Simplicial Sets III

Continuing from last time on simplicial sets, we were going to cover more of this introductory paper starting with Kan complexes. I had planned to finish it but I think I want to take longer on the section calculating \pi in simplicial homotopy and use it as an opportunity to remind myself of basic topology. Be warned, dear reader, that this is where the fact that I haven’t touched algebraic topology in many-a-year is going to start showing hard. At some point I might reread Hatcher, which was the book I first learned some algebraic topology from a good 14 years ago (I’m not that old, I just learned this stuff young which is a bit of a long story).

First, we introduce the notion of the kth horn \Lambda^n_k of the n simplex \Delta^n which is the complex generated by removing the interior of the simplex and the kth face. So, for example, \Lambda^2_0 will be the complex given by [0,1,2] and [0,1],[0,2]. Now, it took me an amusingly long time to understand why the 0-th face was [1,2] but I finally realized that it’s the 0-th face because d_0 [0,1,2] = [1,2]. Well, at least I understand that the face maps return the opposing face now.

Now a Kan complex is a simplicial object X, i.e. functor X^{op} \to C, such that every morphism \Lambda^n_k \to X can be extended to \Delta^n \to X. Okay, well that’s all well and good but what intuitively does that mean? As far as I can tell, it means that the behavior at all horns must determine the behavior when the horns are completed in the natural way. This seems like a kind of coherence condition, but I don’t really understand what it buys you other than being useful in the simplicial homotopy bits we’ll be getting to. On the other hand, if I put on my category theory hat it makes sense in terms of saying that all composable pairs of arrows should have a corresponding arrow that, while not “equal” to the composition, is related to it by a morphism at the next level up. Also, looking this and related sections I think my intuition here is right and Kan complexes give us a way to get a handle on higher-dimensional categorical structure.

To start discussing simplicial homotopy, first we need to describe what a path means for simplicial sets. Well, we already have a nice convenient simplex to represent the interval: \Delta^1 which is [0,1]. So we can say that a path in X is a simplicial map p : \Delta^1 \to X. Then we have that d_0 p is the initial point of the path and d_1 p is the end point. Now, two vertices are in the same path component of X if there’s a path between them in the above sense.

One thing that struck me as odd, which apparently should be a bit alarming, is that since a path has to be between exactly two vertices and not a longer path then it seems like not all of the vertices in [[0,1],[0,2]] would be connected to each other. This is where it’s important that we’re restricting ourselves to the Kan complexes, because with those this example can’t happen because if we have something like the above then it’s a map from a horn to X, which must be able to be extended to a map from the corresponding \Delta^n to X which necessarily has all the “connecting” faces. For example, in the case above if there are two connecting paths that correspond to [[0,1],[0,2]] then since this horn corresponds to \Delta^2 then there must be another path corresponding to the image of [1,2].

The simplest notion of homotopy in this setting is that two maps f : X \to Y and g : X \to Y are homotopic when there is a simplicial map H : X \times I \to Y such that the normal restriction conditions (H = f at 0 and H = g at 1) apply. There’s another, more complicated, definition of homotopy he gives but when I look at it it just seems like a more elaborate version of saying that “simplicial maps are natural transformations”, but that’s probably my bias coming through.

At this point, we just have section 9 left and I think I’ll cover that later this week.


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