The next thing I’d like to do on this blog is to understand the derived ${\infty}$-category of an abelian category.

Given an abelian category ${\mathcal{A}}$ with enough projectives, this is a stable ${\infty}$-category ${D^-(\mathcal{A})}$ with a special universal property. This universal property is specific to the ${\infty}$-categorical case: in the ordinary derived category of an abelian category (which is the homotopy category of ${D^-(\mathcal{A})}$), forming cofibers is not quite the natural process it is in ${D^-(\mathcal{A})}$ (in which it is a type of colimit), and one cannot expect the same results.

For instance, ${\mathcal{A}}$, and given a triangulated category ${\mathcal{T}}$ and a functor ${\mathcal{A} \rightarrow \mathcal{T}}$ taking exact sequences in ${\mathcal{A}}$ to triangles in ${\mathcal{T}}$, we might want there to be an extended functor

$\displaystyle D_{ord}^b(\mathcal{A}) \rightarrow \mathcal{T},$

where ${D_{ord}^b(\mathcal{A})}$ is the ordinary (1-categorical) bounded derived category of ${\mathcal{A}}$. We might expect this by the following rough intuition: given an object ${X}$ of ${D^b(\mathcal{A})}$ we can represent it as obtained from objects ${A_1, \dots, A_n}$ in ${\mathcal{A}}$ by taking a finite number of cofibers and shifts. As such, we should take the image of ${X}$ to be the appropriate combination of cofibers and shifts in ${\mathcal{T}}$ of the images of ${A_1, \dots, A_n}$. Unfortunately, this does not determine a functor because cofibers are not functorial or unique up to unique isomorphism at the level of a trinagulated category.

The derived ${\infty}$-category, though, has a universal property which, among other things, makes very apparent the existence of derived functors, and which makes it very easy to map out of it. One formulation of it is specific to the nonnegative case: ${D_{\geq 0}(\mathcal{A})}$ is obtained from the category of projective objects in ${\mathcal{A}}$ by freely adjoining geometric realizations. In other words:

Theorem 1 (Lurie) Let ${\mathcal{A}}$ be an abelian category with enough projectives, which form a subcategory ${\mathcal{P}}$. Then ${D_{\geq 0}(\mathcal{A})}$ has the following property. Let ${\mathcal{C}}$ be any ${\infty}$-category with geometric realizations; then there is an equivalence

$\displaystyle \mathrm{Fun}(\mathcal{P}, \mathcal{C}) \simeq \mathrm{Fun}'( D_{\geq 0}(\mathcal{A}), \mathcal{C})$

between the ${\infty}$-categories of functors ${\mathcal{P} \rightarrow \mathcal{C}}$ and geometric realization-preserving functors ${D_{\geq 0}(\mathcal{A}) \rightarrow \mathcal{C}}$.

This is a somewhat strange (and non-abelian) universal property at first sight (though, for what it’s worth, there is another more natural one to be discussed later). I’d like to spend the next couple of posts understanding why this is such a natural universal property (and, for one thing, why projective objects make an appearance); the answer is that it is an expression of the Dold-Kan correspondence. First, we’ll need to spend some time on the actual definition of this category.

1. The ${\infty}$-category of chain complexes

Let ${\mathcal{A}}$ be an abelian category, or more generally an additive category. Then one can contemplate a category ${\mathrm{Ch}(\mathcal{A})}$ of ${\mathcal{A}}$-valued chain complexes. Observe that given ${A_\bullet, B_\bullet \in \mathrm{Ch}(\mathcal{A})}$, there is a natural chain complex (in abelian groups) of maps

$\displaystyle \underline{Hom}(A_\bullet, B_\bullet),$

such that ${\underline{Hom}(A_\bullet, B_\bullet)_n = \prod \hom_{\mathcal{A}}(A_i, B_{i+n})}$. In other words, ${\mathrm{Ch}(\mathcal{A})}$ naturally acquires the structure of a differential graded category: there is a chain complex, rather than simply a set, of maps between any two objects. One can recover the set of maps between any two objects by taking ${\pi_0}$ of this chain complex.

A differential graded category can be used to manufacture a simplicial category. Namely, we have an equivalence of categories

$\displaystyle \mathrm{DK}: \mathrm{Ch}_{\geq 0}( \mathbf{Ab}) \simeq \mathrm{Fun}(\Delta^{op}, \mathbf{Ab})$

between nonnegatively graded chain complexes of abelian groups and simplicial abelian groups. (This is the Dold-Kan correspondence in the classical form.) This construction is lax monoidal via the Alexander-Whitney maps (though not actually monoidal), and consequently, given a category enriched over ${\mathrm{Ch}_{\geq 0}(\mathbf{Ab})}$, we can get a category enriched over simplicial abelian groups (in particular, over Kan complexes).

It thus follows that, for any additive category ${\mathcal{A}}$, we can make ${\mathrm{Ch}(\mathcal{A})}$ enriched over ${\mathrm{Ch}(\mathbf{Ab})}$, and applying the truncation functor ${\tau_{\geq 0}}$, we can make ${\mathrm{Ch}(\mathcal{A})}$ enriched over ${\mathrm{Ch}_{\geq 0}(\mathbf{Ab})}$. In view of the Dold-Kan equivalence, we find:

Proposition 2 There is a natural structure on ${\mathrm{Ch}(\mathcal{A})}$ of a simplicial category.

This is exactly what we want from an ${\infty}$-category: we want a mapping space (i.e., a Kan complex or topological space) between any two objects. So we can take the homotopy coherent nerve of this, to get an ${\infty}$-category of chain complexes in ${\mathcal{A}}$, which I’ll write as ${\mathbf{N}( \mathrm{Ch}(\mathcal{A}))}$ to avoid confusion.

We can say a little more.

Proposition 3 ${\mathbf{N}(\mathrm{Ch}(\mathcal{A}))}$ is a stable ${\infty}$-category, whose suspension functor is given by shifting by ${1}$.

This is somewhat interesting—we get the classical shift functor out of the ${\infty}$-categorical context. Also, we get the classical fact that the homotopycategory of chain complexes is a triangulated category. (After localizing this at the quasi-isomorphisms, one gets to the classical derived category.)

How might we prove such a result? We’re going to try to understand what pushouts and pull-backs look like in ${\mathbf{N}(\mathrm{Ch}(\mathcal{A}))}$. We will make a simplifying assumption: ${\mathcal{A}}$ is a full subcategory of an abelian category ${\mathcal{T}}$, and ${\mathcal{A}}$ is idempotent complete and contains pull-backs under ${\mathcal{T}}$-epimorphisms. The particular examples we have in mind are either an abelian category or the category of projective (or injective) objects in an abelian category, so we could just restrict to those cases.

First, observe that chain homotopy equivalences are actually equivalences in ${\mathbf{N}(\mathrm{Ch}(\mathcal{A}))}$—in fact, we can think of the 2-morphisms in ${\mathbf{N}(\mathrm{Ch}(\mathcal{A}))}$ as being given by chain homotopies (between the 1-morphisms, which are ordinary morphisms of chain complexes).

So let’s consider a square in ${\mathbf{N}(\mathrm{Ch}(\mathcal{A}))}$.

This is, in particular, a homotopy commutative square of chain complexes, and by replacing it with a weakly equivalent square we may assume that it is commutative on the nose. We want to show that it is a homotopy push-out if and only if it is a homotopy pull-back. If we replace ${B_\bullet \rightarrow D_\bullet}$ by something up to homotopy, we can assume that it is degreewise split (e.g. replace it by the mapping cylinder). In this case, we have to show that the square is homotopy cartesian if and only if it is homotopy cocartesian.

However, the point is that being homotopy cartesian in an ${\infty}$-category can be tested on the level of hom-spaces. That is,

needs to be a homotopy pull-back of topological spaces for any ${T_\bullet \in \mathrm{Ch}_{\geq 0}}$. (Here ${\hom}$ means the simplicial hom.) Since ${B_\bullet \rightarrow D_\bullet}$ is a degreewise split surjection, one can show that the map ${\hom(T_\bullet, B_\bullet) \rightarrow \hom(T_\bullet, D_\bullet)}$ is a Kan fibration: this corresponds to the fact that a surjection of simplicial abelian groups is a Kan fibration. So another way of saying this is that we have a homotopy equivalence of Kan complexes

$\displaystyle \hom(T_\bullet, A_\bullet) \simeq\hom(T_\bullet, C_\bullet) \times_{\hom(T_\bullet, D_\bullet)} \hom(T_\bullet, B_\bullet),$

for every ${T_\bullet}$. This is equivalent to saying that the chain complexes of morphisms

$\displaystyle \underline{Hom}(T_\bullet, A_\bullet) \rightarrow \underline{Hom}(T_\bullet, C_\bullet) \times_{\underline{Hom}(T_\bullet, D_\bullet)} \underline{Hom}(T_\bullet, B_\bullet)$

is a weak equivalence (of chain complexes of abelian groups), for every ${T_\bullet}$: that is, by the Dold-Kan correspondence in nonnegative homological degrees, and by shifting in general.

Since ${\mathcal{A}}$ admits pull-backs under surjections, this is equivalent to the condition that the map

$\displaystyle A_\bullet \rightarrow C_\bullet \times_{D_\bullet} B_\bullet$

be a homotopy equivalence (i.e., an equivalence in this ${\infty}$-category). In other words, we find that ${C_\bullet \times_{D_\bullet} B_\bullet}$ is the homotopy pull-back. Replacing the square by an equivalent one, we may assume that this is true up to isomorphism: that is, that ${A \simeq B_\bullet \times_{C_\bullet} D_\bullet}$.

The condition that the analogous square be a push-out is that

$\displaystyle B_\bullet \sqcup_{A_\bullet} C_\bullet \rightarrow D_\bullet$

be a homotopy equivalence. But if ${B_\bullet \rightarrow C_\bullet}$ is surjective and ${A_\bullet \rightarrow C_\bullet \times_{D_\bullet} B_\bullet}$ is an isomorphism, then ${B_\bullet \sqcup_{A_\bullet} C_\bullet \rightarrow D_\bullet }$ is an isomorphism. So we find that a pull-back square is a push-out square. The converse is similar.

Technically, we should also show the existence of finite limits and colimits, and pointedness. But this reduces to the existence of coproducts and products (which is straightforward—they are as in the 1-categorical case) and the zero object is the zero complex.

2. Towards the derived ${\infty}$-category

The classical derived category of an abelian category ${\mathcal{A}}$ is usually constructed in two steps. First, one constructs the homotopy category of chain complexes in ${\mathcal{A}}$: the objects are chain complexes and morphisms are homotopy classes of maps. This is already a triangulated category, but it’s not quite what we want. Then, one inverts the quasi-isomorphisms to get the derived category, and shows that this localization process preserves the triangulatedness.

That’s not the best approach in this context: while formally adjoining inverses can be done in ${\infty}$-categories, the philosophy that seems to predominate is that we should find localizations as subcategories. This is the philosophy of Bousfield localization, in which one localizes the stable homotopy category at morphisms inducing isomorphisms in ${E}$-homology for some spectrum ${E}$. This localization is equivalent to the subcategory of “${E}$-local objects” in the original stable homotopy category.

More to the point, there is an alternative description of the classical derived category, valid when there are enough projectives:

Description: The derived category (bounded-below) of ${\mathcal{A}}$ is the homotopy category of the category of chain complexes of projectives in ${\mathcal{A}}$.

This is nice, because a quasi-isomorphism of projectives is automatically a homotopy equivalence. So there’s no “formal inversion” of morphisms necessary: one just restricts to a subcategory. This motivates the following definition:

Definition 4 (Lurie) Given an abelian category ${\mathcal{A}}$ with enough projectives, let ${\mathcal{P}}$ be the subcategory of projectives. Then we define the derived ${\infty}$-category

$\displaystyle D^-(\mathcal{A}) = \mathbf{N}( \mathrm{Ch}_{\gg - \infty}(\mathcal{A}))$

to be the nerve of the subcategory of ${\mathbf{N}( \mathrm{Ch}_{\gg - \infty}(\mathcal{A}))}$ consisting of bounded-below complexes.

It is evident from this definition (and unraveling of what the morphism spaces are in ${\mathbf{N}( \mathrm{Ch}(\mathcal{A}))}$) that the homotopy category of this coincides with the second description of the classical derived category.

The derived ${\infty}$-category, as stated above, has a powerful universal property, which makes it much easier to map out than its 1-categorical shadow, the classical derived category. In the next post, we’ll see how this works.