Nonabelian derived functors

• How nonabelian derived functors used to be constructed
• Presheaves preserving finite products
• Projective objects
• Abelian derived functors

It’s worth talking about how derived functors were originally constructed. Let $\mathcal{A}, \mathcal{B}$ be abelian categories, and let $F \colon \mathcal{A} \to \mathcal{B}$ is a functor. For $a \in \mathcal{A}$ an object, we can find an injective resolution $$ 0 \to a \to M_0 \to M_1 \to \dotsb. $$ Then we can apply $F$ and look at the cohomology.

How nonabelian derived functors used to be constructed#

Now what if $\mathcal{A}$ is not abelian? Let $\mathcal{A}^+$ be the preadditive category freely generated by $\mathcal{A}$, defined by $$ \mathcal{A}^+(a, b) = \mathbb{Z}[\mathcal{A}(a,b)]. $$

Definition 1. A class $\mathcal{M}$ of objects is a class of injective models if

• every object $a \in \mathcal{A}^+$ admits a resolution by $\mathcal{M}$, $$ 0 \to a \to M_0 \to M_1 \to \dotsb, $$
• and for every $N \in \mathcal{M}$, the sequence $$ 0 \leftarrow \Hom(a, N) \leftarrow \Hom(M_0, N) \leftarrow \dotsb $$ is exact.

Example 2. If $\mathcal{M}$ is the class of injective objects in an abelian category $\mathcal{A}^+$, then we get the normal derived functors.

Example 3. Assume $F \colon \mathcal{C} \to \mathcal{D}$ is a functor with right adjoint $G \colon \mathcal{D} \to \mathcal{C}$. Then for every object $c \in \mathcal{C}$, there is a cosimplicial resolution $$ X^n = (GF)^{n+1} c $$ where the unit map and counit map defines the face and degenracy maps. In many cases, the limit is going to recover $c$. If this is the case, then the image of $G$ in $C$ will form a class of injective models.

This is incredibly ad hoc, but this is what you had to do.

Presheaves preserving finite products#

In essence, in the abelian setting we doing is constructing a Kan extension $$ \begin{CD} \mathcal{C} _ \mathrm{inj} @>>> \mathcal{D} \br @VVV @VVV \br D(\mathcal{C}) @>>> D(\mathcal{D}) \end{CD} $$ and in the nonabelian setting, we are making a similar extension $$ \begin{CD} \mathcal{M} @>>> \mathcal{D} \br @VVV @VVV \br \mathcal{C}^\Delta @>>> D(\mathcal{D}). \end{CD} $$ We will try to do something similar.

Let’s work with right exact functors and left derived functors. Let $\mathcal{C}$ be an $\infty$-category, and denote by $\mathcal{P}(\mathcal{C})$ the $\infty$-category of presheaves on $\mathcal{C}$. Let $$ j \colon \mathcal{C} \hookrightarrow \mathcal{P}(\mathcal{C}) $$ be the Yoneda embedding. Recall the following fact.

Proposition 4. The category $\mathcal{P}(\mathcal{C})$ is the universal cocompletion of $\mathcal{C}$, meaning that if $\mathcal{D}$ is any cocomplete $\infty$-category with a functor $\mathcal{C} \to \mathcal{D}$, this uniquely extends to a functor $$ \mathcal{P}(\mathcal{C}) \to \mathcal{D} $$ preserving colimits.

Lemma 5 (HTT, Lemma 5.5.8.13). If $\mathcal{C}$ and $X \in \mathcal{P}(\mathcal{C})$, then there exists a simplicial object $Y \colon N(\Delta)^\mathrm{op} \to \mathcal{P}(\mathcal{C})$ such that

• the colimit of $Y_ \ast$ is equivalent to $X$,
• for $n \ge 0$, each $Y_n$ is a small coproduct of objects in the essential image of the Yoneda embedding.

Definition 6. Suppose $\mathcal{C}$ has finite coproducts. Then we define $$ \mathcal{P} _ \Sigma(\mathcal{C}) \hookrightarrow \mathcal{P}(\mathcal{C}) $$ as the full subcategory of functors preserving finite products.

Lemma 7. The Yoneda embedding factors through this inclusion.

Because $\mathcal{C}$ is small, we can just think of $j(C)$ as $\Hom(-, C)$.

Proposition 8 (HTT, Proposition 5.5.8.15). If $\mathcal{C}$ is an $\infty$-category with finite coproducts, and $\mathcal{D}$ is an $\infty$-category with filtered colimits and geometric realizations, denote by $$ \mathsf{Fun} _ \Sigma(\mathcal{P} _ \Sigma(\mathcal{C}), \mathcal{D}) $$ the full subcategory of functors preserving filtered colimits and geometric realizations. Then the functor $$ j^\ast \colon \mathsf{Fun} _ \Sigma(\mathcal{P} _ \Sigma(\mathcal{C}), \mathcal{D}) \to \mathsf{Fun}(\mathcal{C}, \mathcal{D}) $$ given by precomposition with $j$ is an equivalence. If $\mathcal{D}$ has finite products, then $g \colon \mathcal{P} _ \Sigma(\mathcal{C}) \to \mathcal{D}$ preserves small colimits if and only if $g \circ j$ preserves finite coproducts.

Here, geometric realization means colimits over diagrams like $$ N(\Delta)^\mathrm{op} \to \mathcal{C}. $$ This term is justified by the following fact: if you have a simplicial set, consider as a simplicial object in $\mathcal{S}$, and then take the colimit, this is really the geometric realization.

Projective objects#

Definition 9. Let $\mathcal{C}$ be an $\infty$-category that admits geometric realization of simplicial objects (resp., filtered colimits). Then $P \in \mathcal{C}$ is projective (resp., compact) if the functor corepresented by $P$ commutes with geometric realization (resp, filtered limits).

Definition 10. A collection $S$ of objects of $\mathcal{C}$ is a set of compact projective generators if

• everything in $S$ in compact projective,
• the full subcategory spanned by $S$ is essentially small,
• the category $\mathcal{C}$ is generated by $S$ under small colimits.

Example 11. The category $\mathcal{S}$ is compactly projectively generated by finite sets. This is because filtered colimits can get you to arbitrary discrete spaces, and then you can do geometric realization.

Proposition 12 (HTT, Proposition 5.5.8.22). Let $\mathcal{C}$ be an $\infty$-category with finite coproducts. and let $\mathcal{D}$ be an $\infty$-category with filtered colimits and geometric realization. Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor, and let $$ F \colon \mathcal{P} _ \Sigma(\mathcal{C}) \to \mathcal{D} $$ the left derived functor, meaning the functor $F$ satisfying $f = F \circ j$. Then $F$ is an equivalence if and only if

1. the functor $f$ is fully faithful and
2. the essential image of $f$ consists of compact projective objects,
3. the category $\mathcal{D}$ is generated by the essential image of $f$ under filtered colimits and geometric realization.

Proposition 13 (Dold–Kan, Quillen, etc.). Suppose $\mathcal{C}$ has enough compact projective objects. Then there is an equivalence $$ D(\mathcal{C}) \simeq \mathcal{P}^{\Delta^\mathrm{op}, \mathrm{filt}}(\mathcal{C}^\mathrm{comp,proj}) \simeq \mathcal{P} _ \Sigma(\mathcal{C}). $$ If $\mathcal{C}$ is an abelian category, this is $K(\mathcal{C}^\mathrm{comp,proj})$.

Abelian derived functors#

Let’s see how this works with abelian categories. Let $\mathcal{A}$ be an abelian $1$-category. The usual definition is that $P \in \mathcal{A}$ is projective if and only if $\Hom(P, -)$ is exact.

Lemma 14. An object $P \in \mathcal{A}$ is projective if and only if $\Hom(P, -)$ commutes with geometric realization of simplicial objects.

So Dold–Kan just gives a nonnegatively graded chain complexes in $\mathcal{A}$, and then the derived functor is a functor on that category.