From: Ulrik Buchholtz <ulrikbu...@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: Connected 1-Types
Date: Mon, 31 Oct 2016 08:00:54 -0700 (PDT) [thread overview]
Message-ID: <bbdb58df-4192-4e2d-88bf-402bc52d710a@googlegroups.com> (raw)
In-Reply-To: <a5f9acf7-e01e-48be-92b1-720d1bb316a9@googlegroups.com>
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On Monday, October 31, 2016 at 3:42:05 PM UTC+1, Валерий Исаев wrote:
>
> Let's consider the type of pointed connected 1-types, that is PC1T = Σ (A
> : 1-Type) (a₀ : A) (isConnected A), where isConnected A = (a a' : A) → ∥ a
> ≡ a' ∥.
> This type is equivalent to the type of groups (this construction uses
> HITs). This implies that it is a 1-type.
> Is there a way to prove directly (without HITs) that PC1T is a 1-type?
>
Sure, it suffices to show that any identity type (BG = BH) is a 0-type.
This type is a sub-type of the hom-type hom(B,H) = (BG →* BH), which is
easily seen to be a set: see Section 4 of this handout:
http://www.mathematik.tu-darmstadt.de/~buchholtz/higher-groups.pdf
> Also, is it true for the type of connected 1-types (C1T = C1T = Σ (A :
> 1-Type) (isConnected A)) or merely inhabited connected 1-types (IC1T = Σ (A
> : 1-Type ) (∥ A ∥ × isConnected A))?
>
No, connected 1-types, i.e., connected groupoids, are no simpler than
general groupoids.
Are there analogous theorems for n-types with n > 1?
>
See the handout :)
Cheers,
Ulrik
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next prev parent reply other threads:[~2016-10-31 15:00 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2016-10-31 14:42 Валерий Исаев
2016-10-31 15:00 ` Ulrik Buchholtz [this message]
2016-10-31 15:15 ` [HoTT] " Joyal, André
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