Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] Homotopy type of simply connected spaces.
@ 2019-01-10 20:36 Brian Sanderson
  2019-01-10 21:11 ` Michael Shulman
  0 siblings, 1 reply; 4+ messages in thread
From: Brian Sanderson @ 2019-01-10 20:36 UTC (permalink / raw)
  To: Homotopy Type Theory

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The type of a simply connected space would seem to make it just a set as 
any two paths with the same endpoints would be homotopic. I see that there 
would not be a continuous function from the space of pairs of paths to 
homotopies between them. What would the type of a simply connected space 
look like? Can I say in type theory any two equalities are equal without 
having a function?

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* Re: [HoTT] Homotopy type of simply connected spaces.
  2019-01-10 20:36 [HoTT] Homotopy type of simply connected spaces Brian Sanderson
@ 2019-01-10 21:11 ` Michael Shulman
  2019-01-11 11:49   ` Brian Sanderson
  0 siblings, 1 reply; 4+ messages in thread
From: Michael Shulman @ 2019-01-10 21:11 UTC (permalink / raw)
  To: Brian Sanderson; +Cc: Homotopy Type Theory

Yes, you have to truncate the equality.  See section 7.5 of the HoTT
Book, and also Exercise 7.6.

On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson
<brianjsanderson@gmail.com> wrote:
>
> The type of a simply connected space would seem to make it just a set as any two paths with the same endpoints would be homotopic. I see that there would not be a continuous function from the space of pairs of paths to homotopies between them. What would the type of a simply connected space look like? Can I say in type theory any two equalities are equal without having a function?
>
> --
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
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* Re: [HoTT] Homotopy type of simply connected spaces.
  2019-01-10 21:11 ` Michael Shulman
@ 2019-01-11 11:49   ` Brian Sanderson
  2019-01-11 12:01     ` Cory Knapp
  0 siblings, 1 reply; 4+ messages in thread
From: Brian Sanderson @ 2019-01-11 11:49 UTC (permalink / raw)
  To: Homotopy Type Theory

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Thanks for the references. So am I allowed to say a type is simply 
connected if any two paths are equal, or is that a meta statement which has 
no meaning within type theory.

On Thursday, 10 January 2019 21:12:13 UTC, Michael Shulman wrote:
>
> Yes, you have to truncate the equality.  See section 7.5 of the HoTT 
> Book, and also Exercise 7.6. 
>
> On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson 
> <brianjs...@gmail.com <javascript:>> wrote: 
> > 
> > The type of a simply connected space would seem to make it just a set as 
> any two paths with the same endpoints would be homotopic. I see that there 
> would not be a continuous function from the space of pairs of paths to 
> homotopies between them. What would the type of a simply connected space 
> look like? Can I say in type theory any two equalities are equal without 
> having a function? 
> > 
> > -- 
> > You received this message because you are subscribed to the Google 
> Groups "Homotopy Type Theory" group. 
> > To unsubscribe from this group and stop receiving emails from it, send 
> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com <javascript:>. 
>
> > For more options, visit https://groups.google.com/d/optout. 
>

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* Re: [HoTT] Homotopy type of simply connected spaces.
  2019-01-11 11:49   ` Brian Sanderson
@ 2019-01-11 12:01     ` Cory Knapp
  0 siblings, 0 replies; 4+ messages in thread
From: Cory Knapp @ 2019-01-11 12:01 UTC (permalink / raw)
  To: Brian Sanderson; +Cc: Homotopy Type Theory

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Using the language of the hott book, a type is simply connected if there
*merely exists* a homotopy between any two paths.

On Fri, Jan 11, 2019, 11:49 Brian Sanderson <brianjsanderson@gmail.com>
wrote:

> Thanks for the references. So am I allowed to say a type is simply
> connected if any two paths are equal, or is that a meta statement which has
> no meaning within type theory.
>
>
> On Thursday, 10 January 2019 21:12:13 UTC, Michael Shulman wrote:
>>
>> Yes, you have to truncate the equality.  See section 7.5 of the HoTT
>> Book, and also Exercise 7.6.
>>
>> On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson
>> <brianjs...@gmail.com> wrote:
>> >
>> > The type of a simply connected space would seem to make it just a set
>> as any two paths with the same endpoints would be homotopic. I see that
>> there would not be a continuous function from the space of pairs of paths
>> to homotopies between them. What would the type of a simply connected space
>> look like? Can I say in type theory any two equalities are equal without
>> having a function?
>> >
>> > --
>> > You received this message because you are subscribed to the Google
>> Groups "Homotopy Type Theory" group.
>> > To unsubscribe from this group and stop receiving emails from it, send
>> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
>> > For more options, visit https://groups.google.com/d/optout.
>>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
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>

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2019-01-10 20:36 [HoTT] Homotopy type of simply connected spaces Brian Sanderson
2019-01-10 21:11 ` Michael Shulman
2019-01-11 11:49   ` Brian Sanderson
2019-01-11 12:01     ` Cory Knapp

Discussion of Homotopy Type Theory and Univalent Foundations

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