There are a lot of forms of the Brown representability theorem, which all basically assert that a functor on a suitable homotopy category which plays well with arbitrary coproducts and satisfies a weak condition on push-outs, is representable.

The form proved by Brown was the following. Let ${\mathbf{hCW}}$ be the homotopy category of pointed CW complexes

Theorem 1 (Brown, c. 1950) Let ${F: \mathbf{hCW} \rightarrow \mathbf{Sets}}$ be a contravariant functor such that ${F}$ sends coproducts to products. Suppose that if ${(X_1, X_2, A) \subset X}$ is a proper triad—i.e., that ${(X_1, A)}$ and ${(X_2, A)}$ were CW pairs with ${X_1 \cap X_2 = A, X_1 \cup X_2 = X}$—then the map

$\displaystyle F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)$

is surjective. Then ${F}$ is representable.

In his original article, Brown used this theorem immediately to show that Eilenberg-MacLane spaces ${K(A, n)}$ existed (as representing objects for ordinary singular cohomology) and that the classifying space ${BG}$ of a topological group existed (as the space representing the functor assigning to ${X}$ the set of principal ${G}$-bundles on ${X}$, up to isomorphism.

There is something funny about the statement in the Brown representability theorem. The initial condition, that ${F}$ send coproducts to products, makes sense for any representable functor. But that ${F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)}$ be surjective, as opposed to bijective? Note that ${X}$ is the push-out of ${X_1 \sqcup_A X_2}$ in the category of spaces.

The problem is that ${X}$ is not the push-out ${X_1 \sqcup_A X_2}$ in ${\mathbf{hCW}}$, the homotopy category! In fact, if ${Z}$ is any CW complex, and we have maps ${f: X_1 \rightarrow Z, g: X_2 \rightarrow Z}$ whose restrictions to ${A}$ are homotopic, then it is true that there is a ${h: X \rightarrow Z}$ whose restrictions to ${X_1, X_2}$ are homotopic to ${f, g}$. This is an exercise in the homotopy extension property for a CW pair. So any representable functor satisfies the above surjectivity. However, such an ${h}$ might not be unique up to homotopy, which means that the push-out is not the push-out in ${\mathbf{hCW}}$.

For example, we could take ${X = \Sigma Y}$ for some space ${Y}$, and let ${X_1, X_2}$ be the upper and lower cones ${C_+ Y, C_- Y}$. Then we see that ${X = \Sigma Y}$ is the push-out of ${C_+ Y, C_- Y}$ along the middle ${Y}$. But ${X}$ is far from the push-out in the homotopy category. In fact, ${C_+ Y, C_- Y}$ are trivial in the homotopy category, so the push-out there is a contractible space.

The point is that there is some subtlety, and the notion of “colimit” in homotopy theory is often replaced by a more refined notion, that of a homotopy colimit. But this notion deserves a separate post. Anyway, it falls out that ${X}$ is also a homotopy push-out of ${X_1, X_2}$ along their intersection ${A}$ in this case.

I blogged a while earlier about the proof of this classical result. The proof, though, is not fundamentally topological. Namely, given such a functor ${F}$, one tries to construct a space ${Y}$ (starting from some base space ${X}$) and an element ${\eta \in F(Y)}$ such that the induced natural transformation

$\displaystyle \hom_{\mathbf{hCW}}(S^n, Y) \stackrel{\eta}{\rightarrow} F(Y)$

is an isomorphism for all the spheres. Since CW complexes are built from spheres, this implies the result (with some work). And this ${Y}$, in turn, can be constructed by attaching lots of cells to a space ${X}$, first to make ${F}$ of it really big (by coproducting it with lots of things) and then small enough (by attaching lots of cells). And over and over.

1. Triangulated categories

What people realized about the proof of Brown representability is that the techniques are not fundamentally topological; they’re categorical. The idea is:

1. Pick a small collection ${R}$ of generating objects (in this case, the spheres). The goal is to find an object in the category such that maps from elements of ${R}$ into this object are the same as ${F}$ applied to elements of ${R}$.
2. To do this, start by taking a big coproduct of elements of ${R}$ to make ${F}$ of it really large. Then, attach things to add relations.

So Amnon Neeman realized (in his paper on Grothendieck duality) that this result could be applied to triangulated categories. Namely, here’s the statement:

Theorem 2 (Neeman) Let ${\mathcal{T}}$ be a triangulated category which is compactly generated and admits infinite direct sums. Let ${G: \mathcal{T} \rightarrow \mathbf{Ab}}$ be a contravariant, cohomological functor that commutes with direct sums. Then ${G}$ is representable.

Let me explain what this means. I won’t define a triangulated category, but a triangulated category is called compactly generated if there is a set of compact objects (i.e., such that homming out of them commutes with direct sums) which generates the category in the sense that every nonzero object admits a nonzero map from one of them. A functor ${G: \mathcal{T} \rightarrow \mathbf{Ab}}$ is called cohomological if it sends exact triangles to half-exact sequences. So if ${A \rightarrow B \rightarrow C \rightarrow A[1]}$ is a distinguished triangle, then

$\displaystyle G(C) \rightarrow G(B) \rightarrow G(A)$

is required to be exact. One can see that a “long exact sequence” follows from this, by rotating triangles; I won’t spell out the details here.

Also, this is not the strongest form of the result, but I’m not going to worry too much about that here.

The choice of name seemed very odd to me when I learned about this result, because the hypotheses seem completely different. But they aren’t! On the one hand, the category ${\mathbf{hCW}}$ is not a triangulated category—it’s not even additive. (The usual “linearization” of this category is the so-called stable homotopy category, for which Brown representability in the topological form can be stated, and in this case follows from Neeman’s theorem.)

On the other hand, there are many formal similarities. In the topological version of Brown representability, the functor was required to commute with coproducts, possibly infinite; this is analogous to how in Neeman’s version the functor is required to preserve infinite direct sums. The more subtle point is that homotopy push-out squares in topological spaces are analogous to exact triangles in triangulated categories. Well, at least in the sense that in practice, both are special cases of a higher categorical colimits (at least if your triangulated category comes from a higher category).

Anyway, so what am I saying? In the topological case, we required that

$\displaystyle F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)$

be surjective for every homotopy push-out square

In Neeman’s version, the hypotheses that ${G}$ send exact triangles to half-exact sequences is the analog. And, amazingly, the proof is analogous: one starts by finding an “approximate” representing object and element for ${G}$, say ${W}$, such that

$\displaystyle \hom(\cdot, W) \rightarrow F(\cdot)$

is surjective for all ${\cdot }$ in a generating set. Then, by taking the homotopy cokernel via a map from a bunch of things in the generating set—the homological algebra version of “attaching cells”—one refines ${W}$ to make the above map bijective for all ${W}$ in the generating set. Then, one shows that this is enough to have a representing object.

2. The meaning of this analogy

What I said earlier was definitely an analogy rather than a formal statement. What does it mean to say that exact triangles and homotopy push-out squares are analogous?

Let ${\mathcal{C}}$ be an ${(\infty, 1)}$-category (I’ll just say ${\infty}$-category henceforth). But in fact the theorem to follow is a statement about the homotopy category of ${\mathcal{C}}$, so if you don’t like ${\infty}$-categories, the important thing to know is that underlying every ${\infty}$-category is a homotopy category. The homotopy category is what remains when you throw away all the higher categorical data. It’s kind of like how when you have a scheme of finite type over a field, you can throw away all the nilpotents by taking its reduction, and then get essentially a variety.

The following appears in 1.4 of “Higher Algebra“:

Theorem 3 (Lurie) Let ${\mathcal{C}}$ be a ${\infty}$-category which is cocomplete and which contains a collection of objects ${\left\{S_\alpha\right\}}$ which generates the ${\infty}$-category ${\mathcal{C}}$ under colimits. Suppose moreover that the ${\left\{S_\alpha\right\}}$ are compact objects and that they admit cogroup structures in the homotopy category ${\mathbf{h}\mathcal{C}}$.Then a functor ${F: \mathbf{h}\mathcal{C}^{op} \rightarrow \mathbf{Sets}}$ is representable if and only if it sends coproducts into products, and if whenever

is a push-out square in ${\mathcal{C}}$, then ${F(D) \rightarrow F(B) \times_{F(A)} F(C)}$ is surjective.

So what is this saying? Well, first we need the ${\infty}$-category to be generated by a bunch of cogroup objects, which is pretty special—kind of like pointed spaces, which have the spheres. (I haven’t defined what “generated” means, I suppose. Let’s say that they detect equivalences.)

This result generalizes Brown’s original result. In fact, the category of pointed CW complexes is such a category (well, let’s say connected pointed spaces), because it is generated by the spheres; this is essentially Whitehead’s classical theorem. The point is that the surjectivity in the above claim is precisely the surjectivity condition made in Brown’s original result. This essentially amounts to the observation that “colimits” in the ${\infty}$-category of spaces are really homotopy colimits. So saying that a diagram is a “push-out” in spaces means that you can think of ${(C, B, A)}$ as a “CW triad” whose union is ${D}$.

OK, I haven’t explained how Lurie’s result generalizes Neeman’s. Actually, I don’t know if it does, because I don’t know if every triangulated category can be obtained naturally from an ${\infty}$-category. But here is what is true.

In any ${\infty}$-category, there is a notion of cofiber sequence: basically, this means a push-out via a point. Since a push-out in an ${\infty}$-category is what you should think of as a homotopy push-out, this is consonant with traditional topology. This only depends up to homotopy, so a pair of composable morphisms in the homotopy category can be called a cofiber sequence as well.

As I stated earlier, you’re supposed to think of triangles in a triangulated category as some sort of replacement for cofibers. In fact, here’s the main result:

Let ${\mathcal{C}}$ be a stable ${\infty}$-category. I won’t explain what this is—it’s a fairly short condition on cofiber sequences—but we can just assume that a lot of ${\infty}$-categories, such as the ${\infty}$-category of chain complexes, turn out to be stable. (But that of spaces is not; you have to linearize, to the category of spectra, to get something stable.)

Theorem 4 (Lurie) Let ${\mathcal{C} }$ be a stable ${\infty}$-category. Then the homotopy category is naturally triangulated.

Essentially, the idea is that triangles are supposed to be cofiber sequences. So a sequence ${A \rightarrow B \rightarrow C \rightarrow A[1]}$ in the homotopy category is supposed to be an exact triangle if and only if ${C}$ can be realized as the cofiber of ${A \rightarrow B}$ in the ${\infty}$-category ${\mathcal{C}}$. Oops, what is ${A[1]}$? It is the analog of the suspension of ${A}$.

Anyway, all this deserves it’s own post. But the upshot is that many natural examples of triangulated categories, such as the derived category of an appropriate abelian category, can be realized naturally as homotopy categories of stable ${\infty}$-categories. Moreover, for a triangulated category arising as the homotopy category of a nice stable ${\infty}$-category, the conditions of Neeman’s and Lurie’s formulations of Brown representability turn out to be the same! In fact, anything is a cogroup object in a triangulated category, so that’s irrelevant—the point is that being cohomological corresponds to the surjectivity ${F(D) \rightarrow F(B) \times_{F(A)} F(C)}$. And this is because of how triangles were constructed in the homotopy category, as cofibers.