Unbounded twisted complexes

We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category $B$ admitting countable direct sums and shifts. The resulting DG category of unbounded twisted complexes has a fully faithful convolution functor into Mod-$B$ which filters through $B$ if the latter admits change of differential. As an application, we rewrite definitions of $A_{\infty}$-structures in terms of twisted complexes to make them work in an arbitrary monoidal DG category or a DG bicategory.


Introduction
The notion of a twisted complex of objects in a DG category was introduced by Bondal and Kapranov [BK90].It was used as a tool to study and construct DG enhancements of triangulated categories.A one-sided twisted complex over a DG category A can be thought of as a lift to A of a bounded complex of objects in its homotopy category H 0 (A).The lift includes the maps in the complex, the homotopies up to which the consecutive maps compose to zero, and then the higher homotopies.In addition to the complex of objects in H 0 (A), such data specifies a choice of its convolution together with a collection of Postnikov systems computing this convolution [AL21,§2.4].Taking the category of one-sided twisted complexes over A is a DG realisation of taking the triangulated hull of H 0 (A).Now twisted complexes are ubiquitous in working with DG categories and their modules [Kel06] This paper generalises the notion of a twisted complex to include unbounded complexes.The authors came to need it, and expected the generalisation to be straightfoward.It turned out to involve numerous subtleties.The purpose of this short note is to write down these subtleties for the benefit of others.We also give the original application we had in mind: rewriting the definitions of A ∞ -structures [Kel01][LH03] [Lyu03] in terms of twisted complexes.This decouples them from the differential m 1 and allows them to work in an arbitrary monoidal DG category.
We now describe our results in more detail.In §2 we recall the original definition: Definition 1.1 ([BK90]).A twisted complex over a DG category A comprises • ∀ i ∈ Z, an object a i of A, non-zero for only finite number of i, • ∀ i, j ∈ Z, a degree i − j + 1 morphism α ij ∶ a i → a j in A, satisfying (−1) j dα ij + k α kj ○ α ik = 0. (1.1) The twisted complex condition should be thought of as follows.We have Yoneda embedding A ↪ Mod -A.Consider the object ⊕ a i [−i] in Mod -A.The sum ∑ α ij is its degree 1 endomorphism.Let d nat be the natural differential on ⊕ a i [−i].The condition (1.1) is equivalent to d nat + ∑ α ij squaring to zero.
In other words, a twisted complex is the data which modifies d nat to a new differential on ⊕ a i [−i].The resulting new object of Mod -A is called the convolution of the twisted complex (a i , α ij ).In the special case of twisted complexes of form a 0 → a 1 the convolution is simply the cone construction.
Degree n morphisms (a i , α ij ) → (b i , β ij ) of twisted complexes are collections {f ij } of morphisms f ij ∶ a i → b j in A of degree n + i − j.Their composition and differentiation are defined so as to ensure that the convolution becomes a fully faithful embedding of the resulting DG category Tw A into Mod -A, see Defn.2.2.Indeed, the assignment of the module ⊕ a i [−i] to a collection of objects {a i } determines the rest of the definitions of twisted complexes and their morphisms.
These definitions can be replicated for an infinite collection {a i } resulting in a "naive" notion of an unbounded twisted complex {a i , α ij }, see Defn.3.2.The sum ∑ α ij is now infinite and doesn't necessarily define a degree 1 endomorphism of ⊕ a i [−i] in Mod -A, so we impose this as an extra condition.It means that only a finite number α ij ≠ 0 for any i ∈ Z, and similarly for the components f ij of morphisms of twisted complexes.We again have Tw ± naive A ↪ Mod -A.However, often the "naive" category Tw ± naive A is not what we want.Firstly, when A is not small the category Mod -A isn't well-defined.The definition of Tw ± naive A is still valid, but to have the convolution functor we need to enlarge the universe to make A small.More importantly, even small A might admit countable shifted direct sums ⊕ a i [−i] of its objects.
The main subtlety is then that infinite direct sums, unlike finite, do not commute with the Yoneda embedding A ↪ Mod -A (Example 3.1).The direct sum ⊕ a i [−i] assigned to a twisted complex (a i , α ij ) can thus be taken in Mod -A or in A. The former leads to the "naive" category Tw ± naive A, while the latter to a strictly larger category Tw ± A A where infinite number of α ij can be non-zero for any i ∈ Z as long as ∑ α ij is still an endomorphism of ⊕ a i [−i] in A. The difference between Tw ± naive A and Tw ± A A lies only in unbounded twisted complexes.If A admits change of differential (Defn.3.5), the convolution functor takes values in A. All these considerations apply when A = Mod -C for small C (Example 3.3).
This motivates our §3 where we define unbounded twisted complexes relative to an embedding of A into another DG category B: Definition 1.2 (see Definition 3.4).Let A be a DG category with a fully faithful embedding into a DG category B which has countable direct sums and shifts.
An unbounded twisted complex over A relative to B consists of • The twisted complex condition (1.1).
The DG category Tw ± B (A) of unbounded twisted complexes over A relative to B is defined in the unique way which yields fully faithful convolution functor Any B as above admits change of differential if and only if it admits convolutions of unbounded twisted complexes.In such case for any A ⊆ B the convolution functor (1.2) takes values in B (Lemma 3.7).We can thus use unbounded twisted complexes over non-small categories without running into set-theoretic issues.
In §4 we generalise twisted complexes in another direction and define a twisted bicomplex (Defn.4.1) over A. These are bigraded twisted complexes.To work with the unbounded ones, we again fix an embedding of A into a DG category B. We denote the resulting category of unbounded twisted complexes by Twbi ± B (A).A twisted bicomplex is not a twisted complex of its rows or of its columns.It only becomes one after a sign twist.We write this down explicitly as a pair of functors We relate the images of these functors and show that both become isomorphisms if we only work with one-sided twisted complexes and bicomplexes (Prop.4.3).
Finally, in §5 we give the main application we had in mind: to reformulate and generalise the definitions of A ∞ -algebras and modules [LH03,§2] in terms of twisted complexes.This disposes with the necessity to work explicitly with the operation m 1 (the differential) and makes the definitions work in an arbitrary DG monoidal category A (or, more generally, a DG bicategory).
In §5.1 we give the resulting definitions.They all ask for the bar/cobar constructions of the A ∞ -operations to be a twisted complex.Since these constructions involve infinite number of objects, we need the theory of unbounded twisted complexes.Now, in bar constructions there is only a finite number of arrows emerging from each element of the twisted complex.Hence, our definitions of an A ∞ -algebra or an A ∞ -module are independent of the ambient category B we use to define unbounded twisted complexes.In cobar constructions this is no longer the case and the choice of B matters.These definitions are studied further in [AL23] whose §3.2 explains at length how they generalise the classical definitions [LH03, §2] [Lyu03].
In §5.2 we look at twisted complexes of A ∞ -modules.As per §5.1 let Nod ∞ -A be the category of A ∞ -modules over an A ∞ -algebra A in a monoidal DG category A. We define twisted complexes over Nod ∞ -A neither relative to Mod -(Nod ∞ -A) nor to Nod ∞ -A.Instead, we embed A into a cocomplete closed monoidal DG category B with convolutions of unbounded twisted complexes and define twisted complexes relative to Nod ∞ -A B , the category of A ∞ -A-modules in B. As Nod ∞ -A B also admits convolutions of unbounded twisted complexes (Cor.5.14), convolutions of twisted complexes over Nod ∞ -A take values in Nod ∞ -A B .We can always set B = Mod -A with the induced monoidal structure [GKL21, §4.5].However, we may need to choose differently e.g. for A ∞ -modules in a category of A ∞ -modules.
We then use the twisted bicomplex techniques we developed in §4 to prove that a twisted complex of A ∞ -modules defines an A ∞ -module structure on the twisted complex of their underlying objects in a way that gives a fully faithful embedding of the corresponding categories (Prps.5.12).It follows that the DG category Nod ∞ -A of A ∞ -modules over an A ∞ -algebra A is pretriangulated (resp.admits convolutions of unbounded twisted complexes) if and only if DG monoidal category A we work is (resp.does) (Cor.5.13).
In the Appendix we describe a homotopy transfer of structure for A ∞ -modules.We are aware of an alternative definition of the DG category of unbounded twisted complexes in [Gen22].It ignores the subtleties we consider by imposing no finiteness conditions on the differentials α ij in twisted complexes and the components f ij of their morphisms.The resulting category admits no convolution functor and is better suited to purposes different from ours.
Acknowledgements: We would like to thank Sergey Arkhipov, Alexander Efimov, and Dmitri Kaledin for useful discussions.The first author would like to thank Kansas State University for providing a stimulating research environment while working on this paper.The second author would like to offer similar thanks to Cardiff University and to Max-Planck-Institut für Mathematik Bonn.
We summarise the key notions relevant to this paper.Throughout the paper we work in a fixed universe U of sets containing an infinite set.We also fix the base field or commutative ring k we work over.
We define Mod -k to be the category of U-small complexes of k-modules.It is a cocomplete closed symmetric monoidal category with monoidal operation ⊗ k and unit k.A DG category is a category enriched over Mod -k.In particular, any DG category is locally small.If a DG category A is small, we write Mod -A for the DG category of (right) Amodules.These are functors A opp → Mod -k, so Mod -A = DGFun(A opp , Mod -k).Note that if A is not small, then Mod -A doesn't make sense.It isn't even a DG category in the above sense -its morphism spaces are no longer small and hence do not lie in Mod -k.
We can always enlarge our universe U to a universe V where A is small.This enlarges Mod -k and hence Mod -A depends on choice of V.However, in this paper we only work with Mod -A as a target for the convolution of twisted complexes over A. For these purposes, the choice of V doesn't matter -the only part of Mod -A we interact with are countable direct sums of shifts of objects of A with modified differential.
Thus, when A is not small, we mean by Mod -A the module category of A taken in any appropriate enlargement V of U.Moreover, the constructions in this paper, such as that of the category of twisted complexes over A taken relative to a DG category B, were devised precisely to enable us to replace Mod -A with something more approriate when A is not small.

Key isomorphism.
Let A be a DG-category, let E, F ∈ Mod -A and i, j ∈ Z.The theory of twisted complexes [BK90] which we summarise in §2.3 depends crucially on the choice of an isomorphism The simplest such isomorphism is: to be the map which sends any f ∈ Hom p A (E, F ) to itself considered as an element of Hom p−j+i (E[i], F [j]).In other words, forgetting the grading, in every fiber over every a ∈ A the map ψ(f ) is the same map of k-vector spaces as f .Note that ψ is not compatible with the differentials: There are at least two natural ways to fix this.Define to be the maps which send f ∈ Hom p A (E, F ) to (−1) ip ψ(f ) and (−1) i(p−j+i) ψ(f ).The difference between the two lies in whether we multiply i by the degree of f in Hom A (E, F ) or its degree in Hom Both ψ 1 and ψ 2 are isomorphisms of DG k-modules.However, they are incompatible with the composition.By this we mean the following: let E, F, G ∈ Mod -A and i, j, k ∈ Z, then e.g. the isomorphism is not a composition of ψ 1 (E, i, F, j) and ψ 1 (F, j, G, k).On the other hand, ψ, while incompatible with differentials, is compatible with composition.
The theory of twisted complexes and its fundamental definitions depend on the choice of an isomorphism (2.1).The definition of the DG category Tw(A) of twisted complexes over A is set up so as to ensure that there exists a fully faithful functor Tw(A) ↪ Mod -A called convolution, cf.§2.3.This functor is defined using the isomorphism (2.1), thus different choices would lead to different formulas in the definition of Tw(A).
The incompatibility of ψ with differentials introduces in these formulas a simple sign to every appearance of the differential d A of A, cf.(2.3) and (2.5).On the other hand, the incompatibility of ψ 1 and ψ 2 with composition introduces into the same formulas a complicated sign to every composition of two morphisms of A.
We choose to use the graded module isomorphim ψ to identify Hom ) when defining twisted complexes.We fix this choice and use it implicitly in the sections below.

Bounded twisted complexes.
Here we summarise some known facts about the usual, bounded twisted complexes.This notion was originally introduced by Bondal and Kapranov in [BK90]: Definition 2.2.A twisted complex over a DG category A is a collection of • ∀ i ∈ Z, an object a i of A, non-zero for only finite number of i, satisfying the condition Define the DG category Tw(A) of twisted complexes over A by setting where each f ∈ Hom q A (a k , b l ) has degree q + l − k and where d A is the differential on morphisms in A.
This definition ensures that Tw(A) is isomorphic to the full subcategory of Mod -A consisting of the DG A-modules whose underlying graded modules are of form ⊕ i∈Z a i [−i] with only finite number of a i ∈ A non-zero.Indeed: • The twisted complex condition (2.3) is equivalent to Here d nat is its natural differential.• The Hom-complex (2.5) is defined to have the same underlying graded k-module as and the differential (2.5) is defined so as to coincide under this identification with the one obtained on (2.6) by endowing the two direct sums with their new differentials.
We thus have a fully faithful convolution functor Conv∶ Tw(A) ↪ Mod -A which sends each (a i , α ij ) to the A-module ⊕ a i [−i] equipped with the new differential d nat + ∑ α ij .Note, that the existence of this functor can be used as the definition of the category Tw(A) once one fixes the assignment of the graded module ) is a one-sided twisted complex over A, then (a i , α ij ) is a (usual) complex over H 0 (A).Thus one-sided twisted complexes can be considered as homotopy lifts to A of usual complexes in H 0 (A).The full subcategory of Tw(A) consisting of one-sided twisted complexes is called the pretriangulated hull of A and is denoted Pre-Tr(A).We say that a DG category is pretriangulated (resp.strongly pretriangulated) if the natural embedding A ↪ Pre-Tr(A) is a quasi-equivalence (resp.equivalence).The reason for the term "pretriangulated" is that H 0 (Pre-Tr(A)) is the triangulated hull of H 0 (A) in H 0 (Mod -A), or indeed any H 0 (B) for any fully faithful embedding of A into a pretriangulated DG category B.

Unbounded twisted complexes
In this section we generalise the notions in §2.3 to unbounded twisted complexes.The generalisation seems straightforward, but there are subtleties regarding infinite direct sums.Unlike finite direct sums, these are not preserved by all DG-functors.In particular, they are not preserved by the Yoneda embedding which we used implicitly in defining the convolution of a twisted complex.
are two different A-modules, with the former being a strict submodule of the latter.Let b ∈ A, the morphisms from Υ(b) to the former module are the finite sums of b → a i .On the other hand, the morphisms from Υ(b) to the latter are the morphisms from b to ⊕ i∈Z a i , which includes some infinite sums of b → a i .In particular, if b = ⊕ i∈Z a i , then Id b is the infinite sum of Id a i .
To define an unbounded twisted complex of objects {a i } i∈Z of A, we need to choose in which category we take the infinite direct sum ⊕ i∈Z a i [−i].We can always do it in Mod -A.Then, proceeding as before, we arrive at the following definition.In it, we allow infinite number of non-zero objects a i , but then, both for twisted differentials and for the morphisms of twisted complexes, we disallow an infinite number of non-zero maps to emerge from any one object a i : Definition 3.2 ("Naive" version).An unbounded twisted complex over a DG category A consists of

satisfying
• For any i ∈ Z only finite number of α ij are non-zero, • The twisted complex condition (2.3).
Define DG category Tw ± naive (A) of unbounded twisted complexes over A by Hom • where the degree of Hom q A (a k , b l ) is q + l − k and the differential is defined by (2.5).As before, this results in the fully faithful convolution functor Conv∶ Tw ± naive (A) ↪ Mod -A.Apriori, this is the only definition we can make for an arbitrary DG category A. Indeed, unless specifically mentioned otherwise, we write Tw ± (A) for Tw ± naive (A).However, in some cases it is useful to define Tw ± (A) to be bigger than Tw ± naive (A): Example 3.3.Let A = Mod -C for some small DG category C. Assign to a collection {a i } i∈Z the representable A-module Υ(⊕ i∈Z a i [−i]), instead of the nonrepresentable A-module ⊕ i∈Z Υ(a i [−i]).This yields the definition of Tw ± (A) which is analogous to the naive one above, except we do allow infinite number of twisted differentials α ij to emerge from a single object a i as long as ∑ α ij defines an endomorphism of ⊕ i∈Z a i [−i] in A, and similarly for morphisms of twisted complexes.As before, this definition ensures that we have the fully faithful convolution functor Tw ± (A) ↪ Mod -A.However, since A = Mod -C is closed under the change of differential, this convolution filters through the Yoneda embedding.Thus we have the fully faithful functor Conv∶ Tw ± (A) → A.
In fact, it is an equivalence, since it has a right inverse -the tautological embedding A ↪ Tw ± (A) which sends any a ∈ A to itself considered as a trivial twisted complex concentrated in degree zero.We thus see that A = Mod -C is closed under convolutions of all unbounded twisted complexes in Tw ± (A).
Finally, even when A does not admit all small direct sums, there might still be a better category to take these in than Mod -A.For example, A might be a full subcategory of some Mod -C containing some infinite direct sums, but not all of them.Another example, which indeed motivated these considerations, can be found in §5.2.We thus define the following: Definition 3.4.Let A be a DG category with a fully faithful embedding into a DG category B which has countable direct sums and shifts.
An unbounded twisted complex over A relative to B consists of • The twisted complex condition (2.3).
Define DG category Tw ± B (A) of unbounded twisted complexes over A relative to B by setting wih its natural grading and the differential defined by (2.5).
Where the choice of B is clear or was fixed, we shall write Tw ± (A) for Tw ± B (A).As before, our definition ensures that we have a fully faithful convolution functor We do not need to specify which unbounded twisted complexes B admits convolutions of, because for the convolution to be representable the infinite direct sum needs to be taken in B itself.Thus we need to consider unbounded twisted complexes relative to B itself.
If B admits convolutions of unbounded twisted complexes, then the convolution Tw ± B (B) ↪ B is necessarily an equivalence.It is fully faithful and has a right inverse which sends any b ∈ B to itself considered as trivial twisted complex in degree zero.
Lemma 3.7.Let B be a DG-category which admits countable direct sums and shifts.The following are equivalent: (1) B admits change of differential.
(3) The embedding B ↪ Tw ± B (B) which sends any b ∈ B to itself considered as a trivial twisted complex in degree zero is an equivalence.(2) ⇔ (4): The "if" implication is obvious.The "only if' one results from the following commutative triangle of fully faithful functors: Then the complex consisting of b in degree 0 with a single differential f from b to itself is a twisted complex.Its convolution in Mod -B has the same graded module as b and the differential d b + f .Since B admits convolutions of twisted complexes, it is representable.
Finally, for any version of Tw ± (A), we define Tw + (A) and Tw − (A) to be its full subcategories consisting of all bounded above twisted complexes and all bounded below twisted complexes, respectively.We also define Pre-Tr ± (A), Pre-Tr + (A), and Pre-Tr − (A) to be the full subcategories of Tw ± (A), Tw + (A), and Tw − (A) consisting of one-sided twisted complexes.

Twisted bicomplexes
The following is a natural generalisation of the notion of a twisted complex: Definition 4.1.A twisted bicomplex (a ij , α ijkl ) over a DG category A comprises • ∀ i, j ∈ Z, an object a ij of A, non-zero for only finite number of pairs (i, j), Define the DG category Twbi(A) of twisted bicomplexes over A by setting where each f ∈ Hom q A (a kl , b mn ) has degree q where d A is the differential on morphisms in A.
We think of indices i and j of each a ij as the row index and the column index, respectively.We say that a twisted bicomplex is horizontally one-sided (resp.vertically one-sided) if α ijkl = 0 when l ≤ j (resp.k ≤ i).We say that a twisted bicomplex is one-sided if it is both vertically and horizontally one-sided.
The "naive" categories Twbi ± (A), Twbi + (A) and Twbi − (A) of unbounded, unbounded below, and unbounded above twisted bicomplexes, as well as their versions relative to another DG category B are defined similarly to the way they are defined in §3 for twisted complexes.We also use Twbi vos , Twbi hos and Twbi os to denote the full subcategories consisting of one-sided twisted bicomplexes.
Finally, we note that the category of twisted bicomplexes is naturally isomorphic to the category of twisted complexes of twisted complexes, but in two different ways: the complex of sign-twisted rows and the complex of sign-twisted columns.For this result, the twisted complexes over A need to be considered relative to some B which admits the convolutions of unbounded twisted complexes, cf.Lemma 3.7.This is always true when B = Mod -C for some DG-category C. Definition 4.2.Let A be DG category and fix its embedding into a DG-category B which admits convolutions of unbounded twisted complexes.Define ) as follows.Let (E i , α ik ) be an object of Tw ± B (Tw ± B (A)).Write • E i,j for the objects of each E i and α i,jl ∶ E i,j → E i,l for its differentials.
We then define: • Cxrow(E i , α ik ) to be the twisted bicomplex whose ij-th object is E i,j and whose ijkl-th differential is α ik,jl if i ≠ k and α ii,jl + (−1) i α i,jl if i = k.• Cxcol(E i , α ik ) to be the twisted bicomplex whose ij-th object is E j,i and whose ijkl-th differential is α jl,ik if j ≠ l and α jj,ik + (−1) j α j,ik if j = l.
The resulting "monodromy" Cxrow −1 ○ Cxcol of Pre-Tr ± B (Pre-Tr ± B (A)) is a nontrivial autoequivalence which takes a complex of complexes and slices up the resulting bigraded data of objects and differentials in the other direction to produce a different complex of complexes out of the same data, while sign-twisting purely horizontal and vertical differentials.

Application: A ∞ -structures in monoidal DG categories
The main application we had in mind for unbounded twisted complexes is to reformulate and generalise the definitions of A ∞ -algebras and modules [LH03, §2]: we define these structures in an arbitrary DG monoidal category A (or, more generally, a DG bicategory).This disposes with the necessity to work explicitly with the operation m 1 , i.e. the differential.
Traditionally, A ∞ -algebra formalism was defined for objects in the DG category Mod -k of DG complexes of k-modules with its natural monoidal structure given by the tensor product of complexes [LH03,§2].In Mod -k the internal differential of each object, that is -its differential as a complex of k-modules, exists as a degree 1 endomorphism of the object.It can therefore be a part of the definition of an A ∞ -algebra or A ∞ -module in Mod -k.This is no longer true if we work with an arbitraty monoidal DG category A. In Mod -A the internal differentials of objects do not appear as their degree 1 endomorphisms.Moreover, if we wanted to try and set up A ∞ -formalism to work in A itself, its objects do not possess an internal differential.
The language of twisted complexes solves both of these problems.It implicitly embeds the objects of A into Mod -A as Hom-complexes of A. These do have an internal differential: the differential d A of A. The twisted complex condition (2.3) involves d A and makes it possible to define an A ∞ -algebra or module structure on an object a ∈ A while referring explicitly only to operations {m i } i≥2 .
The resulting definitions all ask for the corresponding bar (or cobar) construction of the A ∞ -operations to be a twisted complex.These twisted complexes have to be unbounded, thus necessitating the theory developed in this paper and its subtleties.We note that in the bar constructions there is only a finite number of arrows emerging from each element of the twisted complex.Hence, our definitions of an A ∞ -algebra or an A ∞ -module are independent of the ambient category B we use to define unbounded twisted complexes.On the other hand, in cobar constructions this is no longer the case and the choice of B becomes crucial.We see another example of these subtleties coming into play when we consider twisted complexes of A ∞ -modules in §5.2.
In §5.1 we give the key definitions which are studied further [AL23].An interested reader should consult §3.2 of that paper for further explanation of the way in which these definitions generalise the classical ones in [LH03,§2].
In §5.2 we use the twisted bicomplex techniques we developed in §4 to prove several theorems about twisted complexes of A ∞ -modules.We first relate a twisted complex of A ∞ -modules to an A ∞ -module structure on the twisted complex of their underlying objects.This allows us to show that the DG category Nod ∞ -A of A ∞modules over an A ∞ -algebra A is strongly pretriangulated (resp.pretriangulated) if and only if DG monoidal category A we work in is.Hence if we expand A to Mod -A with the induced monoidal structure [GKL21, §4.5] all the categories of A ∞ -modules over all A ∞ -algebras in it become strongly pretriangulated.5.1.Definitions.Throughout this section we assume that DG monoidal category A we work with comes with a fixed choice of a monoidal embedding A ↪ B into a closed monoidal DG category B which admits convolutions of unbounded twisted complexes.Note, that we can always set B = Mod -A with the induced monoidal structure [GKL21, §4.5], enlarging our universe if necessary when A is not small.All the unbounded twisted complexes over A are then defined relative to this ambient category B.
The condition that B is closed under convolutions of unbounded twisted complexes can be replaced throughout by B being closed under the convolutions of bounded above twisted complexes and/or bounded below twisted complexes.
Definition 5.1.Let A be a monoidal DG category, let A ∈ A and let {m i } i≥2 be a collection of degree 2 − i morphisms A i → A.
The (non-augmented) bar-construction B na ∞ (A) of A is the collection of objects A i+1 for all i ≥ 0 each placed in degree −i and of degree k −1 maps d (i+k)i ∶ A i+k → A i defined by (5.1) (5.2) Definition 5.2.Let A be a monoidal DG category.An A ∞ -algebra (A, m i ) in A is an object A ∈ A equipped with operations m i ∶ A i → A for all i ≥ 2 which are degree 2 − i morphisms in A such that their non-augmented bar-construction B na ∞ (A) is a twisted complex over A. We define morphisms of A ∞ -algebras in A in a similar way: whose bar construction is a closed degree 0 morphism of twisted complexes.
We define left and right A ∞ -modules over such (A, m • ) in a similar way: Definition 5.5.Let (A, m i ) be an A ∞ -algebra in a monoidal DG category A. Let E ∈ A and let {p i } i≥2 be a collection of degree 2 − i morphisms (5.4) Definition 5.6.For E ∈ A and a collection {p i } i≥2 of degree 2 − i morphisms for all i ≥ 0 placed in degree −i and degree 1−k maps A i+k−1 ⊗E → A i−1 ⊗E defined by (5.6) Definition 5.7.Let A be a monoidal DG category and let (A, m i ) be an Definition 5.8.Let A be a monoidal DG category and let (A, m i ) be an A ∞algebra in A. Let (E, p k ) and (F, q k ) be right A ∞ -modules over A in A.
We illustrate the case when f • is of odd degree: The corresponding definition for the left A ∞ -modules differs only in signs: Definition 5.9.Let A be a monoidal DG category and let (A, m i ) be an We define the DG categories of left and right modules over A in the unique way which makes the left and right module bar constructions into faithful DG functors from these categories to Pre-Tr − (A): Definition 5.10.Let A be a monoidal DG category and A be an A ∞ -algebra in A. Define the DG category Nod ∞ -A of right A ∞ -A-modules in A by: • Its objects are right A ∞ -A-modules in A, • For any E, F ∈ Ob Nod ∞ A, the complex Hom • Nod∞A (E, F ) consists of A ∞morphisms f • ∶ E → F with their natural grading.The differential and the composition is defined by composing the corresponding twisted complex morphisms.
• The identity morphism of E ∈ Nod ∞ A is the morphism (f • ) with f 1 = Id E and f ≥2 = 0 whose corresponding twisted complex morphism is Id B∞(E) .The DG category A-Nod ∞ of left A ∞ -A-modules in A is defined analogously.Similar definitions exist for A ∞ -coalgebras and A ∞ -comodules, see [AL23, §6].

Twisted complexes of
A ∞ -modules.The notion of an A ∞ -module over an A ∞ -algebra (A, m • ) in a monoidal DG category A which we defined in §5.1 differs in several ways from the usual notion which corresponds to the case A = Mod -k.
One is that the DG-category of usual A ∞ -modules is strongly pretriangulated, while in our generality Nod ∞ -A doesn't have to be.In this section we show that Nod ∞ -A is strongly pretriangulated if and only if A is strongly pretriangulated.
First, we need to fix our conventions.As in §5.1 we assume that our monoidal DG category A comes with a monoidal embedding into a closed monoidal DG category B which has convolutions of unbounded twisted complexes.Recall that we can always set B = Mod -A with the induced monoidal structure [GKL21, §4.5], We define Tw ± A and Tw ± B relative to B. Thus twisted complexes in Tw ± A and Tw ± B can have infinite number of differentials and/or morphism components emerge from a single object, but only if their sum still defines a morphism in B. By Lemma 3.7, since B admits convolutions of unbounded twisted complexes, the convolution functor Tw ± B ↪ B is an equivalence.
(2) A closed degree zero map of A ∞ -modules φ • ∶ P → Q extending f ; (3) A closed degree zero map of A ∞ -modules ψ • ∶ P → Q extending g; (4) A degree −1 map of A ∞ -modules H • ∶ P → P extending h, such that To prove this, we use the bar-construction as a fully faithful embedding of the DG-category of A ∞ -A-modules into the DG-category of DG-B ∞ A-comodules.See [LH03, §2.3.3].We give a brief summary.Let E and F be two A ∞ -A-modules.Let x • ∶ E → F be a degree n morphism of A ∞ -modules.By definition, it is an arbitrary collection of degree n + 1 − i maps in Mod-k A .Such collection is equivalent to the data of a degree n Mod-k A map because as a graded module B ∞ E is just E ⊗ k B ∞ A. By universal property of free graded comodules, there is a bijective correspondence of k A -module morphisms with the morphisms of DG-B ∞ A-comodules and, conversely, sends any morphism x of DG-B ∞ A-comodules to Here ∆ and ǫ is are the comultiplication and the counit of B ∞ A.
Proof.By our convention, the A ∞ -structure µ • on P is a collection of µ i for i ≥ 2. View it as the data of a degree one A ∞ -morphism µ • ∶ P → P with µ 1 = 0.The differential on the bar construction where μ is the bar construction of the A ∞ -morphism µ • as per (A.1) and d nat is the natural differential on the tensor product P ⊗ k B ∞ A. We then have Conv∶ Tw ±B (A) → Mod -B.We have the commutative square of fully faithful embeddings -functors preserve finite direct sums, we see that on bounded twisted complexes the convolution functor into Mod -B is simply the composition of the usual convolution functor into Mod -A and I * .Observer that setting B = Mod -A recovers the definition of Tw ± naive (A) with the convolution into Mod -A.On the other hand, when we have A = Mod -C for some small DG-category C, setting B = A recovers the category constructed in Example 3.3 with its convolution into Mod -A which filters through A. Definition 3.5.A DG category B admits change of differential if for all b ∈ B and f ∈ Hom 1 B (b, b) with df + f 2 = 0 the module in Mod -B which has the underlying graded module of Hom B (−, b) and the differential d Hom B (−,b) + f is representable.Definition 3.6.A DG category B admits convolutions of unbounded twisted complexes if it admits countable direct sums and shifts and the convolution functor Tw ± B (B) ↪ Mod -B filters through B ↪ Mod -B.

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For any DG-category A with an embedding into B, Tw ± B (A) → Mod -B filters through B ↪ Mod -B.Proof.(1) ⇒ (2): This is the same argument as in Example 3.3.(2) ⇔ (3): The composition B ↪ Tw ± B (B) ↪ Mod -B is the Yoneda embedding.Thus Tw ± B (B) ↪ Mod -B filters through the Yoneda embedding if and only if B ↪ Tw ± B (B) admits a right quasi-inverse.A fully faithful functor admits a right quasi-inverse if and only if it is an equivalence.