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Some properties of connected spaces #1202

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Some properties of connected spaces #1202

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phijor
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@phijor phijor commented Mar 7, 2025

I proved some properties of connected spaces while working on a project; I figured they might be a good fit for this library.

This pull requests contains:

  • proofs that being connected is a proposition
  • a proof that k-connected k-types are contractible
  • a proof of Corollary 7.5.9 in the HoTT book: A type is n-connected iff every map to n-types is constant (896a7e4)
  • a slightly strengthened version of the "only if"-part of the above: A type A is n-connected if the map λ b a → b : B → (A → B) has a section for any n-type B. (ab12f08)
  • a proof that all loop spaces of k+2-connected spaces are merely equivalent
  • a recursive characterization of connectivity similar to Exercise 7.6 of the HoTT book: A type is (k+1)-connected iff it is (k+1)-inhabited and has k-connected paths (c50abab)

phijor added 10 commits March 7, 2025 20:48
@phijor
Being n-connected is a proposition
4a6e564
@phijor
Being an n-connected function is a proposition
5bbed7d
@phijor
A type is n-connected iff every map to an n-type is constant
896a7e4 
This implements the "only if" part of Corollary 7.5.9 of the HoTT book.
@phijor
All k-connected k-types are are contractible
6da4c2c
@phijor
A type is (k+1)-connected iff it is merely inhabited and has k-connec…
9ae95e4 
…ted paths

This implements Exercise 7.6 of the HoTT book.
@phijor
A type is (k+1)-connected iff it is (k+1)-inhabited and has k-connect…
c50abab 
…ed paths
@phijor
All loop spaces of a (k+2)-connected type are merely equivalent
46acd59
@phijor
Avoid univalence in proof of isConnectedFun→isConnected and converse
56289ce
@phijor
Restructure proof of indMapEquiv→conType
ff80c59 
The proof was essentially an application of `isConnectedFun→isConnected`
in disguise.  The new proof hopefully makes it clear how the elimination
principle for connected connected functions is applied.
@phijor
Strengthen indMapEquiv→conType to only require existence of a section
ab12f08
@anshwad10
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a result that I found useful is that a pointed type (X , a) is connected iff (x : X) -> || x = a ||

@phijor
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phijor commented Mar 10, 2025

a result that I found useful is that a pointed type (X , a) is connected iff (x : X) -> || x = a ||

Check out fdaa6cf. Is this what you had in mind?

@anshwad10
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yes thank you

@mortberg
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@aljungstrom Can you please take a look and review this?

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3 participants
@phijor @anshwad10 @mortberg