The semester here finished recently, which means that I will (hopefully!) have more time to update this blog over the next couple of months. I intend to continue the brief series on simplicial methods, which should eventually lead into the cotangent complex soon. I have some other ideas for topics in the near future, but given my recent record at keeping promises, I shall perhaps refrain from divulging the information until I actually have the posts ready!

Recall that last time, we introduced the notion of a simplicial set. As these were presheaves on the category of finite ordered sets (that is, the simplex category), we talked for a while about presheaves in general, on any small category. We did some abstract nonsense and showed in particular that any presheaf is (canonically, in fact!) the colimit of representable presheaves. In our case, that meant that the standard simplexes were enough to generate the entire category of simplicial sets. Today, using this formalism, we are going to see that functors can be defined solely on the standard simplices and thus extended canonically.

Now you might be wondering why the simplex category itself is so special, especially since everything we’ve done so far has been for presheaves on any small category. In homotopy theory, which we haven’t gotten to, simplicial sets have the highly important property of admitting a closed model structure which is Quillen equivalent to the Serre model structure; you might thus wonder which categories of presheaves provide a model for classical homotopy theory in the above sense. I don’t have a complete answer; however, it seems worth mentioning that there is work by Cisinski that does. That is, there is a complete characterization of categories whose presheaf categories admit a model structure Quillen equivalent to spaces. But simplicial sets are (presumably) one of the simplest, and have the advantage of arising in many settings.

Let ${\mathcal{C}}$ be a category, and ${\mathcal{D}}$ a cocomplete category. We are interested in colimit-preserving functors

$\displaystyle \overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathcal{D}.$

Here, as before, ${\hat{\mathcal{C}}}$ is the category of presheaves on ${\mathcal{C}}$.

We shall, in this post, write functors out of a presheaf category with a line above them, and functors just defined out of ${\mathcal{C}}$ without the line. Functors will be in bold.

We have the standard Yoneda embedding ${X \mapsto h_X}$ of ${\mathcal{C} \rightarrow \hat{\mathcal{C}}}$. Thus any such functor ${\overline{F}: \hat{\mathcal{C}} \rightarrow \mathcal{D}}$ determines a functor ${\mathbf{F}: \mathcal{C} \rightarrow \mathcal{D}}$. However, we know that any object of ${\hat{\mathcal{C}}}$ is a colimit of representable presheaves. So any colimit-preserving functor ${\hat{\mathcal{C}} \rightarrow \mathcal{D}}$ is determined by what it does on ${\mathcal{C}}$, embedded in ${\hat{\mathcal{C}}}$ via the Yoneda embedding. Conversely, let ${\mathbf{F}: \mathcal{C} \rightarrow \mathcal{D}}$ be any functor. We want to extend this to a functor ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathcal{D}}$ that preserves colimits. For each presheaf ${G}$, we can write it (by what we did last time) as a colimit of representable presheaves over some category ${\mathcal{D}_G}$ and functor ${\mathcal{D}_G \rightarrow \mathcal{C}}$; if ${G \rightarrow G'}$ is a morphism of presheaves, we get a commutative diagram

So we can define

$\displaystyle \overline{\mathbf{F}}(G) = \varinjlim_{c \in \mathcal{D}_G} \mathbf{F}(c).$

By functoriality of ${\mathcal{D}_G}$, this is a functor. This extends ${\mathbf{F}}$ because for a representable presheaf ${G}$, the associated category ${\mathcal{D}_G}$ has a final object (namely, ${G}$ itself!). We will see that this functor commutes with colimits. In fact:

Proposition 6 If ${\mathcal{D}}$ is cocomplete, there is a natural bijection between left adjoints ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathcal{D}}$ and functors ${\mathbf{F}: \mathcal{C} \rightarrow \mathcal{D}}$, given by restriction.

Proof: Given a left adjoint ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathcal{D}}$, restriction gives a functor ${\mathbf{F}: \mathcal{C} \rightarrow \mathcal{D}}$, and ${\overline{\mathbf{F}}}$ is determined from ${\mathbf{F}}$ as above, because a left adjoint commutes with colimits. Conversely, we need to show that if ${\mathbf{F}: \mathcal{C} \rightarrow \mathcal{D}}$ is any functor, then the functor ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathcal{D}}$ built from it as above is a left adjoint.

So we need to find a right adjoint ${\overline{\mathbf{G}}: \mathcal{D} \rightarrow \hat{\mathcal{C}}}$. We do this by sending ${D \in \mathcal{D}}$ to the presheaf ${X \mapsto \hom_{\mathcal{C}}( \mathbf{F} X, D)}$. We need now to see that ${\overline{\mathbf{F}}, \overline{\mathbf{G}}}$ are indeed adjoints. This follows formally:

From this, we can get a characterization of representable functors on presheaf categories.

Corollary 7 Any contravariant functor ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathbf{Set}}$ that sends colimits to limits is representable.

Proof: Let ${\overline{\mathbf{F}}: \hat{\mathcal{C}} \rightarrow \mathbf{Set}^{op}}$ be a functor that commutes with colimits. Then, as we have seen, ${\overline{\mathbf{F}}}$ has an adjoint ${\overline{\mathbf{G}}: \mathbf{Set}^{op} \rightarrow \hat{\mathcal{C}}}$. Let ${F = \overline{\mathbf{G}}(\ast) \in \mathcal{\hat{C}}}$, where ${\ast}$ is the one-point set. Then we claim that ${F}$ is a universal object. To see this, consider the chain of equalities for any presheaf ${F'}$

1.5. Geometric realization

We recall that there was a functor ${\Delta \rightarrow \mathbf{Top}}$ that sent ${[n]}$ to the topological ${n}$-simplex ${\Delta_n}$. The category ${\textbf{Top}}$ is cocomplete, so it follows that there is induced a unique colimit-preserving functor

$\displaystyle \mathbf{SSet} \rightarrow \textbf{Top}$

that sends the standard ${n}$-simplex ${\Delta[n]_\bullet}$ (i.e., the simplicial set corresponding to ${[n]}$ under the Yoneda embedding) to ${\Delta_n}$, with the maps ${\Delta_n \rightarrow \Delta_m}$ associated to ${[n] \rightarrow [m]}$ as discussed earlier. So, in our previous notation, the functor ${\Delta \rightarrow \mathbf{Top}}$ is ${\mathbf{F}}$, and the extension to ${\mathbf{SSet}}$ is ${\overline{\mathbf{F}}}$.

Definition 8 This functor is called geometric realization. The geometric realization of ${X_\bullet}$ is denoted ${|X|}$.

As a left adjoint, geometric realization commutes with colimits. It is a basic fact, which we do not prove, that geometric realization commutes with finite limits if the limits are taken in the category ofcompactly generated spaces. We can explicitly describe ${|X|}$. Namely, one forms the simplex category, which has objects consisting of all maps

$\displaystyle \Delta[n]_\bullet \rightarrow X_\bullet$

with morphisms corresponding to maps ${\Delta[n]_\bullet \rightarrow \Delta[m]_\bullet}$ fitting into a commutative diagram. Then we can define

$\displaystyle |X| = \varinjlim_{\Delta[n]_\bullet \rightarrow X_\bullet} \Delta_n.$

This functor has a right adjoint. In fact, this adjoint is none another than the singular simplicial set ${\mathrm{Sing} T_\bullet}$ for a topological space ${T}$! To see this, recall that we computed the adjoint to be

$\displaystyle \mathbf{G}T = \{ [n] \mapsto \hom_{\mathbf{Top}}(\mathbf{F}[n], T) \},$

and since ${\mathbf{F}}$ takes ${[n]}$ to ${\Delta_n}$, it is easy to see that this is the singular simplicial set.

Proposition 9 The functors ${|\cdot|: \mathbf{SSet} \rightarrow \mathbf{Top}, \mathrm{Sing}: \mathbf{Top} \rightarrow \mathbf{SSet}}$ form an adjoint pair.

It is these two functors that provide the Quillen equivalence between simplicial sets and spaces. The geometric realization functor is important simply for the structure of simplicial sets, though! The usual definition of a weak equivalence, one component of the model structure, in simplicial sets is to say that a map of simplicial sets is a weak equivalence if and only if the map on geometric realizations is a weak homotopy equivalence (which will in fact be a homotopy equivalence by the Whitehead theorem).