A seminar on HoTT Equivalences

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

  1. Mike Shulman says:

    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.

    • Mike Shulman says:

      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.

  2. andrejbauer says:

    And they are equivalent, correct?

    • Mike Shulman says:

      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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