basics on homotopy cofibers #2171
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My work is in the attached file, and isn't polished. Maybe you can merge the two together? |
Signed-off-by: Ali Caglayan <[email protected]> <!-- ps-id: a015aefa-300b-4406-941b-3fdcd075d7cb -->
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theories/Homotopy/Cofiber.v
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| (** *** Constructors *) | ||
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| (** Any element of [Y] can be included in the cofiber. *) | ||
| Definition cofib {X Y : Type} (f : X -> Y) : Y -> cofiber f |
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We call pFibers constructor pfib so it seemed appropriate to choose this name.
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I guess cf_in would be an alternative, if we decide to use cf as a prefix for various things about cofibers. Either way seems fine.
theories/Homotopy/Cofiber.v
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| := pushr. | ||
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| (** We have a distinguised point in the cofiber, where the image of [f] is contracted to. *) | ||
| Definition apex {X Y : Type} (f : X -> Y) : cofiber f |
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Does cf_apex seem like a better name?
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I think we should indicate that this has to do with cofibers somehow. cf_apex or cf_pt or cofiber_apex or cofiber_pt. Actually, we should probably define the cofiber as a pointed type, and then we can just use pt to refer to this point...
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The choice of base point is not obvious. When X and Y are pointed, then cf_in would take the base point of Y to cf_in (point Y) which is not definitionally cf_apex. So I was thinking of doing the unpointed version first, and choosing a different basepoint generally.
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I'd be inclined to always choose the "apex" as the base point, even when X and Y are pointed, as cf_in would still be canonically pointed. Also, cf_in o f has a canonical homotopy to the constant map at the "apex", whereas if you choose f pt as the basepoint, the homotopy becomes more complicated.
But if you prefer to leave it unpointed, then we could go with a name like cf_apex.
| := pushl tt. | ||
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| (** Given an element [x : X], we have the path [leg] from the [f x] to the [apex]. *) | ||
| Definition leg {X Y : Type} (f : X -> Y) (x : X) : cofib f (f x) = apex f |
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I think we should call this cfglue, like the other paths we have as part of various colimits: cglue, gcglue, pglue, etc. (I'm not sure why for general colimits we use colimp.)
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That's probably a contraction of colimit_path. I was thinking of adding a bit on the geometric picture here. That's the reason I chose the geometric names. For intuition, thinking of a mapping cone as "join"ing a point to the image and then contracting seems reasonable.
Signed-off-by: Ali Caglayan <[email protected]>
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@jdchristensen Do you want me to take a stab at applying Blakers-Massey to get that connectivity result? |
Sure, that would be great! |
Signed-off-by: Ali Caglayan <[email protected]>
Signed-off-by: Ali Caglayan <[email protected]>
Signed-off-by: Ali Caglayan <[email protected]>
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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. |
| @@ -100,3 +105,19 @@ Proof. | |||
| lhs nrapply (ap (cofib f) p^). | |||
| apply leg. | |||
| Defined. | |||
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| (** The cofiber of a surjective map is [0]-connected. This will be generalized this later. *) | |||
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The second sentence of the comment is grammatically incorrect. And maybe should just be removed anyways.
| Definition blakers_massey_po `{Univalence} (m n : trunc_index) | ||
| {X Y Z : Type} (f : X -> Y) (g : X -> Z) | ||
| `{H1 : !IsConnMap m.+1 f} `{H2 : !IsConnMap n.+1 g} | ||
| : IsConnMap (m +2+ n) (pullback_corec (pglue (f:=f) (g:=g))). |
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I haven't looked at the proof of this carefully, but I suspect that parts of it should be separate lemmas.
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