--- title: Cubical Sets date: June 21th, 2021 abbreviations: cube: 🧊 globe: 🌎 yo: よ ---
_Ahem_, please forgive past-me's type theoretic accent. These are, like the normal degeneracies, 2-cubes in which 2 faces are thin (these are the $\lambda i. p\ ik$ faces in the diagram), and the two other faces are the 1-cube we degenerated (the line $p$). Connections are a very natural extension to the theory of Kan cubical sets, since in a sense they say that an $n$-cube is regarded as a degenerate $(n+1)$-cube in all of the possible ways. This extra structure of connections turns out to be very important when considering a category of cubical sets as an alternative to the category of simplicial sets, $\mathbf{sSet}$, when doing homotopy theory. This is because cubes without connection are not a _strict test category_, a property which is... complicated to describe. But _very roughly_, it says that the canonical way of mapping between cubical sets and homotopy types does not preserve products. The perspective we can get that from this particular application of (Kan) cubical sets is that they provide a systematic way to represent the elements of a type ($X_0$), the equalities between elements of that type ($X_1$), the homotopies between equalities in that type ($X_2$), and so forth. In that sense it's not surprising that Kan cubical sets can be used to (almost) model HoTT! Conclusion ---------- I don't know why I write conclusions; These aren't high school essays. However, in this case I do feel compelled to apologise for how technical and disjointed this post was, and how it seems like I needlessly elaborated on things which (to some) might be trivial while not going into enough detail about highly non-trivial things. Like I said in the first paragraph, I was writing this to learn more about cubical sets. So, unlike my other posts, which are explaining concepts I already had an understanding of --- for instance, my last proper post was talking about _my implementation_ of cubical type theory, not cubical type theory in general --- this post is explaining something I had a fuzzy understanding of, and touches on some category-theoretical concepts I didn't have the faintest clue about, like the Yoneda embedding. Several people had tried to explain the Yoneda embedding to me before, but it had never stuck. It was only when I actually wrote out the definition, worked through its effect on objects and maps, and explored a bit of the structure of the unit interval cubical set. I guess explaining something really is the best way to learn it! This was my shortest interval between blog posts maybe.. ever. Don't get used to it! This is the blog post I should've written instead of whatever filler about Axiom J I wrote about last time, but motivation works in mysterious ways when you struggle with depression. In reality, it's not that mysterious --- I'm writing this on the last week of the first academic semester of 2021, which means the deadline anxiety has _finally_ been lifted. God damn, I hate university. References ---------- Since this post is a lot more technical than my others, and it's about something I don't know a lot about, I figured I should cite my sources so you can know I'm not spewing complete baloney. I don't know how people cite things in English-speaking countries, and, to be perfectly honest, I've done a terrible job of keeping track of where I got all this stuff, but here are the papers and pages and textbooks I consulted along the way: The nLab. Seriously. So many nLab pages. I think these three are the ones I visited most often while writing this post, though: - [closed monoidal structure on presheaves](https://ncatlab.org/nlab/show/closed+monoidal+structure+on+presheaves) - Definition of $[X,Y]$ - [fundamental groupoid of a cubical set and the cubical nerve of a groupoid](https://ncatlab.org/nlab/show/fundamental+groupoid+of+a+cubical+set+and+the+cubical+nerve+of+a+groupoid#nerve_functor) - Direct definition of $N^{\le 2}$ - [cubical set](https://ncatlab.org/nlab/show/cubical+set#in_higher_category_theory) - guess :) The following papers: - [A Model of Type Theory in Cubical Sets](http://www.cse.chalmers.se/~coquand/mod1.pdf) - [Cubical Type Theory: a Constructive Interpretation of the Univalence Axiom](https://arxiv.org/abs/1611.02108) - [An Elementary Illustrated Introduction to Simplicial Sets](https://arxiv.org/abs/0809.4221) - [Varieties of Cubical Sets](https://www2.mathematik.tu-darmstadt.de/~buchholtz/varieties-of-cubical-sets.pdf) - [All $(\infty,1)$-toposes have strict univalent universes](https://arxiv.org/abs/1904.07004) The following pages from Kerodon: - [The Homotopy Coherent Nerve](https://kerodon.net/tag/00KM) - [The Nerve of a Groupoid](https://kerodon.net/tag/0035)