Oscar Cunningham<p>Can you define a 'simplicial set of small simplicial sets' by defining Δⁿ → Simp to be the set of small simplicial sets over Δⁿ, i.e. A → Δⁿ?</p><p>Would we then have that the maps B → Simp were in correspondence with the simplicial sets over B, for all B?</p><p><a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="tag">#<span>CategoryTheory</span></a> <a href="https://mathstodon.xyz/tags/SimplicialSets" class="mention hashtag" rel="tag">#<span>SimplicialSets</span></a> <a href="https://mathstodon.xyz/tags/Homotopy" class="mention hashtag" rel="tag">#<span>Homotopy</span></a></p>
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#homotopy
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Refurio Anachro<p>Chris Staecker does humorous videos about calculation machines, but he also does research on digital <a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a>! Wait, what's that?!</p><p>A rather simple example for a digital space is just a digital image. We'll also need a digital sphere, and that's going to be the vertices of an octahedron. We want to do homotopy stuff, so we'll look at maps from any image to such an octahedron.</p><p>We're all used to looking at images, and since it's also where the fun happens, we'll color all the vertices of the octahedron in different colors, and pull those back to the image. So we can see where any pixel position gets mapped to by looking at its color.</p><p>Homotopy is a subject of topology, and that involves stretching. It also involves continuitiy, or a notion of neighbourhood. Both of these must be transported to our digital space and sphere.</p><p>Well, two vertices on an octahedron are neighbours if they are connected by an edge, or, put differently, they are not neighbours if they are opposite of each other. Now, when should we consider pixels on an image to be neghbours? Chris proposes that pixels, drawn as little squares, are considered to be neighbours if they share a vertex. Or an edge, which means that they share two vertices. So, any pixel in the middle of an image has eight neighbours!</p><p>1/3</p>
⎯ΘωΘ⟶<p>I have neglected this channel, sorry lol. If you follow me you will probably find this interesting: <a href="https://arxiv.org/abs/2307.00442" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/2307.00442</span><span class="invisible"></span></a></p><p><a href="https://mathstodon.xyz/tags/categories" class="mention hashtag" rel="tag">#<span>categories</span></a> <a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> <a href="https://mathstodon.xyz/tags/motives" class="mention hashtag" rel="tag">#<span>motives</span></a> <a href="https://mathstodon.xyz/tags/gaugetheory" class="mention hashtag" rel="tag">#<span>gaugetheory</span></a> <a href="https://mathstodon.xyz/tags/QFT" class="mention hashtag" rel="tag">#<span>QFT</span></a></p>
HoldMyType<p>not all equality was proven using reflexivity. My understanding of the matter is that is has to do with the placement of the forall (x : A) quantifier. It is permissible to move one of the x's to the top level (based path induction), but not both. (This is somewhat obscured by the reuse of variable names.) There is also a geometric intuition, which is that when both or one endpoints of the path are free (inner-quantification), then I can contract the path into nothingness. But I have a difficult time mapping this onto any sort of rigorous argument.<br /><a href="http://blog.ezyang.com/2013/06/homotopy-type-theory-chapter-one/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">http://</span><span class="ellipsis">blog.ezyang.com/2013/06/homoto</span><span class="invisible">py-type-theory-chapter-one/</span></a><br /><a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> folks any clue on it?</p>
RanaldClouston<p>On this week's <a href="https://fediscience.org/tags/blog" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>blog</span></a> , a bit late as I work around the start of teaching this semester, I write about the fascinating PhD thesis, 'On the homotopy groups of spheres in homotopy type theory' <a href="https://updatedscholar.blogspot.com/2023/02/discussing-on-homotopy-groups-of.html" rel="nofollow noopener noreferrer" target="_blank"><span class="invisible">https://</span><span class="ellipsis">updatedscholar.blogspot.com/20</span><span class="invisible">23/02/discussing-on-homotopy-groups-of.html</span></a> <a href="https://fediscience.org/tags/Hott" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Hott</span></a> <a href="https://fediscience.org/tags/TypeTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>TypeTheory</span></a> <a href="https://fediscience.org/tags/Homotopy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Homotopy</span></a></p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/Reference" class="mention hashtag" rel="tag">#<span>Reference</span></a> request: I am attending a learning seminar on <a href="https://mathstodon.xyz/tags/Floer" class="mention hashtag" rel="tag">#<span>Floer</span></a> <a href="https://mathstodon.xyz/tags/Homotopy" class="mention hashtag" rel="tag">#<span>Homotopy</span></a> theory this semester. Coming from the symplectic side of things, I could use a nice accessible reference on the <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="tag">#<span>algebra</span></a> side, specifically on stable ∞ -<a href="https://mathstodon.xyz/tags/categories" class="mention hashtag" rel="tag">#<span>categories</span></a> and the category of spectra in particular. <br />I already have Chapter 1 of *Higher Algebra* by Jacob Lurie. <br />Anybody have any other good suggestions? (Ideally ones that do not require the whole kitchen sink of model categories.)<br />Thanks in advance! 🙂 :k5:<br /><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a></p>
Colin the Mathmo<p><span class="h-card" translate="no"><a href="https://mathstodon.xyz/@buchholtz" class="u-url mention">@<span>buchholtz</span></a></span> It's worth putting <a href="https://mathstodon.xyz/tags/HashTags" class="mention hashtag" rel="tag">#<span>HashTags</span></a> in your posts to help people find relevant conversations:</p><p><a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> <a href="https://mathstodon.xyz/tags/TypeTheory" class="mention hashtag" rel="tag">#<span>TypeTheory</span></a> </p><p>Give it time, but people will find each other.</p><p>And welcome!</p>
⎯ΘωΘ⟶<p><a href="https://mathstodon.xyz/tags/arbitrage" class="mention hashtag" rel="tag">#<span>arbitrage</span></a> <a href="https://mathstodon.xyz/tags/pricing" class="mention hashtag" rel="tag">#<span>pricing</span></a> <a href="https://mathstodon.xyz/tags/theory" class="mention hashtag" rel="tag">#<span>theory</span></a> <a href="https://mathstodon.xyz/tags/decision" class="mention hashtag" rel="tag">#<span>decision</span></a> <a href="https://mathstodon.xyz/tags/praxis" class="mention hashtag" rel="tag">#<span>praxis</span></a> <a href="https://mathstodon.xyz/tags/holonomy" class="mention hashtag" rel="tag">#<span>holonomy</span></a> <a href="https://mathstodon.xyz/tags/information" class="mention hashtag" rel="tag">#<span>information</span></a> <a href="https://mathstodon.xyz/tags/manifold" class="mention hashtag" rel="tag">#<span>manifold</span></a> <a href="https://mathstodon.xyz/tags/curvature" class="mention hashtag" rel="tag">#<span>curvature</span></a> <a href="https://mathstodon.xyz/tags/noncommutative" class="mention hashtag" rel="tag">#<span>noncommutative</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="tag">#<span>geometry</span></a> <a href="https://mathstodon.xyz/tags/modal" class="mention hashtag" rel="tag">#<span>modal</span></a> <a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> <a href="https://mathstodon.xyz/tags/type" class="mention hashtag" rel="tag">#<span>type</span></a></p>
⎯ΘωΘ⟶<p>imagine a <a href="https://mathstodon.xyz/tags/modal" class="mention hashtag" rel="tag">#<span>modal</span></a> <a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> <a href="https://mathstodon.xyz/tags/type" class="mention hashtag" rel="tag">#<span>type</span></a> <a href="https://mathstodon.xyz/tags/theory" class="mention hashtag" rel="tag">#<span>theory</span></a> based on those modalities: <a href="https://mathstodon.xyz/tags/knowledge" class="mention hashtag" rel="tag">#<span>knowledge</span></a>, <a href="https://mathstodon.xyz/tags/belief" class="mention hashtag" rel="tag">#<span>belief</span></a>, and <a href="https://mathstodon.xyz/tags/perception" class="mention hashtag" rel="tag">#<span>perception</span></a></p>
mug<p><a href="https://mathstodon.xyz/tags/Introduction" class="mention hashtag" rel="tag">#<span>Introduction</span></a> </p><p>Hello folks! I'm an undergraduate student in mathematics interested in abstract homotopy theory, and category theory at large! Aside from that, in my spare time I love programming, reading books and listening some good music!</p><p>I'm also into vegetarianism, philosophy, open source, looking forward to learn more about socialism and a variety of other topics :)</p><p><a href="https://mathstodon.xyz/tags/maths" class="mention hashtag" rel="tag">#<span>maths</span></a> <a href="https://mathstodon.xyz/tags/categoryTheory" class="mention hashtag" rel="tag">#<span>categoryTheory</span></a> <a href="https://mathstodon.xyz/tags/homotopy" class="mention hashtag" rel="tag">#<span>homotopy</span></a> <a href="https://mathstodon.xyz/tags/programming" class="mention hashtag" rel="tag">#<span>programming</span></a> <a href="https://mathstodon.xyz/tags/music" class="mention hashtag" rel="tag">#<span>music</span></a> <a href="https://mathstodon.xyz/tags/books" class="mention hashtag" rel="tag">#<span>books</span></a> <a href="https://mathstodon.xyz/tags/openSource" class="mention hashtag" rel="tag">#<span>openSource</span></a> <a href="https://mathstodon.xyz/tags/philosophy" class="mention hashtag" rel="tag">#<span>philosophy</span></a> <a href="https://mathstodon.xyz/tags/vegetarian" class="mention hashtag" rel="tag">#<span>vegetarian</span></a> <a href="https://mathstodon.xyz/tags/socialism" class="mention hashtag" rel="tag">#<span>socialism</span></a></p>
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