用户: Scavenger/Infinity Category and Higher Algebra/Stable Infinity Categories

1Basic Definitions and Properties

Notion 1.1. A zero object of an $\infty$-category is an object that is simultaneously initial and final. An $\infty$-category is called pointed if it admits a zero object. We often denote the zero object by $0$.

Let $\mathcal{C}$ be a pointed $\infty$-category. A triangle in $\mathcal{C}$ is a commutative square

in $\mathcal{C}$ such that $Z$ is a zero object of $\mathcal{C}$. A triangle is a fiber sequence if it is a pullback square, and a cofiber sequence if it is a pushout square.

Let $X\to Y$ be a morphism in $\mathcal{C}$. A fiber of it is a fiber sequence $W\to X\to Y$, and a cofiber of it is a cofiber sequence $X\to Y\to Z$.

Definition 1.2. An $\infty$-category $\mathcal{C}$ is stable if it is pointed, every morphism admits a fiber and a cofiber, and a triangle in $\mathcal{C}$ is a fiber sequence if and only if it is a cofiber sequence.

Notion 1.3. In a stable $\infty$-category $\mathcal{C}$ we have the notions of the loop and suspension functors as functors $\mathcal{C}\to \mathcal{C}$, defined by the fiber (equivalently, cofiber) sequences $X\to 0\to \Sigma X$ and $\Omega X\to 0\to X$, respectively. These two functors are inverses to each other.

If $n\in {\mathbb{Z}}_{\ge 0}$ we let $X\to X[n]$ denote the $n$th power of the suspension functor, and if $n\in {\mathbb{Z}}_{\le 0}$ we let it denote the $(-n)$th power of the loop functor.

Proposition 1.4. Let $\mathcal{C}$ be a stable $\infty$-category, then the homotopy category $h\mathcal{C}$ is additive, and $X\to X[1]$ gives $h\mathcal{C}$ a structure of a triangulated category, where $X\to Y\to Z\to X[1]$ is a distinguished triangle if and only if there is a commutative diagram

in $\mathcal{C}$ such that the two squares are pushout squares, and $X\to Y\to Z\to X[1]$ is isomorphic to $X^{\prime }\to Y^{\prime }\to Z^{\prime }\to X^{\prime }[1]$ in $h\mathcal{C}$.

Proposition 1.5. Let $\mathcal{C}$ be a stable $\infty$-category and $K$ a simplicial set, then $\mathrm{Fun}(K,\mathcal{C})$ is stable.

Proposition 1.6. A stable $\infty$-category admits finite limits and colimits, and a commutative square in which is a pullback square if and only if it is a pushout square.

Proposition 1.7. Let $F:\mathcal{C}\to {\mathcal{C}}^{\prime }$ be a functor between stable $\infty$-categories, then it preserves finite limits if and only if it preserves finite colimits if and only if it brings zero objects to zero objects and brings fiber sequences to fiber sequences.

2Homological Algebra

Definition 2.1. Let $\mathcal{D}$ be a triangulated category. A $t$-structure on $\mathcal{D}$ is a pair of full subcategories ${\mathcal{D}}_{\ge 0},{\mathcal{D}}_{\le 0}\subseteq \mathcal{D}$ stable under isomorphisms, such that

(i) For any $X\in {\mathcal{D}}_{\ge 0}$ and $Y\in {\mathcal{D}}_{\le 0}$, we have ${\mathrm{Mor}}_{\mathcal{D}}(X,Y[-1])=0$.

(ii) ${\mathcal{D}}_{\ge 0}[1]\subseteq {\mathcal{D}}_{\ge 0}$, ${\mathcal{D}}_{\le 0}[-1]\subseteq {\mathcal{D}}_{\le 0}$.

(iii) For any $X\in \mathcal{D}$ there exists a distinguished triangle $X^{\prime }\to X\to X^{\prime \prime }\to$ such that $X^{\prime }\in {\mathcal{D}}_{\ge 0}$ and $X^{\prime \prime }\in {\mathcal{D}}_{\le 0}[-1]$.

In this situation for any $n\in \mathbb{Z}$ we write ${\mathcal{D}}_{\ge n}$ for ${\mathcal{D}}_{\ge 0}[n]$ and ${\mathcal{D}}_{\le n}$ for ${\mathcal{D}}_{\le 0}[n]$.

${\mathcal{D}}_{\ge 0}$ and ${\mathcal{D}}_{\le 0}$ uniquely determine each other, by the orthogonality condition of (i).

Definition 2.2. Let $\mathcal{C}$ be a stable $\infty$-category. A $t$-structure on $\mathcal{C}$ is a $t$-structure on $h\mathcal{C}$.

2.3. Let $\mathcal{C}$ be a stable $\infty$-category, equipped with a $t$-structure. Then for any $n\in \mathbb{Z}$, ${\mathcal{C}}_{\le n}$ is a reflective subcategory and ${\mathcal{C}}_{\ge n}$ is a coreflective subcategory. We let ${\tau }_{\le n}$ denote a left adjoint to the inclusion ${\mathcal{C}}_{\le n}\subseteq \mathcal{C}$ and ${\tau }_{\ge n}$ denote a right adjoint to the inclusion ${\mathcal{C}}_{\ge n}\subseteq \mathcal{C}$.

Both the functors ${\tau }_{\le n},{\tau }_{\ge n}$ map ${\mathcal{C}}_{\le m}$ and ${\mathcal{C}}_{\ge m}$ into themselves, where $m\in \mathbb{Z}$ is an arbitrary integer.

For any $X\in \mathcal{C}$ we have a fiber sequence ${\tau }_{\ge n}X\to X\to {\tau }_{\le n-1}X$.

2.4. Let $\mathcal{C}$ be a stable $\infty$-category equipped with a $t$-structure, let $n,m\in \mathbb{Z}$. We have a natural transformation of functors ${\tau }_{\le m}\circ {\tau }_{\ge n}\to {\tau }_{\ge n}\circ {\tau }_{\le m}$ obtained as follows: using the adjointness of ${\tau }_{\le m}$ this transformation is equivalent to a transformation ${\tau }_{\ge n}\to {\tau }_{\ge n}\circ {\tau }_{\le m}$, which is obtained by applying ${\tau }_{\ge n}$ to the adjunction map $\mathrm{id}\to {\tau }_{\le m}$. This functor is an equivalence.

Definition 2.5. Let $\mathcal{C}$ be a stable $\infty$-category equipped with a $t$-structure. The heart ${\mathcal{C}}^{♡}$ of $\mathcal{C}$ is the full subcategory ${\mathcal{C}}_{\ge 0}\cap {\mathcal{C}}_{\le 0}\subseteq \mathcal{C}$. For each $n\in \mathbb{Z}$ we let ${\pi }_0:\mathcal{C}\to {\mathcal{C}}^{♡}$ denote the functor ${\tau }_{\le 0}\circ {\tau }_{\ge 0}\cong {\tau }_{\ge 0}\circ {\tau }_{\le 0}$, and we let ${\pi }_n:\mathcal{C}\to {\mathcal{C}}^{♡}$ denote the composition ${\pi }_0\circ [-n]$.

Proposition 2.6. The $\infty$-category ${\mathcal{C}}^{♡}$ is equivalent to the nerve of its homotopy category $h{\mathcal{C}}^{♡}$, which is an abelian category.

The functor ${\pi }_0$ is a homological functor, which means that for any fiber sequence $X\to Y\to Z$ in $\mathcal{C}$, the sequence ${\pi }_{0}X\to {\pi }_{0}Y\to {\pi }_{0}Z$ is an exact sequence in ${\mathcal{C}}^{♡}$.

Definition 2.7. Let $\mathcal{C}$ be a stable $\infty$-category. A $t$-structure on $\mathcal{C}$ is said to be nondegenerate if for any $X\in \mathcal{C}$ such that ${\pi }_{n}X=0$ for any $n\in \mathbb{Z}$, we have $X=0$.

If this is the case, then ${\mathcal{C}}_{\ge 0}$ consists of elements $X\in \mathcal{C}$ with vanishing ${\pi }_n$ for $n\in {\mathbb{Z}}_{<0}$, and similarly for ${\mathcal{C}}_{\le 0}$.

A $t$-structure is nondegenerate if and only if ${\bigcap }_{n\in \mathbb{Z}}{\mathcal{C}}_{\ge n}$ and ${\bigcap }_{n\in \mathbb{Z}}{\mathcal{C}}_{\le n}$ are both equal to the full subcategory of zero objects.

3Stablization and Spectra

Definition 3.1. Let $F:\mathcal{C}\to \mathcal{D}$ be a functor between $\infty$-categories. If $\mathcal{C}$ admits pushouts, then $F$ is said to be excisive if it brings pushout squares in $\mathcal{C}$ to pullback squares in $\mathcal{D}$. If $\mathcal{C}$ admits a final object, then $F$ is said to be reduced if it preserves final objects.

If $\mathcal{C}$ admits pushouts, we use $\mathrm{Exc}(\mathcal{C},\mathcal{D})$ to denote the full subcategory of $\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by the excisive functors. If $\mathcal{C}$ admits a final object, we use ${\mathrm{Fun}}_{\ast }(\mathcal{C},\mathcal{D})$ to denote the full subcategory of $\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by the reduced functors. If $\mathcal{C}$ admits pushouts and a final object, we use ${\mathrm{Exc}}_{\ast }(\mathcal{C},\mathcal{D})$ to denote the full subcategory of $\mathrm{Fun}(\mathcal{C},\mathcal{D})$ spanned by the functors that are excisive and reduced.

Notation 3.2. Let ${\mathcal{S}}_{\ast }$ denote the category of pointed objects of $\mathcal{S}$, that is, ${\mathcal{S}}_{\ast }$ is the full subcategory of $\mathrm{Fun}({\Delta }^1,\mathcal{S})$ spanned by those morphisms $X\to Y$ such that $X$ is a final object of $\mathcal{S}$. Let ${\mathcal{S}}^{\mathrm{fin}}$ denote the smallest replete full subcategory of $\mathcal{S}$ which contains the final object and is stable under finite colimits. We will refer to ${\mathcal{S}}^{\mathrm{fin}}$ as the $\infty$-category of finite spaces. We let ${\mathcal{S}}_{\ast }^{\mathrm{fin}}\subseteq {\mathcal{S}}_{\ast }$ be the $\infty$-category of pointed objects of ${\mathcal{S}}^{\mathrm{fin}}$.

${\mathcal{S}}_{\ast }^{\mathrm{fin}}$ is a pointed $\infty$-category which admits pushouts, and thus it has a suspension functor $\Sigma$, taking an object $X$ to the pushout of $0\leftarrow X\to 0$. Let $S^0$ be the Kan complex ${\Delta }^0\coprod {\Delta }^0$, together with an (arbitrarily) specified point, making it into an element of ${\mathcal{S}}_{\ast }^{\mathrm{fin}}$. Let $S^n={\Sigma }^n{S}^0$.

Trick 3.3. Let $\varkappa$ be a strongly inaccessible cardinal that is the size parameter (or universe) we are currently using, or more precisely, in Notation 3.2 by $\mathcal{S}$ we actually mean ${\mathcal{S}}^{\varkappa }$, the $\infty$-category of $\varkappa$-Kan complexes. It is not immediate that ${\mathcal{S}}^{\mathrm{fin},\varkappa }$ is essentially ($\varkappa$-)small. However, we may always choose to work with the size parameters $\varkappa$’s such that there exists a strongly inaccessible cardinal ${\varkappa }_0<\varkappa$, in which case ${\mathcal{S}}^{\mathrm{fin},\varkappa }\stackrel{\sim }{\hookleftarrow }{\mathcal{S}}^{\mathrm{fin},{\varkappa }_0}$ is essentially $\varkappa$-small.

Definition 3.4. Let $\mathcal{C}$ be an $\infty$-category which admits finite limits. A spectrum object of $\mathcal{C}$ is a excisive and reduced functor $X:{\mathcal{S}}_{\ast }^{\mathrm{fin}}\to \mathcal{C}$. We write $\mathrm{Sp}(\mathcal{C})$ for ${\mathrm{Exc}}_{\ast }({\mathcal{S}}_{\ast }^{\mathrm{fin}},\mathcal{C})$.

Proposition 3.5. Let $\mathcal{C}$ be an $\infty$-category which admits finite limits, then $\mathrm{Sp}(\mathcal{C})$ is stable.

Proposition 3.6. Let $\mathcal{C}$ be an $\infty$-category which admits finite limits, let ${\mathcal{C}}_{\ast }$ denote the $\infty$-category of pointed objects of $\mathcal{C}$. Then the forgetful functor ${\mathcal{C}}_{\ast }\to \mathcal{C}$ induces an equivalence $\mathrm{Sp}({\mathcal{C}}_{\ast })\to \mathrm{Sp}(\mathcal{C})$.

Notation 3.7. Let $\mathcal{C}$ be an $\infty$-category which admits finite limits. Let ${\Omega }^{\infty }:\mathrm{Sp}(\mathcal{C})\to \mathcal{C}$ denote the functor given by evaluation at $S^0$. For $n\in \mathbb{Z}$ we let ${\Omega }^{\infty -n}:\mathrm{Sp}(\mathcal{C})\to \mathcal{C}$ denote the functor given by composing ${\Omega }^{\infty }$ with $[n]:\mathrm{Sp}(\mathcal{C})\to \mathrm{Sp}(\mathcal{C})$. If $n\ge 0$, then ${\Omega }^{\infty -n}$ is equivalent to the functor given by evaluation on $S^n$.

Proposition 3.8. Let $\mathcal{C}$ be a stable $\infty$-category, let $\mathcal{D}$ be an $\infty$-category which admits finite limits. Let ${\mathrm{Fun}}^{\prime }(\mathcal{C},\mathrm{Sp}(\mathcal{D}))\subseteq \mathrm{Fun}(\mathcal{C},\mathrm{Sp}(\mathcal{D})),{\mathrm{Fun}}^{\prime }(\mathcal{C},\mathcal{D})\subseteq \mathrm{Fun}(\mathcal{C},\mathcal{D})$ be the full subcategories spanned by the left exact functors (i.e.) those functors that preserve finite limits. Then the composition with ${\Omega }^{\infty }:\mathrm{Sp}(\mathcal{D})\to \mathcal{D}$ induces an equivalence ${\mathrm{Fun}}^{\prime }(\mathcal{C},\mathrm{Sp}(\mathcal{D}))\to {\mathrm{Fun}}^{\prime }(\mathcal{C},\mathcal{D})$.

We now explain in which sense “spectrum object$=$infinite delooping”.

Proposition 3.9. Let $\mathcal{C}$ be a pointed $\infty$-category which admits finite limits, thus there is a natrual loop functor $\Omega :\mathcal{C}\to \mathcal{C}$. Let $I$ be the spine of $N({\mathbb{Z}}_{\le 0})$ (that is, $I$ is a one dimensional simplicial set whose vertices corresponds to elements of ${\mathbb{Z}}_{\le 0}$ and whose nondegenerate edges are $i\to i+1$, where $i<0$), then there exists a limit diagram $I^{◃}\to \mathcal{Q}\mathcal{C}$ depicted as follows:

We now explain this proposition a bit: By virtue of [2] tag 03HV, for any sequence of maps of $\infty$-categories

there is an $\infty$-category denoted $\cdots {\times }_{{\mathcal{C}}_2}^h{\mathcal{C}}_2{\times }_{{\mathcal{C}}_1}^h{\mathcal{C}}_1{\times }_{{\mathcal{C}}_0}^h{\mathcal{C}}_0$ which is a limit of the above diagram in $\mathcal{Q}\mathcal{C}$ whose objects correspond bijectively to sequences of pairs $\{(c_n,{\alpha }_n){\}}_{n\ge 0}$, where $c_n\in {\mathcal{C}}_n$, and each ${\alpha }_n:F_n(c_{n+1})\to c_n$ is an isomorphism in the $\infty$-category ${\mathcal{C}}_n$.

Thus there is an isomorphism $\mathrm{Sp}(\mathcal{C})\stackrel{\sim }{\to }(\cdots {\times }_{\mathcal{C}}^h\mathcal{C}{\times }_{\mathcal{C}}^h\mathcal{C}{\times }_{\mathcal{C}}^h\mathcal{C})$, bringing a spectrum $X$ to a sequence of pairs $\{(c_n,{\alpha }_n){\}}_{n\ge 0}$ such that each $c_n\simeq {\Omega }^{\infty -n}X$ in $\mathcal{C}$.

Definition 3.10. We let $\mathrm{Sp}:=\mathrm{Sp}({\mathcal{S}}_{\ast })$ denote the $\infty$-category of spectra. A spectrum is simply an object of $\mathrm{Sp}$.

Proposition 3.11. Let $\mathcal{C}$ be a presentable $\infty$-category, then the functor ${\Omega }^{\infty }:\mathrm{Sp}(\mathcal{C})\to \mathcal{C}$ admits a left adjoint, which we will denote by ${\Sigma }_{+}^{\infty }$.

Let $\mathrm{Sp}(\mathcal{C}{)}_{\le -1}$ be the full subcategory of $\mathrm{Sp}(\mathcal{C})$ spanned by the objects $X$ such that ${\Omega }^{\infty }X$ is a final object of $\mathcal{C}$. Then this determines an $t$-structure on $\mathrm{Sp}(\mathcal{C})$.

In this case the image of ${\Sigma }_{+}^{\infty }$ is contained in $\mathrm{Sp}(\mathcal{C}{)}_{\ge 0}$.

Notation 3.12. We denote the functor ${\Omega }_{{\mathcal{S}}_{\ast }}^{\infty }:\mathrm{Sp}=\mathrm{Sp}({\mathcal{S}}_{\ast })\to {\mathcal{S}}_{\ast }$ by ${\Omega }_{\ast }^{\infty }$ and denote the composition $\mathrm{Sp}\stackrel{{\Omega }_{\ast }^{\infty }}{\to }{\mathcal{S}}_{\ast }\to \mathcal{S}$ by ${\Omega }^{\infty }$. Both ${\Omega }_{\ast }^{\infty }$ and ${\Omega }^{\infty }$ admit left adjoints, which we will denote by ${\Sigma }^{\infty }$ and ${\Sigma }_{+}^{\infty }$, respectively.

We now take a closer look at the $t$-structure on $\mathrm{Sp}$ given by Proposition 3.11.

3.13. Let ${\mathrm{Sp}}_{\le -1}$ denote the full subcategory of $\mathrm{Sp}$ spanned by the spectra $X$ such that ${\Omega }^{\infty }X$ is contractible, then this determines a $t$-structure on $\mathrm{Sp}$ which is nondegenerate.

Consider the isomorphism $\mathrm{Sp}\stackrel{\sim }{\to }\mathcal{D}:=(\cdots {\times }_{{\mathcal{S}}_{\ast }}^h{\mathcal{S}}_{\ast }{\times }_{{\mathcal{S}}_{\ast }}^h{\mathcal{S}}_{\ast }{\times }_{{\mathcal{S}}_{\ast }}^h{\mathcal{S}}_{\ast })$ given by Proposition 3.9, under which a spectrum $X$ corresponds to a sequence of pointed spaces $X(n)$ together with equivalences $\Omega X(n+1)\sim \Omega X(n)$. For $m\in \mathbb{Z}$, $X\in {\mathrm{Sp}}_{\le m}$ if and only if each $X(n)$ is $(n+m)$-truncated, and $X\in {\mathrm{Sp}}_{\ge m}$ if and only if each $X(n)$ is $(n+m)$-connected. Thus $X\in {\mathrm{Sp}}_{\ge 0}\cap {\mathrm{Sp}}_{\le 0}$ if and only if each $X(n)$ is an Eilenberg-MacLane object of $\mathcal{S}$ of degree $n$. Thus ${\mathrm{Sp}}_{\ge 0}\cap {\mathrm{Sp}}_{\le 0}\simeq N(\mathrm{Ab})$, where $\mathrm{Ab}$ is the category of abelian groups.

In particular, for each $n\in \mathbb{Z}$, ${\pi }_n$ is a functor $\mathrm{Sp}\to N(\mathrm{Ab})$, and if $n\ge 2$, then this functor is equivalent to the composition $\mathrm{Sp}\stackrel{{\Omega }_{\ast }^{\infty }}{\to }{\mathcal{S}}_{\ast }\stackrel{{\pi }_n}{\to }N(\mathrm{Ab})$.

Construction 3.14. Define a spectrum $S:={\Sigma }_{+}^{\infty }{\Delta }^0\in \mathrm{Sp}$, which we will call the sphere spectrum.

The functor ${\pi }_0:\mathrm{Sp}\to N(\mathrm{Ab})$ corresponds to a functor $h\mathrm{Sp}\to \mathrm{Ab}$, and this functor is isomorphic to the functor $h^S$ corepresented by $S$ (recall that $h\mathrm{Sp}$ is an additive category). In particular, ${\pi }_{0}S\simeq \mathbb{Z}$.

4Derived Categories

Construction 4.1. Let $\mathcal{A}$ be an additive category, then the category $\mathrm{Ch}(\mathcal{A})$ of chain complexes in $\mathcal{A}$ has a natural structure of a differential graded category. From now on when we mention $\mathrm{Ch}(\mathcal{A})$ we will always mean the differential graded category and we use $\mathrm{dCh}(\mathcal{A})$ to denote the underlying $1$-category instead. We use ${\mathrm{Ch}}^{-}(\mathcal{A})$ to denote the full subcategory of right bounded chain complexes (i.e. those chain complexes $A_{\bullet }$ such that $A_i=0$ for $i$ sufficiently (negatively) small).

Proposition 4.2. Let $\mathcal{A}$ be an additive category, then the $\infty$-category $N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))$ is stable.

We now investigate fiber sequences in $N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))$.

Proposition 4.3. Let $\mathcal{A}$ be an additive category. Suppose we have a pushout diagram

in $\mathrm{dCh}(\mathcal{A})$ such that $f$ is degreewise split (which means that each of the maps $M_n\to M_n^{\prime }$ admits a left inverse), then this diagram becomes a pushout square in $N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))$.

As a consequence, the triangulated structure on $h{N}_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))\cong K(\mathcal{A})$ coincides with the ordinary triangulated structure on $K(\mathcal{A})$. (See [1] Remark 1.3.2.17 for details.)

Construction 4.4 (right bounded derived $\infty$-category). Let $\mathcal{A}$ be an abelian category with enough projectives. Let ${\mathcal{A}}_{\mathrm{proj}}$ denote the full subcategory of $\mathcal{A}$ spanned by the projective objects. We let ${\mathcal{D}}^{-}(\mathcal{A})$ denote the $\infty$-category $N_{\mathrm{dg}}({\mathrm{Ch}}^{-}({\mathcal{A}}_{\mathrm{proj}}))$ and refer to it as the right bounded derived $\infty$-category of $\mathcal{A}$.

Proposition 4.5. Let $\mathcal{A}$ be an abelian category with enough projectives. Then ${\mathcal{D}}^{-}(\mathcal{A})$ is stable, and its homotopy category is naturally equivalent to the ordinary right bounded derived $1$-category of $\mathcal{A}$. Consequently there is a natural $t$-structure on ${\mathcal{D}}^{-}(\mathcal{A})$, given by the homology groups

Construction 4.6 (unbounded derived $\infty$-category).

Remark 4.7. When we mention a Grothendieck abelian category we always mean the one defined in [3] tag 079B, which is (probably) different from that of [1] Definition 1.3.5.1. However, every Grothendieck abelian category in our sense satisfies the conditions of [1] Definition 1.3.5.1, so don’t worry.

Let $\mathcal{A}$ be a Grothendieck abelian category, then $\mathrm{dCh}(\mathcal{A})$ admits a model structure, where a cofibration is a levelwise monomorphism, and a weak equivalence is a quasi-isomorphim.

Remark 4.8. Thanks to [1] Lemma 1.3.5.11, every fibrant chain complex in the above model structure is K-injective, in the sense of [3] 070H.

We let $\mathrm{Ch}(\mathcal{A}{)}^{\circ }\subseteq \mathrm{Ch}(\mathcal{A})$ denote the full subcategory spanned by the fibrant objects (which are automatically cofibrant). We let $\mathcal{D}(\mathcal{A})$ denote the $\infty$-category $N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}{)}^{\circ })$, and refer to it as the derived $\infty$-category of $\mathcal{A}$.

Proposition 4.9. Let $\mathcal{A}$ be a Grothendieck abelian category, then $\mathcal{D}(\mathcal{A})$ is stable, and its homotopy category is naturally equivalent to the ordinary unbounded derived $1$-category of $\mathcal{A}$. Consequently there is a natural $t$-structure on $\mathcal{D}(\mathcal{A})$, given by the homology groups.

Proposition 4.10 ([1] Proposition 1.3.5.13, Proposition 1.3.5.24). Let $\mathcal{A}$ be a Grothendieck abelian category, then $\mathcal{D}(\mathcal{A})\subseteq N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))$ is a reflective subcategory. We let $L$ denote a left adjoint to the inclusion functor $\mathcal{D}(\mathcal{A})\hookrightarrow N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))$. Then the composition ${\mathcal{D}}^{-}(\mathcal{A})\hookrightarrow N_{\mathrm{dg}}(\mathrm{Ch}(\mathcal{A}))\stackrel{L}{\to }\mathcal{D}(\mathcal{A})$ is fully faithful, and its essential image is those essentially right bounded chain complexes.

References

[1]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/
[2]	Jacob Lurie, “Kerodon”, https://kerodon.net
[3]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu