t-structures and Dold–Kan

• t-structures on a triangulated categories
• t-structures on ∞-categories
• The Dold–Kan correspondence

Let’s start with t-structures on triangulated categories.

t-structures on a triangulated categories#

We use the “incorrect” grading convention, because complexes are homological so the maps go $A_n \to A_{n-1}$. But at least the shift $[1]$ is correct.

Definition 1. Let $\mathcal{D}$ be a triangulated category. A t-structure on $\mathcal{D}$ is a pair of full subcategories $(\mathcal{D}_ {\le 0}, \mathcal{D}_ {\ge 0})$, satisfying

• if $X \in \mathcal{D}_ {\ge 1}$ and $Y \in \mathcal{D}_ {\le 0}$ then $\Hom_\mathcal{D}(X, Y) = 0$,
• $\mathcal{D}_ {\ge 1} \subseteq \mathcal{D}_ {\ge 0}$ and $\mathcal{D}_ {\le 0} \subseteq \mathcal{D}_ {\le 1}$,
• for every $X$ there exists a decomposition $$ X^\prime \to X \to X^{\prime\prime} \to X[1] $$ such that $X^\prime \in \mathcal{D}_ {\ge 1}$ and $X^{\prime\prime} \in \mathcal{D}_ {\le 0}$.

Definition 2. For $(\mathcal{D}_ {\ge 0}, \mathcal{D}_ {\le 0})$ a t-structure on a triangulated category $\mathcal{D}$, we define its heart as $$ \mathcal{D}^\heartsuit = \mathcal{D}_ {\le 0} \cap \mathcal{D}_ {\ge 0}. $$

Proposition 3. $\mathcal{D}^\heartsuit$ is necessarily abelian.

We won’t prove this proposition, but this is fun to see.

Example 4. If $\mathcal{A}$ is an abelian category, the derived category $\mathcal{D}^\flat(\mathcal{A})$ has a natural t-structure given by, $X \in \mathcal{D}_ {\ge 0}$ if and only if $H_i(X) = 0$ for all $i \lt 0$ and $X \in \mathcal{D}_ {\le 0}$ if and only if $H_i(X) = 0$ for all $i \gt 0$.

Why is this a t-structure? Let’s first show that if $X \in \mathcal{D}_ {\ge 1}$ and $Y \in \mathcal{D}_ {\le 0}$ then any map $X \to Y$ is zero. This is not so obvious, because the other direction is not true: we have $$ \Hom_ {\mathcal{D}(\mathbb{Z})} (\mathbb{Z}/2\mathbb{Z}[0], \mathbb{Z}[1]) = \Ext_\mathbb{Z}^1(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}. $$ So why is this fact true? The point is that when we try to choose an injective resolution $Y \to I$, we can do this so that $I_ {\le 0} = 0$ at the level of terms. Then we see that $$ \Hom_ \mathcal{D}(X, Y) = \Hom_ {K(\mathcal{A})}(X, I) = 0. $$

The second condition is pretty obvious. Now let’s think about how truncation works. Given $X$ a complex, we define $$ \tau_ {\ge 1} X = \dotsb \to X_2 \to \ker d_1 \to 0 \to 0 \to \dotsb $$ and $$ \tau_ {\ge 1} X = \dotsb \to 0 \to 0 \to \coker d_1 \to X_{-1} \to \dotsb. $$

We note that the homology functor is given by $$ H_n = \tau_ {\ge n} \tau_ {\le n} = \tau_ {\le n} \tau_ {\ge n}. $$ Then we see that $$ H_0 \colon \mathcal{D} \to \mathcal{D}_ {\le 0} \cap \mathcal{D}_ {\ge 0} = \mathcal{D}^\heartsuit. $$ In fact, we see that $\mathcal{D}^\heartsuit$ is just the original abelian category $\mathcal{A}$.

Remark 5. It is not true that $\mathcal{D} = \mathcal{D}(\mathcal{D}^\heartsuit)$ in general.

Proposition 6. Let $(\mathcal{D}_ {\ge 0}, \mathcal{D}_ {\le 0})$ be a t-structure.

1. $\mathcal{D}_ {\ge n} \hookrightarrow \mathcal{D}$ has a right adjoint $\tau_{\ge n} \colon \mathcal{D} \to \mathcal{D}_ {\ge n}$, so there is a canonical morphism $\tau_ {\ge n} X \to X$.
2. Dually, $\mathcal{D}_ {\le n} \to \mathcal{D}$ has a left adjoint $\tau_ {\le n} \colon \mathcal{D} \to \mathcal{D}_ {\le n}$, so there is a canonical morphism $X \to \tau_ {\le n} X$.
3. There is a natural transformation $\delta \colon \tau_ {\le n} \to \tau_ {\ge n+1}(-) [1]$ such that for every $X \in \mathcal{D}$ the sequence $$ \tau_ {\ge n+1} X \to X \to \tau_ {\le n} X \xrightarrow{\delta} (\tau_ {\ge n+1} X)[1] $$ is a distinguished triangle.
4. $\tau_ {\ge n}$ is just $[n] \circ \tau_ {\ge 0} \circ [-n]$.

t-structures on ∞-categories#

This is not so bad, because a t-structure on an ∞-category on an ordinary category. Recall that the homotopy category of a stable category is naturally a triangulated category.

Definition 7. If $\mathcal{C}$ is a stable ∞-category, a t-structure on $\mathcal{C}$ is a t-structure on $\operatorname{h}\mathcal{C}$. We then set $\mathcal{C}_ {\ge 0}$ to denote the full subcategory of objects spanned by those landing in $(\operatorname{h}\mathcal{C})_ {\ge 0}$, and $\operatorname{C}_ {\le 0}$ similarly.

Proposition 8. Let $\mathcal{C}$ be a stable ∞-category with t-structure. Then subcategories $\mathcal{C}_ {\le n} \hookrightarrow \mathcal{C}$ are localizations, i.e., has a right adjoint.

Corollary 9. For a t-structure, $\mathcal{C}_ {\le 0}$ are stable under limits (existing in $\mathcal{C}$), and similarly $\mathcal{C}_ {\ge 0}$ are stable under colimits (existing in $\mathcal{C}$).

Proposition 10. The natural transformation $\theta \colon \tau_ {\le m} \tau_ {\ge n} \to \tau_ {\ge n} \tau_ {\le m}$ is an equivalence.

Definition 11. We define $\mathcal{C}^\heartsuit = \mathcal{C}_ {\le 0} \cap \mathcal{C}_ {\ge 0}$ and the functors $$ \pi_n = \tau_ {\le 0} \tau_ {\ge 0} [-n] \colon \mathcal{C} \to \mathcal{C}^\heartsuit. $$

Note that if $X, Y \in \mathcal{C}^\heartsuit$ then we have $$ \pi_n \operatorname{Map}_ \mathcal{C}(X, Y) = \Ext_ {\operatorname{h}\mathcal{C}}^{-n}(X, Y) = 0 $$ for all $n \gt 0$. This shows that $\mathcal{C}^\heartsuit$ is an ordinary category. Then the functor $$ \mathcal{C}^\heartsuit \to (\operatorname{h}\mathcal{C})^\heartsuit $$ is an equivalence, so $\mathcal{C}^\heartsuit$ is a good old abelian category.

The Dold–Kan correspondence#

The proof in Lurie is quite clever. Let $\mathcal{A}$ be an abelian category. There is a category of chain complexes $$ \operatorname{Ch}(\mathcal{A})_ {\ge 0} $$ whose negative degree objects are literally zero.

Definition 12. For a nonnegatively supported chain complex $A \in \operatorname{Ch}(\mathcal{A})_ {\ge 0}$, we define a simplicial object $\operatorname{DK}(A)$ as

• $\operatorname{DK}(A) = \bigoplus_{\alpha \colon [n] \twoheadrightarrow [k]} A_k$,
• $\beta \colon [n^\prime] \to [n]$ in $\Delta$ induces a map $\beta^\ast \colon \operatorname{DK}_ n(A) \to \operatorname{DK}_ {n^\prime}(A)$ given by $f_{\alpha,\alpha^\prime} \colon A_k \to A_{k^\prime}$ defined as follows:
  • if $k = k^\prime$ and $\alpha \circ \beta = \alpha^\prime$ then $f_{\alpha,\alpha^\prime} = \id$;
  • if $k^\prime = k-1$ and $\alpha \circ \beta = \delta^0 \circ \alpha^\prime$ then $f_{\alpha,\alpha^\prime} = d_k$;
  • otherwise, $f_{\alpha,\alpha^\prime} = 0$.

This defines a functor $$ \operatorname{DK} \colon \operatorname{Ch}(\mathcal{A})_ {\ge 0} \to \operatorname{Fun}(\Delta^\mathrm{op}, \mathcal{A}). $$

Theorem 13 (Dold–Kan). If $\mathcal{A}$ is an abelian category (even just an additive category that is idempotently complete), then the functor $$ \operatorname{DK} \colon \operatorname{Ch}(\mathcal{A})_ {\ge 0} \to \operatorname{Fun}(\Delta^\mathrm{op}, \mathcal{A}) $$ is an equivalence.

Idempotently complete just means that idempotent maps have kernels and cokernels. We will prove this by writing down an inverse.

Definition 14. If $\mathcal{A}$ is abelian and $A_\bullet \colon \Delta^\mathrm{op} \to \mathcal{A}$ is a simplicial object, then we define a chain complex $N_\bullet(A) \in \operatorname{Ch}(\mathcal{A})_ {\ge 0}$ as $$ N_n(A) = \ker \biggl( A_n \to \bigoplus_ {1 \le i \le n} A_{n-1} \biggr) $$ defined by $\lbrace d_i \rbrace_ {1 \le i \le n}$. The differential is given by $$ d_0 \colon N_n(A) \to N_ {n-1}(A). $$

Let’s compute one direction of this. What happens if we take the normalized chain complex of $\operatorname{DK}(A)$. The claim is that $$ N_n(\operatorname{DK}(A)) = A_n. $$ The left hand side is given by the kernel of $$ \bigoplus_{\alpha \colon [n] \twoheadrightarrow [k]} A_k \to \bigoplus_ {1 \le i \le n} \bigoplus_{\alpha^\prime \colon [n-1] \twoheadrightarrow [k]} A_{k^\prime}. $$ Note that for every $\alpha \colon [n] \twoheadrightarrow [k]$ with $k \lt n$, there exists a $i \neq 0$ such that the composition $$ [n-1] \xrightarrow{\delta^i} [n] \to [k] $$ is surjective. So everything other than $\alpha = \id$ doesn’t show up in the kernel. Then we can check that the differential is the differential. The conclusion is that $$ N_\bullet(\operatorname{DK}(A)) \to A $$ is an isomorphism.

The other direction doesn’t require us to do any computation.

Proposition 15. Assume $\mathcal{A}$ is an abelian category. Then $$ u \colon \id \to N_\bullet \circ \operatorname{DK} $$ is the unit of an adjunction.

Here is something we can show for abelian groups.

Lemma 16. We have $\operatorname{DK} \circ N_\ast = \mathrm{id}$ for $\mathcal{A} = \mathsf{Ab}$.

Now here is the clever step. The idea is to embed $\mathcal{A}$ in the presheaf category on $\mathcal{A}$. Write $\mathcal{A}^\prime = \operatorname{Fun}(\mathcal{A}^\mathrm{op}, \mathsf{Ab})$, and there is a functor $$ j \colon \mathcal{A} \to \mathcal{A}^\prime; \quad A \mapsto \Hom(-, A) $$ that is fully faithful. Look at the following diagram. $$ \begin{CD} \operatorname{Ch}(\mathcal{A})_ {\ge 0} @>>> \operatorname{Fun}(\Delta^\mathrm{op}, \mathcal{A}) \br @VVV @VVV \br \operatorname{Ch}(\mathcal{A}^\prime)_ {\ge 0} @>>> \operatorname{Fun}(\Delta^\mathrm{op}, \mathcal{A}^\prime) \end{CD} $$ Now the bottom horizontal is an equivalence by checking on values. The vertical maps are fully faithful, so we see that the upper horizontal $\operatorname{DK}_ \mathcal{A}$ is fully faithful.

What remains to show is, given any $A_\bullet \colon \Delta^\mathrm{op} \to \mathcal{A}$ a simplicial object, that $N_\ast(j(A))$ is in the essential image of $j$. This follows from the fact that $\mathcal{A}$ is idempotent complete.