A seminar on HoTT Equivalences

Posted on 7 December 2011 by Andrej Bauer

I recorded our local Seminar on foundations in which I talked about the notion of equivalence in HoTT:

Hopefully some people will find some use for it. It is pretty slowly going, and it might motivate some of the strange things going on in Equivalences.v.

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5 Responses to A seminar on HoTT Equivalences

Mike Shulman says:

8 December 2011 at 14:32

It’s surprisingly hard to prove that is_equiv and is_adjoint_equiv are equivalent! I end up fooling around with lots of transport along funexts and needing new lemmas about transporting in iterated total spaces, and I’ve never been sufficiently motivated to make it work.

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- Mike Shulman says:
  
  16 December 2011 at 03:00
  
  Well, I finally managed to prove this. I was going about it too directly — turns out it’s easier to massage them both step-by-step through a series of basic equivalences until they come out the same. See this file.
  
  I also finally proved (same file) that the other possible definition, called is_hiso in Equivalences.v, is equivalent to is_equiv. That one is significantly easier. Of course, both proofs use function extensionality.
  
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  - andrejbauer says:
    
    16 December 2011 at 04:53
    
    Excellent, thank you!
    
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andrejbauer says:

12 December 2011 at 11:14

And they are equivalent, correct?

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- Mike Shulman says:
  
  12 December 2011 at 12:53
  
  Well, since is_equiv is a prop and we have maps in both directions, their being equivalent is equivalent to is_adjoint_equiv being a prop. That doesn’t seem to be any easier to prove in Coq, but homotopical and higher-categorical intuition says that it’s true.
  
  For instance, we could consider the theory of “marked simplicial sets” in section 3.1 of Higher Topos Theory, which are simplicial sets equipped with a subclass of “marked” 1-simplices denoting “equivalences” (thus a higher-categorical version of “categories with weak equivalences”). Any quasicategory $X$ has an underlying marked simplicial set $X^\natural$ in which all the equivalences are marked, and any simplicial set $A$ gives rise to a marked one $A^\sharp$ in which all edges are marked.
  
  Let $\Delta^1$ denote the 1-simplex, and let $E$ denote a simplicial set such that maps $E\to X^\natural$ consist exactly of the data corresponding to is_adjoint_equiv. Thus $E$ has two 0-simplices $x$ and $y$ , two 1-simplices $f\colon x\to y$ and $g\colon y\to x$ , 2-simplices $g\circ f\to 1_x$ and $f\circ g \to 1_y$ , and one 3-simplex exhibiting one triangle identity between $f$ and $g$ . Then the induced inclusion $(\Delta^1)^\sharp \to E^\sharp$ is “marked anodyne” ( $E$ can be obtained from $\Delta^1$ by gluing on two horns). Since the fibrant marked simplicial sets are just the objects $X^\natural$ for $X$ a quasicategory, this implies that the induced map of simplicial sets $Map(E^\sharp,X) \to Map((\Delta^1)^\sharp,X)$ is a trivial fibration. Hence, any equivalence in a quasicategory $X$ has a contractible space of extensions to adjoint equivalence data (the corresponding fiber of this map).
  
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