So we’re still not quite finishing this introduction to simplicial sets, but we’ve almost gotten to section 9 which is about calculating homotopy groups for Kan complexes. First, let’s remind ourselves of what homotopy groups are in ordinary algebraic topology. Conceptually, these groups are an algrebraic invariant that allows us to describe properties about a space. Specifically, each tells us the homotopy-equivalence classes of ways to map an dimensional sphere into a space. In the case of the fundamental group, (which is also often called in the literature, for reasons I don’t entirely understand), this tells us the way that loops behave in the space. For example, in a plane with no holes then every loop at a base point can be deformed into *every other* loop with the same basepoint. Just squish and stretch and drag it around and you’ll have your homotopy between them. On the other hand, if you have a hole in the plane suddenly everything gets much stranger.

You might look at this and guess that there’ll be two possibilities corresponding to whether the loop avoids the hole or gets “snagged” on the hole, and you’d almost be right. It’s actually a bit more complicated because you can avoid the hole, but there’s a distinction about how many times you’re wrapped around the hole and whether you’re wrapped clockwise or counterclockwise around the hole. This means that the fundamental group, regardless of basepoint, on the plane with a hole will be the integers. The group operation being addition, with addition pictorially representing the operating of winding *clockwise* (adding a positive number) or winding *counterclockwise* (adding a negative number).

Now this is as much as I truly remember from many years ago, and I’ll be honest that I have trouble visualizing homotopy groups for any space that’s particularly complicated or above . I’m still thinking that I might want to blog a rereading of sections of Hatcher as a quick re-introduction of algebraic topology for myself. From what I understand, though, all homotopy groups above are abelian because of the Eckmann-Hilton argument.

So that’s all well and good, but how does this relate to Kan complexes and doing homotopy theory with *those*? Well, first we need to cover something I skipped in section 8, relative homotopy. When I first was reading I thought this section was just a side note but, hah, it’s necessary to understand the definitions of homotopy groups.

First, we need to discuss the idea of simplicial subset. is a simplicial subset of when for all , is a subcomplex of and the face maps and degeneracy maps agree on both and .

As an aside, since we know that simplicial sets are functors from then I wonder if there’s a sense in which a simplicial subset is a subobject in the actual functor category? From what I remember, a functor is a subfunctor when there is a natural transformation where every component of the transformation is a monic. If we’re dealing with simplicial *sets*, then the component monics are just going to be 1-1 functions between sets. That seems too general, though, because if we’re arguing that the face and degeneracy maps must be *equal* as opposed to just commuting with the inclusion. On the other hand, maybe that’s what “agree” *really* means in the definition of subcomplex and I’m pretty sure commuting with the inclusion is the definition of it being a natural transformation. I guess I’ve kinda convinved myself now that simplicial subsets are just subfunctors. Also, in general, what are the properties of this functor category? I know that a lot of functor categories tend to have nice properties and, in particular for simplicial *sets*, we’re dealing with a category of presheaves so this must be a topos, right? If it’s a topos, then there must be a subobject classifier. Recalling how presheaf topoi work, then the subobject classifier is the functor generated by sieves. I’m thinking that I need to sit down with Sheaves In Geometry and Logic to refresh myself on how all of this works. I’m almost certain they talk about this particular example at some point in the book, though I’ve only skimmed through the book.

Moving on from that digression, we can define *Kan pairs* which are a pair of a simplicial set and which is a simplicial subset of that both satisfy the Kan condition. We can define relative homotopy, then, for a Kan pair as being a homotopy such that the restriction to can be factored as where is the projection and . What does this *mean*, though? I think the geometric intuition is that it’s a homotopy of maps between and such that all deformations leave alone, i.e. the image of stays the same as you vary the deformation parameter . Just like the other definition of homotopy, this is an equivalence relation when dealing with Kan complexes.

Now, I think we have all the pieces we need to actual describe the homotopy groups for simplicial sets and we can tackle that in what *will* be the last post on this paper because my attention span is starting to wane rapidly. I never thought I’d squeeze a few thousand words out of a 60 page paper, but there ya go.

(I’m still reading the rest of this, but want to respond to your thoughts on π1 vs π0 while I have it fresh in my mind)

It’s kinda common, especially when you start diving into the simplicial language, to identify πk with the connected components of an appropriate derived space. To see how this might work, the key idea is i.e. to see that forms a topological space of its own, where points are loops from a basepoint. In this loop space, paths are exactly homotopies between homotopic loops — so pathconnected components come out as exactly the homotopy classes of loops.

Thus, by writing ∑ for the looping operation, you get things like π_0 ∑^k X = π_k X; so if you understand connected components sufficiently well for arbitrary spaces, this lets you understand all homotopy groups with just one fundamental operation, hiding all that algebra inside the topological space (a loop space has the group operation you expect from a fundamental group as a multiplicative structure on the space itself)…

Also, your subfunctor thoughts are right on the money; it helps me to remind myself of the categorical idea of a subobject: an equivalence class of monics under the equivalence relation of factoring through each other. These factorings are basically ways to change your names for the elements in a (concrete) subobject, and so by identifying some a with f(a) under f monic, you get an identification of the subobject with the stuff inside where after the identification, those commutative diagrams turn into equalities.

All of this for concrete categories — i.e. things that look like sets with structures. For odder categories, you might not have the elements metaphor around; but the properties of monics make everything work out anyway.