Let ${\mathcal{A}}$ be an abelian category with enough projectives. In the previous post, we described the definition of the derived ${\infty}$-category ${D^-(\mathcal{A})}$ of ${\mathcal{A}}$. As a simplicial category, this consisted of bounded-below complexes of projectives, and the space of morphisms between two complexes ${A_\bullet, B_\bullet}$ was obtained by taking the chain complex of maps ${\underline{Hom}(A_\bullet, B_\bullet)}$ between ${A_\bullet, B_\bullet}$ and turning that into a space (by truncation ${\tau_{\geq 0}}$ and the Dold-Kan correspondence).

Last time, we proved most of the following result:

Theorem 5 ${D^-(\mathcal{A})}$ is a stable ${\infty}$-category whose suspension functor is given by shifting by ${1}$. ${D^-(\mathcal{A})}$ has a ${t}$-structure whose heart is ${\mathcal{A}}$, and the homotopy category of ${D^-(\mathcal{A})}$ is the usual derived category.

Note for instance that this means that ${\mathcal{A}}$ sits as a full subcategory inside ${D^-(\mathcal{A})}$: that is, there is a full subcategory ${{D}^-(\mathcal{A})^{\heartsuit}}$ (the “heart”) of ${D^-(\mathcal{A})}$ (spanned by those complexes homologically concentrated in degree zero).

This heart has the property that the mapping spaces in ${D^-(\mathcal{A})^{\heartsuit}}$ are discrete, and the functor

$\displaystyle \pi_0: D^-(\mathcal{A}) \rightarrow \mathcal{A}$

restricts to an equivalence ${D^-(\mathcal{A})^{\heartsuit} \rightarrow \mathcal{A}}$; one can prove this by examining the chain complex of maps between two complexes homologically concentrated in degree zero. The inverse to this equivalence runs ${\mathcal{A} \rightarrow D^-(\mathcal{A})^{\heartsuit}}$, and it sends an element of ${\mathcal{A}}$ to a projective resolution. This is functorial in the ${\infty}$-categorical sense.

Most of the above theorem is exactly the same as the description of the ordinary derived category of ${\mathcal{A}}$ (i.e., the homotopy category of ${D^-(\mathcal{A})}$), The goal of this post is to describe what’s special to the ${\infty}$-categorical setting: that there is a universal property. I will start with the universal property for the subcategory ${D_{\geq 0}(\mathcal{A})}$.

Theorem 6 ${D_{\geq 0}(\mathcal{A})}$ is the ${\infty}$-category obtained from ${\mathcal{P} \subset \mathcal{A}}$ (the projective objects) by freely adding geometric realizations.

The purpose of this post is to sketch a proof of the above theorem, and to explain what it means.

1. Freely adding colimits

Let ${\mathcal{C}}$ be an ordinary category, and ${\mathcal{D}}$ a cocomplete category. Let ${P(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{op}, \mathbf{Sets})}$ be the category of presheaves on ${\mathcal{C}}$. Then it is a classical fact that we have an equivalence of categories

$\displaystyle \mathrm{Fun}^{L}( P(\mathcal{C}), \mathcal{D}) \simeq \mathrm{Fun}(\mathcal{C}, \mathcal{D}) ;$

that is, to give a functor ${\mathcal{C}\rightarrow \mathcal{D}}$ is equivalent to giving a colimit-preserving (equivalently, left adjoint) functor ${P(\mathcal{C}) \rightarrow \mathcal{D}}$. In other words, ${P(\mathcal{C})}$ is the free cocompletion of ${\mathcal{C}}$, obtained by adding all colimits freely. The Yoneda embedding

$\displaystyle \mathcal{C} \rightarrow P(\mathcal{C})$

is the universal imbedding of ${\mathcal{C}}$ into a cocomplete category.

There is an analogous result in the ${\infty}$-categorical setting. Let ${\mathcal{C}}$ be an ${\infty}$-category. We can define the analogous ${\infty}$-category

$\displaystyle P(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{op}, \mathcal{S}),$

where ${\mathcal{S}}$ is the ${\infty}$-category of spaces. Then, if ${\mathcal{D}}$ is any cocomplete ${\infty}$-category, there is an equivalence of ${\infty}$-categories,

$\displaystyle \mathrm{Fun}^L(P(\mathcal{C}), \mathcal{D}) \simeq \mathrm{Fun}(\mathcal{C}, \mathcal{D})$

between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and colimit-preserving functors ${P(\mathcal{C}) \rightarrow \mathcal{D}}$.

We will need a variant of the above construction, when we freely add only some colimits rather than all of them. For simplicity, we state the next result for geometric realizations.

Proposition 7 Let ${\mathcal{C}}$ be an ${\infty}$-category. Then there exists an ${\infty}$-category ${\mathcal{C}'}$ admitting geometric realizations, together with an inclusion ${\mathcal{C} \rightarrow \mathcal{C}'}$ with the property that for any ${\infty}$-category ${\mathcal{D}}$ admitting geometric realizations, we have an equivalence

$\displaystyle \mathrm{Fun}^{\prime}(\mathcal{C}', \mathcal{D}) \simeq \mathrm{Fun}(\mathcal{C}, \mathcal{D})$

between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and functors ${\mathcal{C}' \rightarrow \mathcal{D}}$ that preserve geometric realizations.

The strategy is to start by taking ${\mathcal{C}'}$ to be the smallest sub${\infty}$-category of ${P(\mathcal{C})}$ which contains the image of the Yoneda imbedding ${\mathcal{C} \rightarrow P(\mathcal{C})}$ and which is closed under taking geometric realizations. Then, clearly ${\mathcal{C}'}$ contains ${\mathcal{C}}$; to complete the proof, one has to show that a functor

$\displaystyle \mathcal{C}' \rightarrow \mathcal{D}$

is a left Kan extension of its restriction to ${\mathcal{C}}$ if and only if it preserves geometric realizations. (“Kan extension” seems to be one of the magic words one has to throw around in this business, to make the proofs work efficiently.) In fact, one has an equivalence between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and left Kan extensions ${\mathcal{C}' \rightarrow \mathcal{D}}$, provided that left Kan extensions exist.

So let’s prove this precisely:

1. A functor ${\mathcal{C} \rightarrow \mathcal{D}}$ admits a left Kan extension (if ${\mathcal{D}}$ admits geometric realizations).
2. A functor ${\mathcal{C}' \rightarrow \mathcal{D}}$ is a left Kan extension of its restriction to ${\mathcal{C}}$ precisely if it preserves geometric realizations.

How can we prove this? The strategy, for 1, to show that the left Kan extension exists, is to note that it would exist perfectly well if we knew ${\mathcal{D}}$ was cocomplete. But we can find a colimit-preserving imbedding

$\displaystyle \mathcal{D} \rightarrow \overline{\mathcal{D}}$

into a cocomplete ${\infty}$-category ${\overline{\mathcal{D}}}$ (for instance, the opposite to the Yoneda imbedding), and any functor ${f: \mathcal{C} \rightarrow \mathcal{D}}$ admits a left Kan extension to ${F: \mathcal{C}'\rightarrow\overline{ \mathcal{D}}}$. But, since ${\mathcal{C}'}$ is generated under geometric realizations from ${\mathcal{C}}$, one can then check that ${F(\mathcal{C}')}$ is contained (up to isomorphism) in ${\mathcal{D} \subset \overline{\mathcal{D}}}$ itself. This means that the left Kan extension exists, and in fact is a special case of a left Kan extension

$\displaystyle P(\mathcal{C}) \rightarrow \overline{ \mathcal{D}}.$

In particular, we’ve established the first step. Observe that the Kan extension constructed in this way preserves geometric realizations, because the functor ${P(\mathcal{C}) \rightarrow \overline{\mathcal{D}}}$ was constructed so as to preserve colimits. (This is part of the yoga of the Yoneda lemma: Yoneda extensions are left Kan extensions, and preserve colimits.)

Now we need to establish the second step. That is, if ${\mathcal{C}' \rightarrow \mathcal{D}}$ is any functor preserving geometric realizations, then it is a left Kan extension of its restriction to ${\mathcal{C}}$. We can do this by observing that a left Kan extension of its restriction exists, by the previous step, and said left Kan extension preserves geometric realizations.

I’ve sort of sketched this business; it’s in 5.3 of “Higher Topos Theory.” I’m not sure how much going into these details would be relevant to a blog setting.

Definition 8 Given an ${\infty}$-category ${\mathcal{C}}$, we let ${P_{\Sigma}(\mathcal{C})}$ be the ${\infty}$-category obtained by freely adding geometric realizations, as above.

2. Proof of the universal property

With the general formalism done, we can turn to the main goal, which was to say that ${D_{\geq 0}(\mathcal{A})}$ was obtained by adding geometric realizations to the ${\infty}$-category (even ordinary category) of projectives ${\mathcal{P} \subset \mathcal{A}}$.

How might we prove this? Roughly, the intuition is that ${D_{\geq 0}(\mathcal{A})}$ consists of nonnegatively graded chain complexes with elements in ${\mathcal{P}}$ and these are obtained from simplicial objects by Dold-Kan.

To make it precise, let’s first construct a functor. One can show that ${D_{\geq 0}(\mathcal{A})}$ admits geometric realizations. Let’s accept this—the idea is that it certainly admits finite colimits (being stable) but taking higher and higher stages of a geometric realization only affect the “tail end” (the connectivity increases), so ${D_{\geq 0}(\mathcal{A})}$ turns out to have geometric realizations. (This is a “left completeness” condition.)

Anyway, there is certainly an imbedding

$\displaystyle \mathcal{P} \hookrightarrow D_{\geq 0}(\mathcal{A}),$

and since the latter ${\infty}$-category admits geometric realizations, we get a geometric-realization-preserving functor

$\displaystyle F: P_{\Sigma}(\mathcal{P}) \rightarrow D_{\geq 0}(\mathcal{A}),$

which we have to prove is an equivalence.

To prove that ${F}$ is an equivalence, we have to show that ${F}$ is fully faithful and essentially surjective. Let’s start with full faithfulness. Certainly ${F|_{\mathcal{P}}}$ is fully faithful—${\mathcal{P}}$ is a full subcategory of ${D_{\geq 0}(\mathcal{A})}$.

Proposition 9 ${F: P_{\Sigma}(\mathcal{P}) \rightarrow D_{\geq 0}(\mathcal{A})}$ is fully faithful.

Proof: Let’s start by settling a special case of full faithfulness.

Given an object ${x \in \mathcal{P}}$, we can ask for the collection of ${y \in P_{\Sigma}(\mathcal{P})}$ such that

$\displaystyle F: \hom_{P_{\Sigma}}(x,y) \rightarrow \hom_{D_{\geq 0}(\mathcal{A})}(Fx, Fy)$

is an equivalence. As we’ve seen, this contains all ${y \in \mathcal{P}}$. The idea is that ${x}$ is a “projective” object, so this collection of ${y}$ will be stable under geometric realizations, and thus contain all of ${P_{\Sigma}(\mathcal{P})}$. If we can prove it, then we will be done with full faithfulness: in fact, the collection of all ${x}$ for which

$\displaystyle \hom_{P_{\Sigma}}(x,y) \rightarrow \hom_{D_{\geq 0}(\mathcal{A})}(Fx, Fy)$

is an equivalence (for all ${y}$) is closed under geometric realizations.

Why is this collection of objects ${y}$ stable? We have to show that the functors into spaces ${\hom_{P_\Sigma(\mathcal{P})}(x, \cdot)}$ and ${\hom_{D_{\geq 0}(\mathcal{A})}(Fx, \cdot)}$ commute with geometric realizations. In ${P_{\Sigma}(\mathcal{P})}$ (which is contained in a presheaf category), that’s a special case of the Yoneda principle: homming out of a representable functor commutes with colimits in the presheaf category (e.g., geometric realizations), because homming out of a representable functor is equivalently evaluation. In the case of ${Fx \in D_{\geq 0}(\mathcal{A})}$, it is a little more subtle.

The point is that ${Fx}$ is concentrated in degree zero: it is the complex with some (projective) ${X}$ in degree zero and zero elsewhere. Given a complex ${Y_\bullet}$, we have that the internal (chain complex) homs are given by

$\displaystyle \underline{Hom}(Fx, Y_\bullet)_n = \hom_{\mathcal{A}}(X, Y_n).$

Consider now a simplicial object ${Q_\bullet}$ in ${D_{\geq 0}(\mathcal{A})}$. For simplicity, assume that it is a simplicial object in the 1-category of chain complexes (i.e., on the nose, not up to coherent homotopy); this can always be arranged suitably, possibly by changing the category up to equivalence. So ${Q_\bullet}$ is really a simplicial chain complex, or a double chain complex ${Q_{\bullet, \bullet}}$. The geometric realization, spaces or chain complexes, corresponds to taking the total complex. But this is all preserved under taking ${\hom(X, \cdot)_\cdot}$. $\Box$

The above proof was somewhat handwavy. One key point near the end is that the functor from nonnegatively graded chain complexes (i.e., simplicial abelian groups) to spaces preserves geometric realizations. This is an illustration of the fact that simplicial abelian groups are monadic over spaces.

Anyway, the point of the above argument is that we get a fully faithful functor

$\displaystyle P_{\Sigma}(\mathcal{P}) \rightarrow D_{\geq 0}(\mathcal{A}),$

and the last thing to check is that it is essentially surjective. But this is the Dold-Kan correspondence. Given an object in ${D_{\geq 0}(\mathcal{A})}$, it is a chain complex of projectives. Dold-Kan allows us to say that it corresponds to a simplicial object in ${\mathcal{P}}$, and in fact the object is a geometric realization of that simplicial object. So ${D_{\geq 0}(\mathcal{A})}$ is generated under geometric realizations by ${\mathcal{P}}$, which means that

$\displaystyle P_{\Sigma}(\mathcal{P}) \rightarrow D_{\geq 0}(\mathcal{A})$

is essentially surjective, and hence an equivalence.