In slice filtration, if we have a cofibration with first and last term are both n-slices, it is an easy argument that the middle term is also a n-slice, right?

@skd K(2) kills EM-spectra so it's enough to see that it kills KO. There's lots of ways to see this; for example, you can reduce to checking it for KU, and hence for the Adams summand, which is E(1).

@MingcongZeng This is false even for the trivial

group

Er... Wait maybe I was wrong hold on

It should hold for trivial group I suppose?

(Yeah sorry I made a silly mistake)

The thing I'm worried about is that P^n doesn't preserve cofiber sequences

for example: is it even true for 0-slices? Why is an extension of Mackey functors with objective restrictions also of this form?

Hmm

**injective

I think it does

Does what?

we can make a standard argument in short exact sequence to show that an extension of 2 mackey functros whose restrictions are injective also shares this property. But maybe I am drunk and could not make this right...

Drunk you is doin better than tired me

maybe thr statement is true... But for some reason I don't see how to get around the fact that P^n doesn't preserve fiber sequences (as a functor back to spectra)

The argument for n-slice is closed under extension in my mind is the following: clearly >= n is preserved by extension, so the problem is <=n. By a lemma(4.14 in one of those versions) in HHR, for showing X <= n, it is equivalent to show that for a slice cells S>n, [S,X]^G is trivial. and we form the LES using [S,-]^G applying to our extension and we get 0 -> something -> 0, so it should be.

sounds pretty convincing to me :)

Ok bed time

Thanks for the conversation Dylan. I am debugging something, and now I believe that the bug isn't here.

@DylanWilson makes sense, thanks

A framing of the 0-manifold $\{pt\}$ is the homotopy class of a map $pt\to BO$. So two framings are the same if the two points are in the same path component of $BO$. But that thing is connceted, so there's only one framing

But there should be two framings, right?

A framing is a choice of nullhomotopy of the classifying map for the stable normal bundle. Those are parameterized by maps to O in this case, not BO. And O has two path components

Oh yeah, I'm confusing stuff. Thanks

do we really need that the unit is cofibrant in proposition 3.2 here: ncatlab.org/nlab/show/monoidal+model+category ? I thought that one introduces that second axiom in the definition of monoidal model category precisely not to have to require that the unit is cofibrant...?

@GeoffroyHorel ah, that's true. I was focusing on the first part actually. thanks!

@QiaochuYuan: your blog post about fire got posted to Hacker News, and has plenty of comments there: news.ycombinator.com/item?id=13361199

@OmarAntolín-Camarena HOLY MOLY!

Does anybody know the relationship between reduced and unreduced parametrized homology theories in the parametrized setting à la May-Sigurdsson with maps to a reference space $B$? Is it just a copy of the "ordinary cohomology theory" on the reference space $B$. How do you go between reduced and unreduced theories using the base chanage functors?