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. (more…)

I have to apologize for this post: while I have, as of late, been making efforts to make this blog more interesting and useful to outsiders (especially now that I have somewhat more readers than before), the present post will be a somewhat detailed walk-through of one of the first important results in BBD, and, as homological algebra, it is slightly on the technical side. Readers unfamiliar with the material may wish to skim the main result and skip the proof (or just read BBD for it).

1. A cohomological functor

We saw that a ${t}$-structure ${(\mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0})}$ on a triangulated category ${\mathcal{D}}$ always implies that ${\mathcal{D}}$ contains an abelian category ${\mathcal{C} = \mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 0}}$, called the heart of ${\mathcal{D}}$. More is true, however:

Theorem 6 The functor ${H^0: \mathcal{D} \rightarrow \mathcal{C}}$ given by ${X \mapsto H^0(X) = \tau_{\geq 0} \tau_{\leq 0} X}$ is a cohomological functor.

This is, of course, familiar from the case when ${\mathcal{D}}$ is the derived category and then these are just the ordinary cohomology functors. In other words, it is a generalization of the long exact sequence in cohomology from a short exact sequence (triangle) of complexes. (more…)

I’ve been reading the Beilinson-Bernstein-Deligne (BBD) paper “Faisceaux pervers” as part of my summer project. To help myself understand it, I thought I would try to do a few blog posts about the material. The paper is ultimately about perverse sheaves, which are suitable complexes of sheaves on a (stratified) topological space satisfying certain cohomological conditions. The surprise is that perverse sheaves actually form an abelian category. The reason behind this is a bit of homological algebra, which I would like to discuss today to start.

1. t-structures

The first important notion in BBD is that of a ${t}$-structure. If ${\mathcal{A}}$ is an abelian category, and ${\mathbf{D}(\mathcal{A})}$ its derived category, we know that there is a notion of an object in positive degrees and one in negative degrees. There are two subcategories ${\mathbf{D}_{\geq 0}(\mathcal{A}), \mathbf{D}_{\leq 0}(\mathcal{A}) \subset \mathbf{D}(\mathcal{A})}$. Namely, a complex ${K^\bullet}$ can be said to be in ${\mathbf{D}_{\geq 0}(\mathcal{A})}$ if it has no cohomology in negative degrees, and similarly in ${\mathbf{D}_{\leq 0}(\mathcal{A})}$ if it has no cohomology in positive degrees. This is equivalent to saying that ${K^\bullet}$ is isomorphic to a complex all of whose nonzero terms are in nonnegative (resp. nonpositive) degrees because of the truncation functors ${\tau_{\geq 0}, \tau_{\leq 0}}$.

A ${t}$-structure is an axiomatization of this observation for the derived category.

Let ${\mathcal{D}}$ be a triangulated category. A ${t}$-structure on ${\mathcal{D}}$ is the data of two full subcategories ${\mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0}}$. We can define ${\mathcal{D}_{\leq n}}$ as ${\mathcal{D}_{\leq 0}[-n]}$ and similarly ${\mathcal{D}_{\geq 0} = \mathcal{D}_{\geq 0}[-n]}$, in accordance with what happens for the derived category.

The following axioms are required:

1. ${\mathcal{D}_{\leq 0} }$ is closed under translation ${X \mapsto X}$.
2. If ${X \in \mathcal{D}_{\leq 0}}$, ${Y \in \mathcal{D}_{\geq 1}}$, then ${\hom_{\mathcal{D}}(X, Y) = 0}$.
3. If ${X \in \mathcal{D}}$, there is a triangle $\displaystyle A \rightarrow X \rightarrow B \rightarrow X$

such that ${A \in \mathcal{D}_{\leq 0}}$ and ${B \in \mathcal{D}_{\geq 1}}$.

The axioms for a ${t}$-structure are simple, but they turn out to be quite powerful, because the axioms for a triangulated category are themselves quite powerful. (more…)