This post is an expansion of all the details I did not have a good feeling for when I started with for page 7 of Goerss-Jardine, where the geometric realization of simplicial sets is introduced.
The construction works by constructing a few helpful categories, and using their properties along the way. Especially after unpacking the categorical results G-J rely on, there are quite a few categories floating around. I shall try to be very explicit about which category is which, and how they work.
As we recall, simplicial sets are contravariant functors from the category of ordinal numbers to the category of sets. We introduce the simplex category of a simplicial set with objects (simplices) given by maps and a map from to being given by a map in such that .
Interpreting each simplicial simplex as a simplicial set, this can be thought of as the subcategory of the slice category over spanned by maps from the simplicial simplices. Any map between simplicial simplices is determined completely by a unique ordinal number map that induces it.
The proof of the isomorphism
taken over all simplices in the simplex category of proceeds using the theorem that every functor from a small category to sets is a colimit of representable functors.
To unpack this statement, we turn to, say, Awodey, where this is Proposition 8.10.
Proposition 8.10 (Steve Awodey, Category Theory)
For any small category , every object P in the functor category is a colimit of representable functors using the Yoneda embedding .
Specifically, we can choose an index category and a functor such that .
We start by introducing the category of elements or Grothendieck category of P. This will be our index category for the colimit, and is often written . This category has objects , and arrows given by arrows in such that .
This is almost like a category of pointed sets, only that we also care about the action of P while we’re at it. Since is a small category, so is , and there is a projection functor given by forgetting about the point, so sending and preserving the arrow forming a morphism of elements.
We also recall the Yoneda embedding in order to progress nicely here:
. The Yoneda lemma tells us that for contravariant , , naturally in and , and works by assigning to the natural transformation which has components . These are defined by their actions on actual maps and we set .
In other words, a contravariant functor to sets corresponds to natural transformations from the Yoneda embedding to the functor itself. An element in the functor image gives rise to a particular such natural transformation by using the element as a test case, and using the functor to make the test runnable.
Now, returning to the proof of 8.10, we need to build a colimit that will work as our colimit presentation of P. We will do this by using and . Indeed, for an object , by the Yoneda lemma this corresponds to some natural transformation . These natural transformations form a subcategory of the slice category of over P by the naturality of the Yoneda construction.
So we can build a cocone by taking the map from to be the natural transformation . This is a colim because if we had some other cocone with components , we can produce the unique natural transformation by defined by and recall that natural transformations are, by the Yoneda lemma, the same as elements of .
But what does all this mean!?
This was probably all about as headache inducing as anything else to do with Yoneda’s lemma. It’s a result that both in its proof and its applications tends to climb the ladder of nested categorical constructs pretty high.
So let’s get back to the simplicial case, and tease out what this implies for our reading. is a simplicial set, hence a contravariant set-valued functor on . So is our small category.
The category of elements, is the category where objects are for . Morphisms track what happens to under faces and degeneracies. Thus, specifically, by the arguments in example 1.7, we can put in bijective correspondence with the collection of all simplicial n-simplices of X, in the sense that we associate to each the simplicial map as defined on page 6.
Thus, the index category is clear. The category of elements for a simplicial set really is “just” the category of simplices . The representable functors are the for , and so are all on the shape . But these are just the collections of standard simplicial n-simplices.
So our simplicial set is the colimit of simplical n-simplices under the conditions that they obey the face and degeneracy maps in the original simplicial set.
Let’s work this all through on an example. Consider the interval I, given by non-degenerate simplices a, b, x, subject to and . Maps from the simplicial n-simplex to I can hit one of these three simplices, or any degeneracy of these.
There are two maps , namely and . Thus, we start the construction of our colimit by introducing cells and . Any degeneracies are adsorbed by the fact that a simplicial simplex is a simplicial map, hence and so on.
There are three maps , namely:
These three choices all come with boundary maps that have to be taken into account in the colimit. Specifically, the two first maps here, the ones that hit degenerate simplices, factor through and respectively. There is a map (in the category ) given by . This map corresponds to the map (as a map of simplicial simplices), which embeds the simplex into . Thus, this embedding along degeneracies is a map in the index category of the colimit, and therefore leads to an identification of any simplex with all its degeneracies in the colimit. The non-degenerate case gives us a simplex .
Similarly, the colimit forces the identifications and , finishing the description of as a colimit.
Now, we have a nice candidate for the realization of a single simplicial n-simplex: the topological n-simplex defined in Example 1.1. This is used to define the geometric realization of a simplicial set through “simply” forcing the realization functor to be compatible with this colimit structure. Thus, we define the geometric realization of a simplicial set to be the colimit of the topological n-simplices over the same index category that we used to display as a colimit.
Well, what about adjointness?
The first really nice property about realization is that it is left adjoint to the singular functor. Proving this is a sequence in abstract symbol manipulation and remembering what everything at each step of the way actually means. Thus, say we have a simplicial set X and a topological space Y. Adjointness means there is an isomorphism
which is natural in both X and Y.
Now, let’s consider the left hand side.
by definition of the geometric realization.
because contravariant representable functors map colimits to limits. This is Awodey 5.29, and sits very well with our intuition from sets, vector spaces, and just about anywhere else.
Now, remember that a map in is a simplex in the singular set , and that as we saw above, a simplicial set is the colimit of its simplices. Thus, we get
and then finally, we can re-introduce the colimit inside the morphism set
Each step along the way is natural, which finishes the proof.