Reeding List 2022-01-03

One thing that I found interesting was the idea of an Eilenberg-Zilber Category, which captures a lot of the structure that makes \(\Delta\) so nice to work with. The core idea is to take a category \(A\), and identify two subcategories \(A_{+}\) and \(A_{-}\), which are meant to generalize the face and degeneracy maps. We then add on a function \(dim : A \to \mathbb{N}\), which gives us a way to talk about the "dimension" of a given object.

Next, we define all the properties that make \(A_{+}\) and \(A_{-}\) tick. First, if \(f : X \to Y\) is an isomorphism, then \(dim(X) = dim(Y)\), \(f \in A_{+}\), and \(f \in A_{-}\). The first condition is pretty self explanatory: isomorphic things have the same dimension. The other two are a bit more subtle: we want \(A_{+}\) to contain things that can increase the dimension, and \(A_{-}\) to contain the things that can lower it, but for them to be proper subcategories they have to contain identity morphisms as well. For most "shape" categories (\(\Delta\), \(\square\)), the only isomorphisms are identity morphisms, but I'd be interested to know if there are shape categories where this isn't true.

If \(f : X \to Y \in A_{+}\) (and it's not an isomorphism) then it should raise the dimension: \(dim(X) < dim(Y)\). The same holds for \(A_{-}\), except for it should lower the dimension. These two properties really give \(A_{+}\) and \(A_{-}\) their vibes.

We also have some factorization properties that are very reminiscent to \(\Delta\): for any morphism \(f : X \to Y\), there is a unique factorization \(f = p \cdot i\) such that \(p \in A_{-}\) and \(i \in A_{+}\). I like to think of this as stating that every morphism can be built by injecting into some high dimensional thing, and then getting rid of all the structure we end up not needing.

Finally, we have a classification property: If \(p : X \to Y \in A_{-}\), then \(p\) has a section \(i : Y \to X\) such that \(p \cdot i = id\). Furthermore, we can determine the equality of two morphisms in \(A_{-}\) by comparing their set of sections. This one is a bit hard to understand without drawing out a bunch of pictures yourself, but I'll do my best! The idea here is that a section of \(p\) selects out the data that \(p\) doesn't throw away. Then, if \(p, p' \in A_{-}\) don't throw away the same stuff, then they must be equal.

Onto something a bit less abstract: the nerve of a category, denoted \(Nerve(\mathcal{C})\). This is something I've come across before, but I still love it. The idea is that we can build a simplicial set from any category by considering chains of morphisms of length \(n\) to be the \(n\text{-dimensional}\) data. Our face maps then witness the fact that we can get \(n\) length chains from \(n+1\) length ones by composing adjacent morphisms, and the degeneracy maps witness the fact that we can build \(n+1\) length chains from \(n\) length ones by adding in identity morphisms.

The reason this is so delightful for me is that it gives us an algebraic way of thinking about something that feels like it should be super spatial. This is a really important tool to be able to understand some of the concepts in later sections.

Next, we start to look at some constructions. In my opinion these are phrased in a somewhat confusing way, but the vibes of the ideas are doable. We start by defining the boundary of the standard n-simplex \(\partial \Delta^{n}\), which we can think of as an n-dimensional triangle without the inside filled in. From an algebraic perspective, we can think of a map \(\partial \Delta^{n} \to Nerve(\mathcal{C})\) as picking out a (potentially non-commuting) diagram involving \(n\) morphisms. Of particular interest are diagrams of the form \(\partial \Delta^{2} \to Nerve(\mathcal{C})\), which pick out diagrams of the form:

We then can say such a diagram \(d : \partial \Delta^{n} \to Nerve(\mathcal{C})\) commutes if we can find a map \(f : \Delta^n \to X\) such that we the following diagram commutes:

We can think of this as "filling in" the diagram \(d\), and the commuting condition ensures that the diagram is involved with both maps. We call the existence of such maps a lifting property.

Next, we define "horns", which we can think of as an n-dimensional simplex that's missing one of it's faces. We denote these \(\Lambda^{n}_{k}\), which is a really lovely piece of notation. It looks like a little triangle that's missing a side!

From an algebraic perspective, the easiest way to think about horns is via \(\Lambda^{2}_1 \to Nerve(\mathcal{C})\), which gives us a diagram of the following shape:

Interestingly enough, composition gives us a way to fill this diagram in! We can represent this as another lifting property:

We can think of this as filling in the bottom part of the triangle formed by \(f : X \to Y\) and \(g : Y \to Z\) with \(f \cdot g : X \to Z\).

Next, we define a notion of "spine". There is a nice geometric interpretation to this, but my tikz skills aren't quite there! Luckily, there's a nice algebraic interpretation too! A map \(Sp^{n} \to \textit{Nerve}\left(\mathcal{C}\right)\) picks out a chain of morphisms, but ignores their composites. Note that \(\Lambda^{2}_{1}\) is the same as \(Sp^{2}\).

To tie this week up, we define the Grothendieck-Segal condition, which characterizes the simplicial sets that define 1-categories. Luckily, this condition is very easy to state! We just need a bijection between \(\textit{Hom}\left(\Delta^{n}, X\right)\) and \(\textit{Hom}\left(Sp^{n}, X\right)\). We can think of this as a sort of "associativity law"; in that every chain of morphisms has a unique composite.

That's it for this week! Next time we get to dig into model categories proper.