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. (more…)

Last time, we were discussing the category ${\mathbf{PT}}$ whose objects are pointed topological spaces and whose morphisms are pointed homotopy classes of basepoint-preserving maps. It turns out that the homotopy groups are functors from this category ${\mathbf{PT}}$ to the category of groups.

The homotopy group functors ${\pi_n}$ are, however, representable. They are representable by ${(S^n, s_0)}$, where ${s_0}$ is a base-point; this is equivalently ${I^n/\partial I^n, \partial I^n}$ for ${I^n}$ the ${n}$-cube and ${\partial I^n}$ the boundary. The fact that these are functors to the category of groups is equivalent to saying that ${(S^n, s_0)}$ is a cogroup object in ${\mathbf{PT}}$.

But why should ${(S^n, s_0)}$ be a cogroup object? To answer this question, let us consider a pair of adjoint functors on ${\mathbf{PT}}$.
(more…)