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.

• andrejbauer says:

Excellent, thank you!

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).