I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent complex, partially because I’ve been busy doing other things–namely, trying to learn about the foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent complex thing will get finished as a short expository note on my website.

One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many such concepts, to replace them with the newer ones.

Sheaves

An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working out the exercises there. But as I have more recently discovered, many of the methods there are not the appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines sheafification (and many other things) on a topological space using stalks. However, on a site this is meaningless because there is no analogous notion in general.

The stalk of a sheaf (or presheaf) on a space $X$ at a point corresponds to the inverse image functor via the inclusion $\{\ast\} \to X$. The analogy in the theory of sites would be the inverse image via a morphism from the site with one point (or something equivalent to this). It turns out, fortunately, in etale cohomology this more general notion does make sense, if $\{\ast\}$ is taken to be the spectrum of a separably closed field. So, if $X$ is a scheme, it is not topological points $\{\ast \} \to X$ that lead to the stalk functors in etale cohomology, but the morphisms $\mathrm{Spec} K \to X$ for $K$ a separably closed field (e.g. the separable closure of the residue fields of the topological points).

It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of sets and the given topos. The direct and inverse image functors obtained from maps $\mathrm{Spec} K \to X$ show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves. Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far from understanding the story here though.

Derived functors

Another notion that I’ve reformulated in my mind is that of a derived functor. From the homological algebra textbooks that I read, I learned to think of derived functors as somewhat ad hoc but highly useful means of dealing with the fact that natural additive functors didn’t preserve exact sequences. In other words, it was a way of making long exact sequences to extend almost-short ones. It just happened that they turned out to be fantastically useful.

On the other hand, I was then immensely confused when I learned about Quillen functors between model categories, and that they too admit a form of derived functors. Given model categories $\mathcal{A}, \mathcal{B}$ and a Quillen adjunction $F, G: \mathcal{A} \rightleftarrows \mathcal{B}$, one can define a derived adjunction $\mathrm{Ho}(\mathcal{A}) \rightleftarrows \mathrm{Ho}(\mathcal{B})$ on the homotopy categories.

Now, Quillen’s notion of a derived functor is about extending, say, $F: \mathcal{A} \to \mathcal{B}$ to a functor on the level of the homotopy category. Recall that the homotopy category of a model category is what one gets by inverting the weak equivalences. But you can’t do that–the problem is that Quillen functors generally don’t preserve weak equivalences! What they do preserve, however, are weak equivalences between cofibrant objects. I was told that this was the non-abelian generalization of the Cartan-Eilenberg definition.

So the idea is that $F$ doesn’t behave nicely with respect to weak equivalences in general. To derive it,  then, let’s restrict to the cofibrant objects, where it is still better behaved—and since cofibrant objects are enough for the homotopy category, nothing goes wrong. Here, it does. The technical formulation is that, while $F: \mathcal{A} \to \mathcal{B}$ cannot induce a functor on the homotopy categories directly, the derived functor is the thing that comes closest to doing so. In other words, it’s a Kan extension of the composite from $\mathcal{A}$ to $\text{Ho}\mathcal{B}$ through the homotopy category of $\mathcal{A}$.

What does this have to do with long exact sequences though? Well, let’s recall how to derive a right-exact covariant functor $F$ between abelian categories (with enough projectives, say): namely, given some object $X$, you consider a projective resolution $P_\bullet \to X$, apply $F$ to $P_\bullet$, and take the homology of that. The reason that this works is that, while $F$ does not preserve homology equivalences of chain complexes, it does preserves homology equivalences of projective (bounded-below) chain complexes! Indeed, such a homology equivalence turns out necessarily to be a chain homotopy equivalence, and it’s easy to see that any additive functor preserves those.

There’s a clear analogy here. In fact, it seems that the main difference between the two notions of derived functor (Quillen and Cartan-Eilenberg) go away if, instead of taking the homology of $F(P_\bullet)$, one simply considers the chain complex $F(P_\bullet)$ as the “total” derived functor of $F$. As said above, this chain complex is well-defined up to homology equivalence, or quasi-isomorphism.

This view of derived functors is that afforded by the derived category. And now the analogy becomes clearer: we’re given some functor $F: \mathcal{A} \to \mathcal{B}$, and clearly it becomes a functor between the chain complex categories. Now we want a functor between the (bounded-below, say) derived categories $D^+(\mathcal{A}) \to D^+(\mathcal{B})$. As with Quillen functors, the obvious approach fails: the functor $F$ will not generally preserve quasi-isomorphisms since it is probably not exact. But, it preserves quasi-isomorphisms between chain complexes of projectives. So this is just how to construct one: given, more generally, a chain complex $X_\bullet$, find a “projective resolution” $P_\bullet \to X_\bullet$, and consider $F(P_\bullet)$ as an object of the derived category. The whole point is that the derived functor between derived categories is just a Kan extension, in the same way.

In fact, this is more than an analogy. At least for the category $\mathcal{A}$ of modules over a ring, there is in fact a model structure on chain complexes in $\mathcal{A}$ where the weak equivalences are the quasi-isomorphisms and the fibrations are the surjections. The homotopy category, sure enough, turns out to be the derived category. The cofibrant objects turn out to include the bounded-below complexes of projectives! So when “deriving” functors, we are really doing the same thing.

Historical remarks

Coincidentally, my order of learning these topics (which is pure coincidence) parallels the way in which these concepts were discovered. Derived categories were introduced by Grothendieck and Verdier in an attempt to improve homological algebra to the point where a duality theory for coherent sheaves, generalizing classical Serre duality to the setting of a morphism of schemes. In particular, it came after Cartan-Eilenberg had published their famous treatise on the state of homological algebra then.

I don’t really understand enough about derived categories to explain why they have become so prominent, but perhaps one reason is that the triangulated structure on a derived category means a lot. For instance, there are general (and surprisingly weak) criteria that ensure when an additive functor on a nice triangulated category is representable. It turns out that this is actually an extension of the ideas in the classical topological Brown representability theorem (and is now called the same thing)! In fact, even Grothendieck duality can be derived from this fact.

Similarly, sheaves were introduced by Leray and the 1940’s for topological reasons, and infused into algebraic geometry by Serre in his famous FAC (“Faisceaux algebriques coherents”) paper, where he discovered that many results of classical algebraic geometry could be derived using sheaf theory. But Serre and Leray did everything on a topological space. It was only later that the general theory of sheaves on sites–which soon led to new exotic cohomology theories–was developed.

nPOV

What both these anecdotes have in common is that they realized big and powerful ideas as special cases of bigger and more powerful ideas. These bigger ideas were not simply abstractions for their own sake: as I mentioned, Verdier and Grothendieck had definite aims when introducing the derived category, and the idea of a “site” grew out of the observation (apparently first made by Serre) that the Zariski topology was too coarse for many nice things to hold.

Yet mathematics might not be finished with these concepts. The definition of a triangulated category, for instance, is somewhat cumbersome, and apparently there have been suggestions that it is not the right one in the end. From what I understand, it seems that the notion of a stable $(\infty, 1)$-category is a more natural–in that the axioms are much simpler and feel somehow more right—replacement. For instance, the homotopy category of such an $(\infty,1)$-category turns out to be triangulated. So perhaps triangulated categories such as the derived category are shadows of a deeper, richer higher-categorical structure out there. As the nLab’s point of view says (perhaps somewhat strongly!), “Nothing in mathematics makes sense except in the light of higher category theory.”