basics on homotopy cofibers

[Alizter]

Here is a start on homotopy cofibers. I have more work waiting, but it will be some time before I can clean it up.

@jdchristensen feel free to push any changes as you see fit here. There will probably be some name/style differences with whatever you have.

This builds on

• add functoriality lemmas for pushout #2163

[jdchristensen]

My work is in the attached file, and isn't polished. Maybe you can merge the two together?

Cofiber.txt

[Alizter]

@jdchristensen Do you want me to take a stab at applying Blakers-Massey to get that connectivity result?

[jdchristensen]

@jdchristensen Do you want me to take a stab at applying Blakers-Massey to get that connectivity result?

Sure, that would be great!

[Alizter]

I derived the 8.10.2 HoTT Book Blakers-Massey from the one we had. I'm not yet certain this is enough for the cofiber result, but this is probably something we should cleanup and open a PR for.

[jdchristensen]

I haven't looked at the proof of this carefully, but I suspect that parts of it should be separate lemmas.

[jdchristensen]

I still haven't looked at it carefully, but I'm surprised that it's so long. It should mostly be fairly formal. I'll make a couple of comments for now, but will think more about it later.

[jdchristensen]

Here's an idea that might work. Suppose we have f : X -> Y, g : X -> Z and an equivalence e : X' <~> X. Then you can take a pushout involving X or a pushout involving X'. Then take the corresponding pullbacks P and P', and you get two comparison maps, X -> P and X' -> P'. The claim is that if the first is k-connected, then so is the second. Since this is phrased as a general statement, you should be able to prove it using equivalence induction, i.e. by assuming that e is the identity equivalence, in which case I think it is trivial. (It also shouldn't be too hard to prove this by showing that a square involving X' -> X -> P and X' -> P' -> P commutes.)

Then you apply this to the case where one of them is X and the other is the double-sigma type of Q, and the result you want should follow. (You'd still need to factor out the lemma I suggested in a different comment that says that for the SPushout, the iterated sigma of the spglue map is appropriately connected.)

[jdchristensen]

The idea in the previous comment might work, and might save us the one homotopy in the current proof, but that homotopy is pretty straightforward, so I think it's ok to leave it like it is and not try this idea.

[Alizter]

I've done the review suggestions and factored out some parts of the Blakers-Massey proof. It's possible there are more lemmas to factor out. I added comments explaining what is going on in the proof.

[Alizter]

I managed to prove the connectivity of cofibers using the same argument as the one for suspensions. It didn't use Blakers-Massey and generalised pretty nicely. I'm not sure what we should do with the BM stuff, I struggled to apply it in this case anyway.

[Alizter]

Oh nevermind about my comment about BM I just realised you wanted that to be applied to something else. I'll have a go at that now.

  (** Blakers-Massey implies that the comparison map is highly connected. *)
  Definition isconnected_fiber_to_cofiber (n m : trunc_index)
    {ac : IsConnected n.+1 A} {fc : IsConnMap m.+1 f} (b : B)
    : IsConnMap (m +2+ n) (@fiber_to_cofiber b).
  (** TODO: is m +2+ n correct? *)
  Admitted.

[jdchristensen]

I still haven't looked at it carefully, but I'm surprised that it's so long. It should mostly be fairly formal. I'll make a couple of comments for now, but will think more about it later.