Simplicial Sets V: The Undiscovered Complex

Okay, we’re actually finishing this topic today. Where we left off last time we summarized

• the notion of simplicial subset and how it does means subobject in the functor category
• the idea of Kan pairs, which are a pair $(X,A)$ of a simplicial set $X$ and a simplicial subset $A$ that both satisfy the Kan condition
• relative homotopy for Kan pairs, which is a homotopy of maps such that image of $A$ in the homotopy is constant.
• the basic intuition for what homotopy groups areFirst, we define that a Kan-complex with basepoint is a Kan-pair of complex $X$ and a choice of “point” in $X$, i.e. the image of a simplicial map $\Delta_0 \to X$. We need this because we’ll be defining simplicial homotopy as relative to a basepoint the same way we do with classic topological notions of homotopy. Now, given the notion of Kan-complex with basepoint we can simply say that the homotopy groups $\pi_n(X,*)$ are the set of homotopy equivalence classes of maps $(\partial \Delta^{n+1},*) \to (X,*)$. Now that we’ve built up all the machinery, our definition of simplicial homotopy is basically the same as our definition of homotopy in normal algebraic topology with the simplicial sets $\partial \Delta^{n+1}$ playing the role of the spheres at every dimension $n$.

What this doesn’t obviously have is a group structure, so instead the author introduces a different definition where $\pi_n(X,*)$ is the set of equivalence classes of $n$-simplices $x \in X_n$ with $di x ∈ *$, with the equivalence being homotopy of simplices. Now, what’s the point of this condition on the simplices we consider? If I understand it correctly, it’s essentially saying that the $n$-simplex is a single $n$-cell connected to the vertex $*$. This again would give us something like the spheres. We also haven’t defined homotopy of simplices, however, so let’s do that now. Two $n$ simplices $x,x' \in X^n$ are, essentially, homotopic when there is a simplex $y \in X^{n+1}$ that connects $x$ and $x'$ as its faces. Now my understanding is that the existence of such a $y$ guarantees that you could define the homotopy in the normal sense because you have a way of “sliding” over from $x$ to $x'$ “continuously”. That’s a lot of “scare quotes”, enough to be utterly frightening, but at the moment I’m just talking out my intuition for these things.

Now if I’m understanding the notes correctly, this second definition allows us to define the group structure on $\pi_n(X,*)$ in the following way. Let $x$ and $y$ be two $n$ simplices such that $d_i x = d_i y \in *$ for all $i$. We can consider then a horn $\Lambda^{n+1}_n$ such that $x$ is the $n-1$ face of the horn and y is the $n+1$ face of the horn and we let all the other faces be in $*$. (Note of confusion: why does such a horn necessarily exist? Is it because the faces of $x$ and $y$ are all trivial?) Then, since this is a Kan complex, it has a completion to some $z$ which is the image of $\Delta^{n+1}$ and we then consider the “product” of $x$ and $y$ to be $d_n z$, which interestingly to my category theorist brain is the “composition of arrows” $x$ and $y$. Can someone who knows about both topology and higher-dimensional category theory tell me if that’s a useful comparison or not? With that analogy, though, I can at least see why this would make a group if I see this as “composition”.

Well, that pretty much brings us to the end of this paper. It’s been pretty informative to me but I still think I only conceptually get what’s going on here and not entirely why it’s useful other than higher-dimensional category theory and for models of HoTT.

Can someone maybe educate me on the general advantages of this abstract approach to homotopy or point me to a reference that would help provide that context? It would be greatly appreciated.

Next time I think I’ll talk a bit about a little project I’ve been working on to educate students next term. There’s not a lot to say but I can link to the github repo, at least.

2 thoughts on “Simplicial Sets V: The Undiscovered Complex”

1. So if d_i x = d_i y = const * for all i, then for any face of the horn NOT one of the ones we fill with x and y will have to be something that agrees with x and y on at most a face; possibly even lower dimension than that.

So if we pick const * for all the other faces, then we get agreement along all gluing boundaries “for free”. And that, as I understand it, is why the horn always exists.

As for the intuition, I think “composition” is spot on for a metaphor. It is exactly that for 1-cells (that tend to behave very morphism-like already); and for higher cells you basically take your two cells, identified along a common face, and “compose” them; where the composition really is about sliding from one to the other keeping their commonalities reasonably still.

If you imagine the higher order arrows cases; a homotopy between morphisms is a 2-cell connecting the chain f g to the composition gf. If you step up one dimension, you’ll have two 2-cells — one of them connecting f and g to their composition gf; another connecting i and j to their composition ji = gf. With this setup, the 3-cell then connects the connecting 2-cell expressing the compositional equalities to each other. (does that make sense?)

• Okay, great, that’s actually really helpful. Thanks.