<div dir="ltr"><div>Next week, I plan for us to do what was actually on the schedule for this week, namely covering questions on chapters 3 and 4 and reviewing exercises 3.13, 3.14, and 4.5; please come in with the exercises completed or with questions about how to do them.</div>
<div><br></div><div>I also welcome advice and feedback on the reading group (now or in the future, in person, or by private email), whether that's "I want to get <i>foo</i> out of the group, and I'm not getting it" or "keep doing what you're doing" or "more/fewer exercises" or something else.</div>
<div><br></div><div>Finally, today in the HoTT reading group, we talked about how chapter 4 seems dry to those with no prior knowledge of how the various definitions of equivalence are used in math, but interesting to those who can connect them up to various other bits of math; we talked about various notions of contractibility (inhabited hprop; equivalent to unit; filling of spheres/holes); a bit about truncation; and I brought up the fact that <a href="http://homotopytypetheory.org/2013/10/28/the-truncation-map-_-%E2%84%95-%E2%80%96%E2%84%95%E2%80%96-is-nearly-invertible/">The Truncation Map |_| : ℕ -> ‖ℕ‖ is nearly Invertible</a>, which was perhaps a mistake, given that I botched the presentation. I will leave it up to those interested to read the <a href="http://homotopytypetheory.org/2013/10/28/the-truncation-map-_-%E2%84%95-%E2%80%96%E2%84%95%E2%80%96-is-nearly-invertible/">blog post</a>, <a href="http://red.cs.nott.ac.uk/~ngk/html-trunc-inverse/trunc-inverse.html">Nicolai's Agda code</a>, or <a href="https://gist.github.com/JasonGross/9064514">my own Coq version of it</a>. I think the essence of it is that you can extract information even from contractible types, which I show in <a href="https://gist.github.com/JasonGross/9064636">a much shorter Coq example</a> involving Bool, and two values [x] and [y] where we have [x = y] but we have a (dependent) function [f] with [f x ≠ f y]. (I believe this is related to the difference between having a path between two points in a fiber, and a path between those points which lies over the constant path, and how you can have the former without the latter.)</div>
<div><br></div><div>-Jason</div></div>