378 lines
13 KiB
HTML
378 lines
13 KiB
HTML
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<body>
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<div id="content">
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgcb7a42f">Planning The HoTT Game</a>
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<ul>
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<li><a href="#org86f8abf">Aims of the HoTT Game</a></li>
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<li><a href="#org891bd67">Barriers</a></li>
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<li><a href="#org865979d">Format</a></li>
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<li><a href="#org7cb3a5d">Content</a></li>
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<li><a href="#org313668c">Debriefs</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgcb7a42f" class="outline-2">
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<h2 id="orgcb7a42f">Planning The HoTT Game</h2>
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<div class="outline-text-2" id="text-orgcb7a42f">
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</div>
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<div id="outline-container-org86f8abf" class="outline-3">
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<h3 id="org86f8abf">Aims of the HoTT Game</h3>
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<div class="outline-text-3" id="text-org86f8abf">
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<ul class="org-ul">
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<li>To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT</li>
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<li>[?] Work towards showing an interesting result in HoTT</li>
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<li>Try to balance hiding cubical implementations whilst exploiting their advantages</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org891bd67" class="outline-3">
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<h3 id="org891bd67">Barriers</h3>
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<div class="outline-text-3" id="text-org891bd67">
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<ul class="org-ul">
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<li>HOLD Installation of emacs</li>
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<li>TODO Usage of emacs</li>
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<li>TODO General type theoretic foundations</li>
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<li>TODO Cubical type theory</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org865979d" class="outline-3">
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<h3 id="org865979d">Format</h3>
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<div class="outline-text-3" id="text-org865979d">
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<ul class="org-ul">
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<li>[?] Everything done in .agda files</li>
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<li>Partially written code with spaces for participants to fill in + answer files</li>
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<li>Levels set out with mini-bosses like in Nat Num Game, but with an overall boss</li>
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<li>[?] Side quests</li>
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<li>References to Harper lectures and HoTT book</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org7cb3a5d" class="outline-3">
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<h3 id="org7cb3a5d">Content</h3>
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<div class="outline-text-3" id="text-org7cb3a5d">
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<ul class="org-ul">
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<li>emacs usage</li>
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<li>agda usage
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<ul class="org-ul">
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<li>basic commands (all covered in <a href="https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html">https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html</a>)</li>
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<li>recommend doom emacs</li>
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<li>implicit/explicit arguments</li>
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<li>holes and inferred types</li>
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<li><code>_+_</code> vs <code>plus__</code></li>
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</ul></li>
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<li>type theory basics
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<ul class="org-ul">
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<li>meta (judgemental/definitional) equality vs internal (propositional) equality
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<ul class="org-ul">
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<li>function extensionality</li>
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</ul></li>
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<li>type formation
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<ul class="org-ul">
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<li>inductive types
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<ul class="org-ul">
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<li>(side Q) positive and negative constructions of Pi/Sigma types</li>
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<li><code>data</code> and <code>record</code></li>
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</ul></li>
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</ul></li>
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<li>universes</li>
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<li>recursors / pattern matching</li>
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<li>(side Q) some natural number exercises as early evidence of being able to ’do maths’?</li>
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<li>different notions of equivalence
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<ol class="org-ol">
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<li>fibers contractable</li>
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<li>quasi-inverse</li>
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<li>zig-zag</li>
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</ol></li>
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<li>(side Q) types are infinity groupoids</li>
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<li>extra paths (univalence, fun ext, HITs)</li>
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</ul></li>
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<li>HoTT
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<ul class="org-ul">
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<li>basics
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<ol class="org-ol">
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<li>meta interval, identity type vs path type
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<ul class="org-ul">
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<li>mention identity type for compatability with other sources, but just use path type</li>
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</ul></li>
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<li>path type on other types</li>
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<li>dependent path type PathP vs path over</li>
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<li>univalence</li>
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<li>the (non)-issue of J in (Cu)TT</li>
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<li>isContr, isProp, isSet</li>
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<li>drawing pictures</li>
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</ol></li>
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<li>Structures, using univalence to transport
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<ol class="org-ol">
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<li>transporting results between isomorphic structures</li>
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</ol></li>
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<li>HITs, examples
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<ol class="org-ol">
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<li>the constructed interval</li>
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<li>booleans and covers</li>
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<li>S<sup>n</sup></li>
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<li>S<sup>1</sup> with 2 cw structures equiv</li>
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</ol></li>
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<li>Homotopy n-types
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<ol class="org-ol">
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<li>homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
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<ul class="org-ul">
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<li>in particular sigma types</li>
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</ul></li>
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</ol></li>
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</ul></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org313668c" class="outline-3">
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<h3 id="org313668c">Debriefs</h3>
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<div class="outline-text-3" id="text-org313668c">
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<ul class="org-ul">
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<li>2021 July 15; Homotopy n-types
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<ul class="org-ul">
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<li>watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.</li>
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<li>Harper does product case, claiming sigma case follows analogously,</li>
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<li>attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.</li>
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<li>difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious</li>
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<li>Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath</li>
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</ul></li>
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</ul>
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</div>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: JLH KL</p>
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<p class="date">Created: 2021-07-19 Mon 18:31</p>
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</div>
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</body>
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</html>
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