[HoTT] Localising spheres in HoTT

[HoTT] Localising spheres in HoTT

[Ali Caglayan]

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So I have been reading https://arxiv.org/abs/1807.04155

In it they detail a construction for localising types at a prime. I want to 
construct ΩS²ⁿ⁺¹₍₂₎, this is an important space because it appears in the 
James fibration from classical algebraic topology:

Sⁿ₍₂₎ --> ΩSⁿ⁺¹₍₂₎ --> ΩS²ⁿ⁺¹₍₂₎

This is the fibration that gives the EHP spectral sequence at p=2 and 
allows one to compute homotopy groups (at least 2-primary parts) 
inductively.

So, in HoTT one can hope for a type family ΩS²ⁿ⁺¹₍₂₎ --> U, which has the 
proper fiber and base space. Now as far as I know, the only homotopy 
theoretic fibrations that have been formalised in HoTT are the 
Hopf-fibrations and their general H-space construction.

So this is really two questions:

   1. How to define ΩS²ⁿ⁺¹₍₂₎
   2. How to come up with the HoTT version of the James fibration

Now the first one is kind of already done. Heres how to construct it: 

Let S : N -> N be defined as S(k) = k if k is prime and k =/= 2. S(k) = 1 
otherwise. Now by theorem 4.20, we can define s : N -> N by s(k) = prod 
0<=n<=k S(k). So s(k) is the product of all primes less than k, excluding 2.

Now consider the diagram:
       1        1        1        3         3        3.5     3.5     s(7)
  X ----> X ----> X ----> X ----> X ----> X ----> X ----> X ----> ....

Where the integers k denote the deg(k) map, and X is ΩS²ⁿ⁺¹. The colimit of 
this diagram should be our desired space ΩS²ⁿ⁺¹₍₂₎.

If you recall from the HoTT book, the colimit of a diagram has constructors 
for each node in the graph, and each arrow. I haven't really worked out the 
details but this should give us some higher inductive type which is (at 
least) equivalent to ΩS²ⁿ⁺¹₍₂₎. I'm not even sure this is the kind of HIT 
that you can write down.

Here are my questions:

   1. Can we write down a HIT for a type localised at a prime?
   2. Does anybody know any work on other fibration in HoTT?
   3. Do you think that localisation methods from classical algebraic 
topology will work in HoTT?

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Re: [HoTT] Localising spheres in HoTT

[Michael Shulman]

Localizations at arbitrary families of maps are constructed in
arXiv:1706.07526 using a recursive HIT.  Egbert Rijke has shown that
in the case of localizing at maps between omega-compact types, such as
circles and spheres (therefore including this case), the localization
can be constructed as a non-recursive sequential colimit which is at
least similar to the classical construction as a telescope; but I
don't think this work is published anywhere yet.
On Sat, Aug 11, 2018 at 1:58 PM Ali Caglayan <alizter@gmail.com> wrote:
>
> So I have been reading https://arxiv.org/abs/1807.04155
>
> In it they detail a construction for localising types at a prime. I want to construct ΩS²ⁿ⁺¹₍₂₎, this is an important space because it appears in the James fibration from classical algebraic topology:
>
> Sⁿ₍₂₎ --> ΩSⁿ⁺¹₍₂₎ --> ΩS²ⁿ⁺¹₍₂₎
>
> This is the fibration that gives the EHP spectral sequence at p=2 and allows one to compute homotopy groups (at least 2-primary parts) inductively.
>
> So, in HoTT one can hope for a type family ΩS²ⁿ⁺¹₍₂₎ --> U, which has the proper fiber and base space. Now as far as I know, the only homotopy theoretic fibrations that have been formalised in HoTT are the Hopf-fibrations and their general H-space construction.
>
> So this is really two questions:
>
>    1. How to define ΩS²ⁿ⁺¹₍₂₎
>    2. How to come up with the HoTT version of the James fibration
>
> Now the first one is kind of already done. Heres how to construct it:
>
> Let S : N -> N be defined as S(k) = k if k is prime and k =/= 2. S(k) = 1 otherwise. Now by theorem 4.20, we can define s : N -> N by s(k) = prod 0<=n<=k S(k). So s(k) is the product of all primes less than k, excluding 2.
>
> Now consider the diagram:
>        1        1        1        3         3        3.5     3.5     s(7)
>   X ----> X ----> X ----> X ----> X ----> X ----> X ----> X ----> ....
>
> Where the integers k denote the deg(k) map, and X is ΩS²ⁿ⁺¹. The colimit of this diagram should be our desired space ΩS²ⁿ⁺¹₍₂₎.
>
> If you recall from the HoTT book, the colimit of a diagram has constructors for each node in the graph, and each arrow. I haven't really worked out the details but this should give us some higher inductive type which is (at least) equivalent to ΩS²ⁿ⁺¹₍₂₎. I'm not even sure this is the kind of HIT that you can write down.
>
> Here are my questions:
>
>    1. Can we write down a HIT for a type localised at a prime?
>    2. Does anybody know any work on other fibration in HoTT?
>    3. Do you think that localisation methods from classical algebraic topology will work in HoTT?
>
> --
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> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
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