k)}$ is an approximation.
Is there a reference for this?
And then I stop XD
 
Dylan Wilson
Dylan Wilson
I think you can do this straight from the definition, but if you want you could also use 2.3.3.16. I think in the example at hand the relevant map is maybe the identity...
 
Andrea Marino
Andrea Marino
4:36 PM
Thank you very much Dylan! :)
 
Dylan Wilson
Dylan Wilson
5:11 PM
no problem! :)
 
 
2 hours later…
Jonathan Beardsley
Jonathan Beardsley
7:00 PM
@BrunoStonek you can also get this formula from Riehl and Verity's work on ∞-cosmoi (of which quasicategories are an example). I think this is Propo 5.2.4 as well as the final remarks of this paper: arxiv.org/pdf/1507.01460.pdf
 

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