2.1 Limits and colimits in $\infty $ -categories of $\infty $ -categories

We begin by introducing the following $\infty $ -categories of $\infty $ -categories:

We are further interested in R-linear $\infty $ -categories, where R is an $\mathbb {E}_{\infty }$ -ring spectrum – that is, a commutative algebra object in the symmetric monoidal $\infty $ -category $\operatorname {Sp}$ of spectra. The $\infty $ -category $\mathcal {P}r^{L}$ also admits the structure of a symmetric monoidal $\infty $ -category [Reference LurieLur17, Section 4.8.1]. Given an $\mathbb {E}_{\infty }$ -ring spectrum R, the $\infty $ -category $\operatorname {LMod}_{R}\in \mathcal {P}r^{L}$ of left module-spectra over R is an algebra object of $\mathcal {P}r^{L}$ .

We now recall (in order of appearance) results on

1. i) how to compute limits in $\operatorname {Cat}_{\infty }$ ,

2. ii) how to compute limits and colimits in $\mathcal {P}r^{L}$ , $\mathcal {P}r^{L}_{\operatorname {St}}$ and $\mathcal {P}r^{R}$ , $\mathcal {P}r^{R}_{\operatorname {St}}$ ,

3. iii) how to compute limits and colimits in $\operatorname {LinCat}_{R}$ and

4. iv) how to compute limits and colimits in $\operatorname {St}^{\operatorname {idem}}$ .

i) There is a general formula for limits in $\operatorname {Cat}_{\infty }$ . Let $D:Z\rightarrow \operatorname {Set}_{\Delta }$ be a diagram taking values in $\infty $ -categories. Consider the co-Cartesian fibration $p:X\rightarrow Z$ classified by D. The limit $\infty $ -category $\operatorname {lim}D$ is equivalent to the $\infty $ -category of co-Cartesian sectionsFootnote ² of p [Reference LurieLur09, 3.3.3.2]. If Z is the nerve of a $1$ -category, the model for computing limits in $\operatorname {Cat}_{\infty }$ can be described more explicitly. We can use the relative nerve construction [Reference LurieLur09, 3.2.5.2] for the co-Cartesian fibration classified by D, which is very explicitly defined. We denote this model for the co-Cartesian fibration by $p:\Gamma (D)\rightarrow K$ and call it the (covariant) Grothendieck construction. A more detailed introduction to the relative nerve construction can be found in [Reference ChristChr20 , Section 1.2].

ii) One of the nice features of presentable $\infty $ -categories is that there is an $\infty $ -categorical adjoint functor theorem, which states that a functor between presentable $\infty $ -categories admits a right adjoint if and only if it preserves all colimits, and admits a left adjoint if and only if it is accessibleFootnote ³ and preserves all limits. There thus exists an adjoint equivalence of $\infty $ -categories

$$ \begin{align*} \operatorname{radj}:\mathcal{P}r^{L}\simeq \left(\mathcal{P}r^{R}\right)^{op}:\operatorname{ladj}^{op},\\[-15pt] \end{align*} $$

with the functors $\operatorname {radj},\operatorname {ladj}$ acting as the identity on objects. The functor $\operatorname {radj}$ maps a colimit-preserving functor to its right adjoint and the functor $\operatorname {ladj}$ maps an accessible and limit-preserving functor to its left adjoint. The adjoint equivalence $\operatorname {radj}\dashv \operatorname {ladj}$ also restricts to an adjoint equivalence between $\mathcal {P}r^{L}_{\operatorname {St}}$ and $\left (\mathcal {P}r_{\operatorname {St}}^{R}\right )^{op}$ . The equivalences $\operatorname {radj},\operatorname {ladj}$ preserve all limits and colimits, so that we can exchange the computations of limits and colimits of diagrams of (stable) presentable $\infty $ -categories. For the computation of limits, we can use i) and the fact that that the inclusions $\mathcal {P}^{L}_{\operatorname {St}}\subset \mathcal {P}r^{L}\subset \operatorname {Cat}_{\infty }$ and $\mathcal {P}r^{R}_{\operatorname {St}}\subset \mathcal {P}r^{R}\subset \operatorname {Cat}_{\infty }$ preserve all limits.

iii) The computation of limits and colimits of R-linear $\infty $ -categories reduces to the computation of limits and colimits in $\mathcal {P}r^{L}$ , because the forgetful functor $\operatorname {LMod}_{\operatorname {LMod}_{R}}\left (\mathcal {P}r^{L}\right )\rightarrow \mathcal {P}r^{L}$ preserves all limits and colimits [Reference LurieLur17, 4.2.3.1, 4.2.3.5].

iv) The inclusion functor $\operatorname {St}^{\operatorname {idem}}\subset \operatorname {Cat}_{\infty }$ preserves all limits. The computation of colimits of idempotent stable $\infty $ -categories can be related to the computation of colimits of presentable stable $\infty $ -categories via the colimit-preserving Ind-completion functor $\operatorname {Ind}:\operatorname {St}^{\operatorname {idem}}\rightarrow \mathcal {P}r^{L}_{\operatorname {St}}$ . Given an $\infty $ -category $\mathcal{C}\in \mathcal{P}r^L_{\operatorname {St}}$ , we denote by $\mathcal{C}^c\in \operatorname {St}^{\operatorname {idem}}$ its full subcategory of compact objects. Note that for $\mathcal{C}\in \operatorname {St}^{\operatorname {idem}}$ , there exists an equivalence $\operatorname {Ind}(\mathcal{C})^c\simeq \mathcal{C}$ .

2.2 Modules over ring spectra

Consider the symmetric monoidal $\infty $ -category $\operatorname {Sp}$ of spectra. $\operatorname {Sp}$ is a stable and presentable $\infty $ -category. An $\mathbb {E}_{1}$ -ring spectrum is an object of $\operatorname {Alg}(\operatorname {Sp})$ , the $\infty $ -category of (coherently associative) algebra objects in $\operatorname {Sp}$ . For every such $\mathbb {E}_{1}$ -ring spectrum R, there is a stable and presentable $\infty $ -category $\operatorname {RMod}_{R}$ of right R-modules in $\operatorname {Sp}$ . If R can be enhanced to a commutative algebra object of $\operatorname {Sp}$ – that is, an $\mathbb {E}_{\infty }$ -ring spectrum – then $\operatorname {RMod}_{R}$ inherits the structure of a symmetric monoidal $\infty $ -category. In this case, we can form the $\infty $ -category $\operatorname {Alg}(\operatorname {RMod}_{R})$ of algebra objects in $\operatorname {RMod}_{R}$ . Given $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ , we can again form the $\infty $ -category $\operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ of right A-modules in $\operatorname {RMod}_{R}$ . Alternatively, we can also consider the $\mathbb {E}_{1}$ -ring spectrum $\xi (A)\in \operatorname {Alg}(\operatorname {Sp})$ underlying A, obtained as follows. We consider the forgetful functor $\operatorname {RMod}_{R}\rightarrow \operatorname {Sp}$ , mapping a right R-module to the underlying spectrum. This functor extends to a functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ , which we apply to A. We can form the $\infty $ -category of right modules $\operatorname {RMod}_{\xi (A)}$ over $\xi (A)$ . We will show in Corollary 2.7 that this does not yield a further $\infty $ -category; there exists an equivalence of $\infty $ -categories

$$ \begin{align*} \operatorname{RMod}_{A}(\operatorname{RMod}_{R})\simeq \operatorname{RMod}_{\xi(A)}.\\[-15pt] \end{align*} $$

Let $\mathcal {D}$ be a stable $\infty $ -category and consider any object $X\in \mathcal {D}$ . We can find an $\mathbb {E}_{1}$ -ring spectrum $\operatorname {End}(X)\in \operatorname {Alg}(\operatorname {Sp})$ , called the endomorphism algebra, with the following properties [Reference LurieLur17, 7.1.2.2]:

• • $\pi _{n}\operatorname {End}(X) \simeq \pi _{0}\operatorname {Map}_{\mathcal {D}}(X[n],X)$ for all $n\in \mathbb {Z}$ .

• • The induced ring structure of $\pi _{*}\operatorname {End}(X)$ is determined by the composition of endomorphisms in the homotopy category $\operatorname {Ho}(\mathcal {D})$ .

The algebra object $\operatorname {End}(X)$ is an endomorphism object of X in the sense of [Reference LurieLur17, Section 4.7.1], and its existence expresses the enrichment of the stable $\infty $ -category $\mathcal {D}$ in spectra.

Assume that the stable $\infty $ -category $\mathcal {D}$ is also presentable. An object $X\in \mathcal {D}$ is called a compact generator if

• • X is compact – that is, commutes with filtered colimits – and

• • an object $Y\in \mathcal {D}$ is zero if and only if $\operatorname {Map}_{\mathcal {D}}(X,Y[i])\simeq \ast $ for all $i\in \mathbb {Z}$ .

The importance of this notion is that if X is a compact generator, there exists an equivalence of $\infty $ -categories $\mathcal {D}\simeq \operatorname {RMod}_{\operatorname {End}(X)}$ [Reference LurieLur17, 7.1.2.1].

We now restrict to R-linear $\infty $ -categories where R is an $\mathbb {E}_{\infty }$ -ring spectrum. The most important case will be where $R=k$ is a commutative ring. Suppose that $\mathcal {D}$ is an R-linear $\infty $ -category and $X\in \mathcal {D}$ a compact generator. Lemma 2.5 shows that we can lift $\operatorname {End}(X)$ along the forgetful functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ to an algebra object in $\operatorname {RMod}_{R}$ :

Lemma 2.5. Let R be an $\mathbb {E}_{\infty }$ -ring spectrum. Let $\mathcal {C}$ be a stable and presentable R-linear $\infty $ -category with a compact generator X. Then there exist an algebra object $\operatorname {End}_{R}(X)\in \operatorname {Alg}(\operatorname {RMod}_{R})$ and an equivalence of R-linear $\infty $ -categories

(3) $$ \begin{align} \mathcal{C}\simeq \operatorname{RMod}_{\operatorname{End}_{R}(X)}(\operatorname{RMod}_{R}). \end{align} $$

The algebra object $\operatorname {End}_{R}(X)$ is mapped under the functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ to the endomorphism algebra $\operatorname {End}(X)\in \operatorname {Alg}(\operatorname {Sp})$ .

Proof. The left-tensoring of $\mathcal {C}$ over R determines an R-linear functor

. By the adjoint functor theorem, the functor admits a right adjoint G. We denote $\operatorname {End}_{R}(X):=G(X)\in \operatorname {RMod}_{R}$ . The existence of a lift of $\operatorname {End}_{R}(X)$ to $\operatorname {Alg}(\operatorname {RMod}_{R})$ and of the equivalence (3) follow from [Reference LurieLur17, 4.8.5.8] (compare also the proof of [Reference LurieLur17, 7.1.2.1]). The right adjoint of the composite functor

maps X to the endomorphism object $\operatorname {End}(X)$ . By the universal property of $\operatorname {End}(X)$ and $X\in \mathcal {C}\simeq \operatorname {RMod}_{\xi \left (\operatorname {End}_{R}(X)\right )}(\operatorname {Sp})$ , there exists a morphism $\xi (\operatorname {End}_{R}(X))\rightarrow \operatorname {End}(X)$ in $\operatorname {Alg}(\operatorname {Sp})$ , which is an equivalence on underlying spectra and thus an equivalence of $\mathbb {E}_{1}$ -ring spectra.

Corollary 2.7. Let R be an $\mathbb {E}_{\infty }$ -ring spectrum and set $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ . Then there exists an equivalence of $\infty $ -categories

(4) $$ \begin{align} \operatorname{RMod}_{A}(\operatorname{RMod}_{R})\simeq \operatorname{RMod}_{\xi(A)}, \end{align} $$

where $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(Sp)$ denotes the forgetful functor.

Let R be an $\mathbb {E}_{\infty }$ -ring spectrum. We end this section with a brief discussion of the relation between colimits of algebra objects in $\operatorname {RMod}_{R}$ and the colimits of the corresponding $\infty $ -categories of right modules in $\operatorname {LinCat}_{R}$ . There is a functor $\theta : \operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {LinCat}_{R}$ that assigns to an algebra object $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ the $\infty $ -category $\operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ [Reference LurieLur17, section 4.8.3]. The functor $\theta $ assigns to an edge $\phi :A\rightarrow B$ in $\operatorname {Alg}(\operatorname {RMod}_{R})$ the relative tensor product

using the right A-module structure on B provided by $\phi $ . For all $\phi :A\rightarrow B$ , the functor $\theta (\phi )$ admits a right adjoint, given by the pullback functor $\phi ^{*}:\operatorname {RMod}_{B}(\operatorname {RMod}_{R})\rightarrow \operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ along $\phi $ [Reference LurieLur17, 4.6.2.17]. The functor $\theta $ preserves colimits indexed by contractible simplicial sets (that is, simplicial sets whose geometric realisation is a contractible space), most notably pushouts.

2.3 Differential graded categories and their modules

Let k be a commutative ring. A k-linear dg-category is a $1$ -category enriched in the $1$ -category $\operatorname {Ch}(k)$ of chain complexes of k-modules. Given a dg-category C and two objects $x,y\in C$ , we write $\operatorname {Hom}_{C}(x,y)$ or $\operatorname {Hom}(x,y)$ for the mapping complex. We consider dg-algebras as dg-categories with a single object.

We can identify left A-modules with dg-functors $A\rightarrow \operatorname {Ch}(k)$ , right A-modules with dg-functors $A^{op}\rightarrow \operatorname {Ch}(k)$ and A-B-bimodules with dg-functors $A\otimes B^{\operatorname {op}}\rightarrow \operatorname {Ch}(k)$ . The following definition is thus consistent with Definition 2.8:

Given a dg-category C and an object $~x\in C$ , we denote by $\operatorname {End}^{\operatorname {dg}}(x)$ the endomorphism dg-algebra with underlying chain complex given by $\operatorname {Hom}_{C}(x,x)$ and algebra structure determined by the composition of morphisms in C.

2.4 A model for the derived $\infty $ -category of a dg-algebra

Let A be a k-linear dg-algebra. The $1$ -category $\operatorname {dgMod}(A)_{0}$ underlying the dg-category $\operatorname {dgMod}(A)$ is the $1$ -category with the same objects and with mapping sets given by the $0$ -cycles. This $1$ -category admits the projective model structure, where the weak equivalences are given by quasi-isomorphisms and the fibrations are given by degree-wise surjections. All objects of $\operatorname {dgMod}(A)_{0}$ are fibrant. A description of the cofibrant objects in $\operatorname {dgMod}(A)_{0}$ can be found, for example, in [Reference Barthel, May and RiehlBMR14], where they are called q-semi-projective objects. A right A-module M is cofibrant if and only if

• • the ungraded module $\bigoplus _{i\in \mathbb {Z}}M_{i}$ is a projective right module over the ungraded algebra $\bigoplus _{i\in \mathbb {Z}}A_{i}$ and

• • for all acyclic right A-modules N, the mapping complex $\operatorname {Hom}_{A}(M,N)$ is acyclic.

If $A=k$ is a commutative ring, the cofibrant objects are the complexes of projective k-modules. We denote by $\operatorname {dgMod}(A)^{\circ }\subset \operatorname {dgMod}(A)$ the full dg-subcategory spanned by fibrant-cofibrant objects. We call the dg-nerve $\mathcal {D}(A):= N_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })$ the (unbounded) derived $\infty $ -category of A.

Before we can further discuss the properties of $\mathcal {D}(A)$ , we need to briefly discuss localisations of $\infty $ -categories.

In [Reference LurieLur09], localisations in the sense of Definition 2.13 are simply called localisations. We are, however, interested in a more general class of localisations, which can be characterised by the following universal property:

Definition 2.14. Let $\mathcal {C}$ be an $\infty $ -category and let W be a collection of morphisms in $\mathcal {C}$ . We call an $\infty $ -category $\mathcal {C}^{\prime }$ the $\infty $ -categorical localisation of $\mathcal {C}$ at W if there exists a functor $f:\mathcal {C}\rightarrow \mathcal {C}^{\prime }$ such that for every $\infty $ -category $\mathcal {D}$ , composition with f induces a fully faithful functor

$$ \begin{align*} \chi: \operatorname{Fun}(\mathcal{C}^{\prime},\mathcal{D})\rightarrow \operatorname{Fun}(\mathcal{C},\mathcal{D}),\\[-17pt] \end{align*} $$

whose essential image consists of those functors $F:\mathcal {C}\rightarrow \mathcal {D}$ for which $F(\alpha )$ is an equivalence in $\mathcal {D}$ for all $\alpha \in W$ . In that case, we also write $\mathcal {C}^{\prime }=\mathcal {C}\left [W^{-1}\right ]$ .

It is shown in [Reference LurieLur09, 5.2.7.12] that reflective localisations are localisations in the sense of Definition 2.14. If the collection of morphisms W is closed under homotopy and composition and contains all equivalences in $\mathcal {C}$ , we can regard $\mathcal {C}\left [W^{-1}\right ]$ as a fibrant replacement of $(\mathcal {C},W)$ in the model category of marked simplicial sets (see also the discussion in the beginning of [Reference LurieLur17, Section 4.1.7]).

Our first goal in this section is to prove the following analogue of [Reference LurieLur17, 1.3.5.15], relating the derived $\infty $ -category of A with the $\infty $ -categorical localisation of $\operatorname {dgMod}(A)_{0}$ at the collection of quasi-isomorphisms:

Proposition 2.15. Let A be a dg-algebra and let W denote the collection of quasi-isomorphisms. There exists an equivalence of $\infty $ -categories

$$ \begin{align*} \mathcal{D}(A)\simeq N\left(\operatorname{dgMod}(A)_{0}\right)\left[W^{-1}\right].\\[-17pt] \end{align*} $$

Given a model category C, the $\infty $ -categorical localisation of $N(C)$ at the collection of weak equivalences is called the $\infty $ -category underlying C. We refer to [Reference HinchHin16] for general background. Proposition 2.15 thus shows that the derived $\infty $ -category of A is the $\infty $ -category underlying the model category $\operatorname {dgMod}(A)_{0}$ .

For the proof of Proposition 2.15, we need the following two lemmas.

Lemma 2.16. Let A be a dg-algebra. The inclusion functor $N(\operatorname {dgMod}(A)_{0})\rightarrow N_{\operatorname {dg}}(\operatorname {dgMod}(A))$ induces an equivalence of $\infty $ -categories

$$ \begin{align*} N(\operatorname{dgMod}(A)_{0})\left[H^{-1}\right]\rightarrow N_{\operatorname{dg}}(\operatorname{dgMod}(A)),\\[-17pt] \end{align*} $$

where H is the collection of chain homotopy equivalences.

Lemma 2.17. Let A be a dg-algebra. There exists an equivalence of $\infty $ -categories

$$ \begin{align*} N_{\operatorname{dg}}(\operatorname{dgMod}(A)^{\circ})\simeq N_{\operatorname{dg}}(\operatorname{dgMod}(A))\left[W^{-1}\right]. \end{align*} $$

Proof. We adapt the proofs of [Reference LurieLur17, 1.3.4.6, 1.3.5.12]. We show that the inclusion functor $i:\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}\rightarrow \operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{\operatorname {op}}$ admits a left adjoint which exhibits $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}$ as a reflective localisation at the collection of quasi-isomorphisms. Note that any functor is a localisation if and only if the opposite functor is a localisation. We thus conclude that $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })$ is equivalent as an $\infty $ -category to the localisation of $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))$ at the collection of quasi-isomorphisms.

To verify that $i^{\operatorname {op}}$ is a reflective localisation, we need to show that it admits a left adjoint $G:\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{\operatorname {op}}\rightarrow \operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}$ . To show that W is the collection of quasi-isomorphisms, we need to show by [Reference LurieLur09, 5.2.7.12] that any edge $e:M\rightarrow N$ in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{op}$ is a quasi-isomorphism if and only if $G(e)$ is an equivalence. Consider a trivial fibration $f:Q^{\prime }\rightarrow Q$ in $\operatorname {dgMod}(A)$ given a cofibrant replacement and any $P\in \operatorname {dgMod}(A)^{\circ }$ . [Reference LurieLur09, 5.2.7.8] shows the existence of G, provided that the composition with f induces an isomorphism of spaces

$$ \begin{align*} \operatorname{Map}_{\operatorname{N}_{\operatorname{dg}}(\operatorname{dgMod}(A)^{\circ})}(P,Q^{\prime})\rightarrow \operatorname{Map}_{\operatorname{N}_{\operatorname{dg}}(\operatorname{dgMod}(A))}(P,Q). \end{align*} $$

We deduce this from the assertion that composition with f induces a quasi-isomorphism

(5) $$ \begin{align} \alpha:\operatorname{Hom}_{\operatorname{dgMod}(A)}(P,Q^{\prime})\rightarrow \operatorname{Hom}_{\operatorname{dgMod}(A)}(P,Q). \end{align} $$

The surjectivity of $\alpha $ follows from the lifting property of the cofibration $0\rightarrow P$ with respect to trivial fibrations. The kernel of $\alpha $ is given by $\operatorname {Hom}_{\operatorname {dgMod}(A)}(P,\operatorname {ker}(f))$ . Using the fact that f is a quasi-isomorphism, we deduce that $\operatorname {ker}(f)$ is acyclic. The contractibility of the kernel of $\alpha $ thus follows from the property of P being cofibrant. We can thus deduce the existence of G. We note that G is point-wise given by choosing a cofibrant replacement. Consider an edge $e:M\rightarrow N$ in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{op}$ . If e is a quasi-isomorphism, it follows from Whitehead’s theorem for model categories that $G(e)$ is an equivalence. If $G(e)$ is an equivalence, we have the following commutative diagram in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))$ :

The vertical edges and the upper horizontal edge are quasi-isomorphisms. It follows that e is also a quasi-isomorphism.

Proof of Proposition 2.15.

By Lemmas 2.16 and 2.17, there exists an equivalence of $\infty $ -categories

$$ \begin{align*} \left( N(\operatorname{dgMod}(A))\left[H^{-1}\right]\right)\left[W^{-1}\right]\simeq \mathcal{D}(A). \end{align*} $$

Using the fact that $H\subset W$ , the statement follows.

Let k be a commutative ring. The symmetric monoidal structure of the $1$ -category $\operatorname {Ch}(k)$ can be used to also endow the $\infty $ -category $\mathcal {D}(k)$ with a symmetric monoidal structure. As shown in [Reference LurieLur17, 7.1.4.6], there exists an equivalence of $\infty $ -categories

(6) $$ \begin{align} \operatorname{N}\left(\operatorname{Alg}\left(\operatorname{Ch}^{\otimes}(k)\right)\right)\left[W^{-1}\right] \simeq \operatorname{Alg}(\mathcal{D}(k)). \end{align} $$

The left side of formula (6) is the $\infty $ -categorical localisation of the nerve of the $1$ -category of dg-algebras at the collection of quasi-isomorphisms. The right side is the $\infty $ -category of algebra objects in $\mathcal {D}(k)$ . The equivalence (6) expresses that every dg-algebra can be considered as an algebra object in $\mathcal {D}(k)$ and that every algebra object in $\mathcal {D}(k)$ can be obtained this way (meaning it can be rectified). Unless stated otherwise, we will omit this identification and consider dg-algebras as algebra objects in the symmetric monoidal $\infty $ -category $\mathcal {D}(k)$ .

We can consider k also as an $\mathbb {E}_{\infty }$ -ring spectrum. The $\infty $ -category $\operatorname {RMod}_{k}$ of right modules over k thus inherits a symmetric monoidal structure. The $\infty $ -categories $\mathcal {D}(k)$ and $\operatorname {RMod}_{k}$ are equivalent as symmetric monoidal $\infty $ -categories [Reference LurieLur17, 7.1.2.13].

Let A be a k-linear dg-algebra and X a cofibrant A-module. Consider the Quillen adjunction

(7)

between the tensor functor on the level of chain complexes and the internal Hom functor composed with the forgetful functor $\operatorname {dgMod}(A)\rightarrow \operatorname {dgMod}(k)$ . Given a Quillen adjunction between model categories, there is an associated adjunction between the underlying $\infty $ -categories [Reference Mazel-GeeMG16]. We denote the adjunction of $\infty $ -categories underlying the Quillen adjunction (7) by

(8)

Proposition 2.19. Let A be a k-linear dg-algebra. Using the symmetric monoidal equivalence $\mathcal {D}(k)\simeq \operatorname {RMod}_{k}$ , we can consider $\operatorname {RMod}_{A}\overset {\text {formula~}(4)}{\simeq } \operatorname {RMod}_{A}(\operatorname {RMod}_{k})$ as left-tensored over $\mathcal {D}(k)$ . There exists an equivalence

(9) $$ \begin{align} \mathcal{D}(A)\simeq \operatorname{RMod}_{A} \end{align} $$

of $\infty $ -categories left-tensored over $\mathcal {D}(k)$ .

Proof. Consider the adjunction of $\infty $ -categories underlying the Quillen adjunction . Using the adjunction, it can be directly checked that A is a compact generator of $\mathcal {D}(A)$ . It follows from [Reference LurieLur17, 4.8.5.8] that there exists an equivalence

(10) $$ \begin{align} \mathcal{D}(A)\simeq \operatorname{RMod}_{\operatorname{End}_{k}(A)}(\mathcal{D}(k)) \end{align} $$

of $\infty $ -categories left-tensored over $\mathcal {D}(k)$ , where $\operatorname {End}_{k}(A)\in \operatorname {Alg}(\mathcal {D}(k))$ is the k-linear endomorphism algebra of A (see Remark 2.6). We note that the underlying chain complex satisfies $\operatorname {End}_{k}(A)\simeq \operatorname {RHom}_{A}(A,A)\simeq A$ . By the universal property of $\operatorname {End}_{k}(A)$ , there exists a morphism of dg-algebras $\chi :A\rightarrow \operatorname {End}_{k}(A)$ , whose underlying morphism of chain complexes is induced by the actions $A\otimes _{k}A\rightarrow A$ and $A\otimes _{k} \operatorname {End}_{k}(A)\rightarrow A$ . The latter is induced by the counit of the adjunction and is thus equivalent to the former. It follows that $\chi $ induces a quasi-isomorphism $\operatorname {End}_{k}(A)=\operatorname {RHom}_{A}(A,A)\simeq A$ on underlying chain complexes and is hence a quasi-isomorphism of dg-algebras. In total, we obtain that there also exists an equivalence of k-linear $\infty $ -categories $\operatorname {RMod}_{\operatorname {End}_{k}(A)}(\mathcal {D}(k))\simeq \operatorname {RMod}_{A}(\mathcal {D}(k))\simeq \operatorname {RMod}_{A}(\operatorname {RMod}_{k})$ , which combined with formula (10) shows the statement.

Let $A,B\in \operatorname {Alg}(\mathcal {D}(k))$ be dg-algebras and $F:\operatorname {RMod}_{A}\rightarrow \operatorname {RMod}_{B}$ a k-linear functor. Clearly $F(A)\in \operatorname {RMod}_{B}$ carries the structure of a right B-module. Let $m:A\otimes _{k} A\rightarrow A$ be the multiplication map of A. Using the k-linearity of F, we find an action map

which is part of the datum of a left A-module structure on $F(A)$ . It turns out that both module structures are compatible, so we can endow $F(A)$ with the structure of an A-B-bimodule. In total, we obtain a functor

As shown in [Reference LurieLur17, Section 4.8.4], the functor $\phi $ is an equivalence of $\infty $ -categories. Given a bimodule

, we denote by

a choice of k-linear functor such that

.

Proposition 2.20. Let $A,B$ be dg-algebras and set and consider the functor of $\infty $ -categories underlying the right Quillen functor . There exists a commutative diagram in $\operatorname {LinCat}_{k}$ as follows:

(11)

Proof of Proposition 2.20.

The k-linear functor

is of the form

for

[Reference LurieLur17, Section 4.8.4]. We note that N can be rectified to a strict dg-bimodule and is thus determined by its right B-module structure and its left A-module structure. The right B-module structures of N and M are clearly equivalent. In particular, there exists an equivalence $N\simeq M$ of underlying chain complexes. The left action of A on N is determined by

where m denotes the multiplication of A and is thus equivalent to the given left action of A on the A-B-bimodule M. This shows that $N\simeq M$ as bimodules.

Proof. Proposition 2.20 shows that the functor

is equivalent to

. The right adjoint G of F is given by

. It follows that $\operatorname {RHom}_{A}(X,X)\simeq G(X)=\operatorname {End}_{k}(X)$ in $\mathcal {D}(k)$ (see also the definition of $\operatorname {End}_{k}(X)$ in the proof of Lemma 2.5). Using the fact that $\operatorname {RHom}_{A}(X,X)=\operatorname {Hom}_{A}(X,X)=\operatorname {End}^{\operatorname {dg}}(X)$ and the explicit $\operatorname {Hom}_{A}(X,X)$ -module structure on X, it follows from the universal property of the endomorphism object that there exists a morphism of dg-algebras $\alpha :\operatorname {RHom}_{A}(X,X)\rightarrow \operatorname {End}_{k}(X)$ , which restricts to the quasi-isomorphism on underlying chain complexes and is hence a quasi-isomorphism of dg-algebras.

2.5 Morita theory of dg-categories

Let k be a commutative ring. We denote by $\operatorname {dgCat}_{k}$ the category of k-linear dg-categories. Given a dg-category $C\in \operatorname {dgCat}_{k}$ , the dg-category $\operatorname {dgMod}(C)$ admits a model structure called the projective model structure. We have already encountered this model structure in Section 2.4, in the case where C is a dg-algebra. We define $C^{\operatorname {perf}}$ as the full dg-subcategory of $\operatorname {dgMod}(C)$ spanned by fibrant-cofibrant objects x which are compact in the homotopy category $H^{0}(\operatorname {dgMod}(C))$ – that is,

preserves coproducts. This assignment forms a functor

As shown by Tabuada [Reference TabuadaTab05], the category $\operatorname {dgCat}_{k}$ admits a model structure where the following hold:

• • The weak equivalences are the quasi-eqivalence – that is, dg-functors $F:A\rightarrow B$ such that for all $a,a^{\prime }\in A$ , the morphism between morphism complexes $F(a,a^{\prime }):\operatorname {Hom}_{A}(a,a^{\prime })\rightarrow \operatorname {Hom}_{{B}}(F(a),F(a^{\prime }))$ is a quasi-isomorphism and such that the induced functor on homotopy categories is an equivalence.

• • The fibrations are the dg-functors F such that for all $a,a^{\prime }\in A$ , the morphism between mapping complexes $F(a,a^{\prime }):\operatorname {Hom}_{A}(a,a^{\prime })\rightarrow \operatorname {Hom}_{B}(F(a),F(a^{\prime }))$ is degree-wise surjective and for any isomorphism $b\rightarrow F(a^{\prime })$ in the homotopy category of B, there exists a lift along F to an isomorphism $a\rightarrow a^{\prime }$ in the homotopy category of A.

The $\infty $ -category underlying this model category is given by the $\infty $ -categorical localisation $\operatorname {dgCat}_{k}\left [W^{-1}\right ]$ of the nerve of $\operatorname {dgCat}_{k}$ at the collection W of weak equivalences. This model structure can be further localised at the collection M of Morita equivalences – that is, dg-functors F such that $(F)^{\operatorname {perf}}$ is a quasiequivalence. The resulting model structure is called the Morita model structure. The corresponding localisation functor

$$ \begin{align*} L: \operatorname{dgCat}_{k}\left[W^{-1}\right]\longrightarrow \operatorname{dgCat}_{k}\left[M^{-1}\right]\\[-17pt] \end{align*} $$

exhibits $\operatorname {dgCat}_{k}\left [M^{-1}\right ]$ as a reflective localisation of $\operatorname {dgCat}_{k}\left [W^{-1}\right ]$ and thus preserves colimits. Given $C\in \operatorname {dgCat}_{k}\left [W^{-1}\right ]$ , its image $L(C)$ is quasiequivalent to $C^{\operatorname {perf}}$ . The Morita model structure models the $\infty $ -category of k-linear, stable and idempotent complete $\infty $ -categories, meaning that there exists an equivalence of $\infty $ -categories [Reference CohnCoh13]

(12) $$ \begin{align} \operatorname{dgCat}_{k}\left[M^{-1}\right]\simeq \operatorname{Mod}_{N_{\operatorname{dg}}\left(k^{\operatorname{perf}}\right)}\left(\operatorname{St}^{\operatorname{idem}}\right).\\[-17pt]\nonumber \end{align} $$

The right side describes the $\infty $ -category of modules in the symmetric monoidal category $\operatorname {St}^{\operatorname {idem}}$ over the algebra object $N_{\operatorname {dg}}\left (k^{\operatorname {perf}}\right )$ . The equivalence (12) maps a dg-category C to the dg-nerve of the dg-category $C^{\operatorname {perf}}$ . $\operatorname {Ind}$ -completion provides a further colimit-preserving functor $\operatorname {Ind}:\operatorname {Mod}_{\mathcal {D}(k)^{\operatorname {perf}}}\left (\operatorname {St}^{\operatorname {idem}}\right )\rightarrow \operatorname {LinCat}_{k}$ . In total, we obtain the colimit-preserving functor

(13)

Note that given a dg-algebra A, the derived $\infty $ -category $\mathcal {D}(A)$ is equivalent to the image of A under formula (13), so the notation for the functor is justified. Furthermore, we can compute colimits in $\operatorname {dgCat}_{k}\left [W^{-1}\right ]$ as homotopy colimits in $\operatorname {dgCat}_{k}$ with respect to the quasiequivalence model structure.

2.6 Semiorthogonal decompositions

In this section we discuss semiorthogonal decompositions of stable $\infty $ -categories of length $n\geq 2$ . Some of the treatment is based on the discussion of semiorthogonal decompositions of length $n=2$ in [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKSS21].

The next lemma shows that semiorthogonal decompositions of length n are simply repeated semiorthogonal decompositions of length $2$ .

Proof. For $1\leq i \leq n-1$ and $0\leq j \leq n-i-1$ , denote by $\mathcal {D}^{i}_{j}\subset \operatorname {Fun}\left (\Delta ^{\left \{j,\dotsc ,n-i\right \}},\langle \mathcal {A}_{i},\dotsc ,\mathcal {A}_{n}\rangle \right )$ the full subcategory spanned by diagrams $D^{i}_{j}$ such that

• • $D_{j}^{i}(l)$ lies in $\langle \mathcal {A}_{n-l},\dotsc ,\mathcal {A}_{n}\rangle $ for $j\leq l\leq n-i$ and

• • the cofibre of $D^{i}_{j}(l)\rightarrow D^{i}_{j}(l+1)$ in $\mathcal {V}$ lies in $\mathcal {A}_{n-l+1}$ for all $j\leq l \leq n-i-1$ .

We denote by $r_{i,j}:\mathcal {D}^{i}_{j}\rightarrow \langle \mathcal {A}_{i},\dotsc ,\mathcal {A}_{n}\rangle $ the functor given by the restriction to the vertex $n-i$ . Note that for $i<k \leq n-j-1$ , there is a trivial fibration $\mathcal {D}^{i}_{j}\rightarrow \mathcal {D}^{i}_{n-k}\times _{\left \langle \mathcal {A}_{k},\dotsc ,\mathcal {A}_{n}\right \rangle }\mathcal {D}^{k}_{j}$ .

Now suppose that $(\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n})$ is a semiorthogonal decomposition and let $\mathcal {D}\rightarrow \mathcal {V}$ be the corresponding trivial fibration. Condition i) is immediate. For condition ii), we need to show that $r_{i,n-i-1}$ is a trivial fibration for all $1\leq i \leq n-1$ . Using the fact that pullbacks preserve trivial fibrations, it follows that

(14) $$ \begin{align} \mathcal{D}^{\prime}=\mathcal{D}\times_{\mathcal{V}}\langle \mathcal{A}_{i},\dotsc,\mathcal{A}_{n}\rangle\rightarrow \langle \mathcal{A}_{i},\dotsc,\mathcal{A}_{n}\rangle \end{align} $$

is a trivial fibration. We can describe the elements of $\mathcal {D}^{\prime }$ as the left Kan extensions along the inclusion $\Delta ^{\{0,\dotsc ,n-i\}}\rightarrow \Delta ^{n}$ of elements of $\mathcal {D}^{i}_{0}$ . It thus follows from [Reference LurieLur09, 4.3.2.15] that the restriction functor $\mathcal {D}^{\prime }\rightarrow \mathcal {D}^{i}_{0}$ to $\Delta ^{\{0,\dotsc ,i\}}$ is a trivial fibration. Using the fact that the functor (14) factors through $r_{i,0}:\mathcal {D}^{i}_{0}\rightarrow \langle \mathcal {A}_{i},\dotsc ,\mathcal {A}_{n}\rangle $ , it follows from the $2/3$ property of equivalences that $r_{i,0}$ is also a trivial fibration. The following commutative diagram thus shows that $r_{i,n-i-1}$ is a trivial fibration, showing statement ii):

(15)

We now show that conditions i) and ii) imply that $(\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n})$ is a semiorthogonal decomposition of $\mathcal {V}$ . If $n=2$ , the assertion is obvious. We proceed by induction over n. Assume that $(\mathcal {A}_{2},\dotsc ,\mathcal {A}_{n})$ is a semiorthogonal decomposition of $\langle \mathcal {A}_{2},\dotsc , \mathcal {A}_{n}\rangle $ , meaning that $r_{2,0}$ is a trivial fibration. To show that $(\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n})$ is a semiorthogonal decomposition of $\mathcal {V}=\langle \mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\rangle $ , we need to show that $r_{1,0}$ is also a trivial fibration. Condition ii) implies that $r_{1,n-2}$ is a trivial fibration. Diagram (15) for $i=1$ thus shows that $r^{1}_{0}$ is also a trivial fibration.

As it turns out, the functoriality data involved in the definition of semiorthogonal decompositions of length $2$ are redundant.

A simple source of semiorthogonal decompositions are sequences of functors between stable $\infty $ -categories.

Lemma 2.29. Let $D:\Delta ^{n-1}\rightarrow \operatorname {Cat}_{\infty }$ be a diagram taking values in stable $\infty $ -categories, corresponding to n composable functors

1. 1. The stable $\infty $ -category

   $$ \begin{align*} \{\mathcal{A}_{1},\dotsc,\mathcal{A}_{n}\}:= \operatorname{Fun}_{\Delta^{n-1}}\left(\Delta^{n-1},\Gamma(D)\right) \end{align*} $$
   of sections of the Grothendieck construction $p:\Gamma (D)\rightarrow \Delta ^{n-1}$ (see Section 2.1) admits a semiorthogonal decomposition $(\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n})$ of length n.

2. 2. Let R be an $\mathbb {E}_{\infty }$ -ring spectrum. If each $F_{i}$ is an R-linear functor between R-linear $\infty $ -categories, then the $\infty $ -category $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ further inherits the structure of an R-linear $\infty $ -category such that each inclusion functor $\iota _{i}:\mathcal {A}_{i}\rightarrow \{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ is R-linear.

Proof. We begin by showing statement $1$ . Consider the simplicial set

$$ \begin{align*} Z= \left(\Delta^{0}\times \Delta^{n-1}\right)\amalg_{\Delta^{\{1\}}\times \Delta^{\left\{1,\dotsc,n-1\right\}}} \left( \Delta^{1}\times \Delta^{n-2}\right)\amalg \dotsb \amalg_{\Delta^{\left\{1,\dotsc,n-1\right\}}\times \Delta^{\{1\}}}\left(\Delta^{n-1}\times \Delta^{0}\right). \end{align*} $$

Let $\mathcal {D}^{\prime }$ be the full subcategory of $\operatorname {Fun}(Z,\Gamma (D))$ spanned by diagrams given by right Kan extensions along the inclusion $\Delta ^{n-1}\times \Delta ^{0}\rightarrow Z$ of a diagram in $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ . By [Reference LurieLur09, 4.3.2.15], the restriction functor $\mathcal {D}^{\prime }\rightarrow \{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ to $\Delta ^{0}\times \Delta ^{n-1}$ is a trivial fibration. We can describe the elements of $\mathcal {D}^{\prime }$ up to equivalence as diagrams in $\Gamma (D)$ of the form

such that $a_{i}\in \mathcal {A}_{i}$ . The restriction functor $\mathcal {D}^{\prime }\rightarrow \{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ corresponds in the foregoing description point-wise to the restriction to the rightmost column. The $\infty $ -category $\mathcal {D}$ of Definition 2.25 can be identified with the full subcategory of $\operatorname {Fun}\left (\Delta ^{n-1}\times \Delta ^{n-1},\Gamma (\alpha )\right )$ spanned by left Kan extensions along $Z\rightarrow \Delta ^{n-1}\times \Delta ^{n-1}$ of diagrams lying in $\mathcal {D}^{\prime }$ . It follows that the restriction functor $\mathcal {D}\rightarrow \mathcal {D}^{\prime }$ is a trivial fibration and thus that the restriction functor $\mathcal {D}\rightarrow \{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ is a trivial fibration. This shows statement 1.

We now show statement 2. Consider the diagram of $\infty $ -operads over $\mathcal {LM}^{\otimes }$

exhibiting the functors $F_{i}$ as R-linear. The morphism of $\infty $ -operads

$$ \begin{align*} \operatorname{Fun}_{\Delta^{n-1}}\left(\Delta^{n-1},\Gamma\left(D^{\otimes}\right)\right)\times_{\operatorname{Fun}\left(\Delta^{n-1},\mathcal{LM}^{\otimes}\right)}\mathcal{LM}^{\otimes}\rightarrow \mathcal{LM}^{\otimes} \end{align*} $$

exhibits $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ as left-tensored over

$$ \begin{align*} \mathcal{M}:= \operatorname{Fun}_{\Delta^{n-1}}\left(\Delta^{n-1},\Gamma\left(D^{\otimes}\right)\right)\times_{\operatorname{Fun}\left(\Delta^{n-1},\mathcal{LM}^{\otimes}\right)}\mathcal{LM}^{\otimes} \times_{\mathcal{LM}^{\otimes}}\operatorname{Assoc}^{\otimes}. \end{align*} $$

Let $\tilde {D}^{\otimes }:\Delta ^{n-1}\rightarrow \operatorname {Cat}_{\infty }$ be the constant diagram with value $\operatorname {LMod}_{R}^{\otimes }$ . We find $\mathcal {M}$ to be equivalent as a monoidal $\infty $ -category to

(16) $$ \begin{align} \operatorname{Fun}_{\Delta^{n-1}}\left(\Delta^{n-1},\Gamma\left(\tilde{D}^{\otimes}\right)\right)\times_{\operatorname{Fun}\left(\Delta^{n-1},\operatorname{Assoc}^{\otimes}\right)}\operatorname{Assoc}^{\otimes}. \end{align} $$

Pulling back along the monoidal functor $\operatorname {LMod}_{R}^{\otimes }\rightarrow \mathcal {M}$ and assigning to $x\in \operatorname {LMod}_{R}^{\otimes }$ the constant section in expression (16), we obtain a left-tensoring of $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ over $\operatorname {LMod}_{R}$ . To show that the left-tensoring provides the structure of an R-linear $\infty $ -category, it suffices to show that the monoidal product preserves colimits in the second entry. This follows from the observation that colimits in $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}$ are computed point-wise – that is, the n restriction functors $\{\mathcal {A}_{1},\dotsc ,\mathcal {A}_{n}\}\rightarrow \mathcal {A}_{i}$ preserve colimits.

We also introduce the following notation used later on:

Our next goal is to describe an analogue of the construction of the semiorthogonal decomposition in Lemma 2.29 in the setting of dg-categories, and to show that the resulting $\infty $ -categories with semiorthogonal decompositions are equivalent. For that, we recall the notion of gluing functors of semiorthogonal decompositions of length $2$ (see also [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKSS21]).

The next proposition can be summarised as showing that Cartesian semiorthogonal decompositions of length $2$ are fully determined by their left gluing functor, and dually that co-Cartesian semiorthogonal decompositions of length $2$ are fully determined by their right gluing functor.

We now introduce a dg-analogue of Lemma 2.29: semiorthogonal decompositions arising from upper triangular dg-algebras concentrated on the diagonal and upper minor diagonal.

Definition 2.36. For $1\leq i\leq n$ , let $A_{i}$ be a dg-algebra, and for $1\leq i \leq n-1$ , let $M_{i}$ be an $A_{i}$ - $A_{i+1}$ -bimodule. We denote by

$$ \begin{align*} \mathbf{A}= \begin{pmatrix} A_{1} & M_{1} &0 & \dotsb &0 &0\\ 0 & A_{2} & M_{2}& \dotsb &0 &0\\ 0 & 0 & A_{3}& \dotsb &0 &0\\ \vdots& \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \dotsb & A_{n-1} & M_{n-1}\\ 0 & 0 & 0 & \dotsb & 0 & A_{n} \end{pmatrix} \end{align*} $$

the upper triangular dg-algebra – that is, the dg-algebra with underlying chain complex

$$ \begin{align*} \bigoplus_{1\leq i \leq n}A_{i} \oplus \bigoplus_{1\leq i \leq n-1}M_{i} \end{align*} $$

and multiplication $\cdot $ given by

$$ \begin{align*} a_{i}\cdot a_{j}^{\prime}& =\delta_{i,j}a_{i}a_{j}^{\prime}, & m_{i}\cdot m_{j}^{\prime}&= 0,\\ a_{i} \cdot m_{j}& = \delta_{i,j}a_{i}.m_{j},& m_{j}\cdot a_{i} &=\delta_{j+1,i}m_{j}.a_{i}, \end{align*} $$

where $a_{i}\in A_{i},a_{j}^{\prime }\in A_{j}$ and $m_{i}\in M_{i}, m_{j}^{\prime }\in M_{j}$ , and $\delta _{i,j}$ denotes the Kronecker delta.

Proof. The upper triangular dg-algebra $\bf A$ is quasi-isomorphic to the upper triangular dg-algebra obtained from cofibrantly replacing each $M_{i}$ . We thus assume without loss of generality that the $M_{i}$ are cofibrant bimodules. Consider the morphisms of dg-algebras $v^{i}:A_{i}\rightarrow \bf A$ and $w^{i}:\mathbf {A}\rightarrow A_{i}$ , given on the underlying chain complexes by the inclusion of the direct summand $A_{i}$ and the projection to the summand $A_{i}$ , respectively. The dg-functor and the pullback $\left (w^{i}\right )^{*}$ determine right A-modules $v^{i}_!(A_{i})$ and $\left (w^{i}\right )^{*}(A_{i})$ with underlying chain complexes $A_{i}\oplus M_{i}$ , where we set $M_{n}=0$ , and $A_{i}$ , respectively. The functors $\mathcal {D}\left (v_!^{i}\right )$ and $\mathcal {D}\left (\left (w^{i}\right )^{*}\right )$ both exhibit $\mathcal {D}(A_{i})\subset \mathcal {D}(\bf A)$ as a stable subcategory. For concreteness, we denote the stable subcategories obtained from $\mathcal {D}\left (v_!^{i}\right )$ by $\mathcal {D}(A_{i})_{v}$ and the stable subcategories obtained from $\mathcal {D}\left (\left (w^{i}\right )^{*}\right )$ by $\mathcal {D}(A_{i})_{w}$ . We wish to show that $(\mathcal {D}(A_{1})_{v},\dotsc ,\mathcal {D}(A_{n})_{v})$ is a semiorthogonal decomposition of $\mathcal {D}(\bf A)$ . For that, it suffices to show statements i) and ii) of Lemma 2.27. To show statement ii), it suffices to show conditions $1$ and $2$ of Lemma 2.28 for the pairs of stable subcategories $\mathcal {D}(A_{i})_{v},\langle \mathcal {D}(A_{i+1})_{v},\dotsc ,\mathcal {D}(A_{n})_{v}\rangle \subset \langle \mathcal {D}(A_{i})_{v},\dotsc ,\mathcal {D}(A_{n})_{v}\rangle $ for all $1\leq i \leq n$ .

We compute for an $A_{i}$ -module $N_{i}$ and an $A_{j}$ -module $N_{j}$ the mapping complex

(17) $$ \begin{align} \operatorname{Hom}_{\operatorname{dgMod}(\bf A)}\left(v_!^{i}(N_{i}),v_!^{j}\left(N_{j}\right)\right)\simeq \begin{cases} \operatorname{Hom}_{\operatorname{dgMod}\left(A_{i}\right)}\left(N_{i},N_{j}\right) & \text{if }i=j,\\ \operatorname{Hom}_{\operatorname{dgMod}\left(A_{j}\right)}\left(N_{i}\otimes_{A_{i}} M_{i},N_{j}\right) & \text{if }i+1=j,\\ 0 & \text{else}. \end{cases} \\[-17pt]\nonumber\end{align} $$

This shows condition 1 of Lemma 2.28.

We observe that the datum of a right dg-module N over $\bf A$ is equivalent to the datum of a sequence

where $N_{i}$ is a right $A_{i}\simeq \operatorname {End}^{\operatorname {dg}}\left (\left (w^{i}\right )^{*}(A_{i})\right )$ -module and $f_{i}\in M_{i}(N_{i},N_{i+1})$ . Denote by $N_{\geq i}$ the submodule

of N. We thus find distinguished triangles $N_{\geq i+1}\rightarrow N_{\geq i} \rightarrow N_{i}$ in $\operatorname {dgMod}(\bf A)$ . As shown in [Reference FaonteFao17, Theorem 4.3.1], the image under the dg-nerve of a distinguished triangle in a dg-category can be extended to a fibre and cofibre sequence. We can thus express $N\in \mathcal {D}(\bf A)$ as repeated cofibres of modules $N_{i}\in \mathcal {D}(A_{i})_{w}\subset \mathcal {D}(\bf A)$ with $1\leq i \leq n$ . A simple induction, using the fact that there exist distinguished triangles in $\operatorname {dgMod}(\bf A)$ of the form $N_{i}\rightarrow N_{i}\otimes _{A_{i}} v_!^{i}(A_{i})\rightarrow N_{i}\otimes _{A_{i}} M_{i}$ for $1\leq i\leq n-1$ and $N_{n}\in \mathcal {D}(A_{n})_{w}=\mathcal {D}(A_{n})_{v}$ , thus shows that $N\in \langle \mathcal {D}(A_{1})_{v},\dotsc ,\mathcal {D}(A_{n})_{v}\rangle $ . It follows that statement i) of Lemma 2.27 is fulfilled.

Consider the subalgebra $\mathbf {A}_{\geq i}$ of $\bf A$ with underlying chain complex

$$ \begin{align*} \bigoplus_{i\leq k\leq n}A_{k}\oplus \bigoplus_{i\leq k\leq n-1}M_{k}.\\[-17pt] \end{align*} $$

The fully faithful dg-functor $\operatorname {dgMod}(\mathbf {A}_{\geq i})\rightarrow \operatorname {dgMod}(\bf A)$ induces a fully faithful functor of $\infty $ -categories $\iota :\mathcal {D}(\bf A_{\geq i})\rightarrow \mathcal {D}(\bf A)$ . The foregoing arguments show that the essential image of $\iota $ is $\langle \mathcal {A}_{i},\dotsc ,\mathcal {A}_{n}\rangle $ and can easily be adapted to also show condition $2$ of Lemma 2.28. We have thus proven the existence of the desired semiorthogonal decomposition of $\mathcal {D}(\bf A)$ .

We now determine the ith left gluing functor of $(\mathcal {D}(A_{1})_{v},\dotsc ,\mathcal {D}(A_{n})_{v})$ . Consider the fully faithful left Quillen functor

The right adjoint is given by the Quillen functor

, the restriction of which to $\operatorname {dgMod}\left (\mathbf {A}_{\geq i+1}\right )$ is given by

, which in turn is left adjoint to

. Passing to the underlying adjunctions of $\infty $ -categories of the Quillen adjunctions shows that the ith left gluing functor of $(\mathcal {D}(A_{1}),\dotsc ,\mathcal {D}(A_{n}))$ is given by

.

Proposition 2.39. For $1\leq i\leq n$ , let $A_{i}$ be a dg-algebra, and for $1\leq i \leq n-1$ , let $M_{i}$ be an $A_{i}$ - $A_{i+1}$ -bimodule. Denote by $\bf A$ the upper triangular dg-algebra of Definition 2.36. Consider the diagram $\alpha :\Delta ^{n-1}\rightarrow \operatorname {LinCat}_{k}$ corresponding to

Then there exists an equivalence of $\infty $ -categories

$$ \begin{align*} \mathcal{D}(\textbf{A})\simeq \{\mathcal{D}(A_{1}),\dotsc,\mathcal{D}(A_{n})\} \end{align*} $$

such that for all $1\leq i\leq n$ , the following diagram commutes:

(18)

3.1 An ad hoc categorification

We begin with briefly recalling the concept of a spherical adjunction. Consider an adjunction of stable $\infty $ -categories $F:\mathcal {A}\leftrightarrow \mathcal {B}:G$ . We associate the following endofunctors:

• • The twist functor $T_{\mathcal {A}}$ is defined as the cofibre in the stable $\infty $ -category $\operatorname {Fun}(\mathcal {A},\mathcal {A})$ of the unit map $\operatorname {id}_{\mathcal {A}}\rightarrow GF$ of the adjunction $F\dashv G$ .

• • The cotwist functor $T_{\mathcal {D}}$ is defined as the fibre in the stable $\infty $ -category $\operatorname {Fun}(\mathcal {B},\mathcal {B})$ of the counit map $FG\rightarrow \operatorname {id}_{\mathcal {B}}$ of the adjunction $F\dashv G$ .

The adjunction $F\dashv G$ is called spherical if the functors $T_{\mathcal {A}}$ and $T_{\mathcal {B}}$ are equivalences. In this case, the functor F is also called spherical. A spherical functor F admits repeated left and right adjoints, each given by the composite of F or G with a power of the twist or cotwist functor. For a treatment of spherical adjunctions in the setting of stable $\infty $ -categories, we refer to [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKSS21] and [Reference ChristChr20].

A $2$ -simplicial stable $\infty $ -category is an $(\infty ,2)$ -functor from the $2$ -categorical version of the simplex category to the $(\infty ,2)$ -version $\mathcal {S}t$ of the $\infty $ -category $\operatorname {St}$ of stable $\infty $ -categories. The categorified Dold–Kan correspondence of [Reference DyckerhoffDyc21] is an adjoint equivalence between the $\infty $ -category of bounded-below complexes of stable $\infty $ -categories and the $\infty $ -category of $2$ -simplicial stable $\infty $ -categories. The right adjoint is called the categorified Dold–Kan nerve $\mathcal {N}$ . It generalises the well-known construction from K-theory called the Waldhausen $S_{\bullet }$ -construction. More precisely, given a complex of stable $\infty $ -categories concentrated in degrees $0,1$ , the categorified Dold–Kan nerve recovers Waldhausen’s relative $S_{\bullet }$ -construction. We refer to [Reference DyckerhoffDyc21] for further details.

Let $F:\mathcal {A}\leftrightarrow \mathcal {B}$ be a spherical adjunction. We consider the spherical functor $G:\mathcal {B}\rightarrow \mathcal {A}$ as a complex of stable $\infty $ -categories concentrated in degrees $0,1$ , denoted $G[0]$ . We further denote by $\mathcal {B}[1]$ the complex concentrated in degree $1$ with value $\mathcal {B}$ . Consider the morphism between bounded-below complexes of stable $\infty $ -categories $G[0]\rightarrow \mathcal {B}[1]$ depicted as follows:

Applying the categorified Dold–Kan nerve $\mathcal {N}$ , we obtain a morphism $\phi _{\ast }:\mathcal {N}(G[0])_{\ast }\rightarrow \mathcal {N}(\mathcal {B}[1])_{\ast }$ between the simplicial objects in $\operatorname {St}$ underlying the $2$ -simplicial objects in $\mathcal {S}t$ . Spelling out the definition of the categorified Dold–Kan nerve and the properties of Kan extensions (compare [Reference LurieLur09, 4.3.2.15]), we obtain the following:

We propose that for $n\geq 0$ , the $\infty $ -category $\mathcal {N}(G[0])_{n}$ of n-simplices categorifies the vector space $V_{n+1}$ of sections supported on $n+1$ outgoing rays and the $\infty $ -category $\mathcal {B}\simeq \mathcal {N}(\mathcal {B}[1])_{1}$ of $1$ -simplices categorifies the vector spaces $N_{i}$ of nearby cycles. Accordingly, we call $\mathcal {B}$ the $\infty $ -category of nearby cycles and $\mathcal {A}$ the $\infty $ -category of vanishing cycles of $F\dashv G$ , or simply of F.