Homology, Homotopy and Applications, vol.3, No.1, 2001, pp.1–35

INTRODUCTION TO A-INFINITY ALGEBRAS AND MODULES

DEDICATED TO H. KELLER ON THE OCCASION OF HIS SEVENTY FIFTH BIRTHDAY

BERNHARD KELLER (communicated by Lionel Schwartz) Abstract These are expanded notes of four introductory talks on A_{∞}-

algebras, their modules and their derived categories.

Contents

1 Introduction 1 2 Motivation and topological origin 3 3 A_{∞}-algebras and their morphisms 6 4 A_{∞}-modules and the derived category 14 5 Triangulated structure 17 6 Standard functors 18 7 Twisted objects 21 8 On the proofs 28

1. Introduction

1.1. These notes

These are expanded notes of a minicourse of three lectures given at the Euroconference ‘Homological Invariants in Representation Theory’ in Ioannina, Greece, March 16 to 21, 1999, and of a talk at the Instituto de Matemáticas, UNAM, México, on April 28, 1999. They present basic results on A_{∞}-algebras, their modules and their derived categories.

Partially supported by the European Network ‘Algebraic Lie Representations’, Contract ERB-FMRX-CT97-0100 Received January 17, 2000, revised January 13, 2001; published on January 22, 2001. 2000 Mathematics Subject Classification: 18E30, 16D90, 18G40, 18G10, 55U35 Key words and phrases: A-infinity algebra, Derived category \begin{matrix}c \\ ◯ \end{matrix} 2001, Bernhard Keller. Permission to copy for private use granted.

1.2. History

A_{∞}-spaces and A_{∞}-algebras (= sha algebras = strongly homotopy associative algebras) were invented at the beginning of the sixties by J. Stasheff [45] [46] as a tool in the study of ‘group-like’ topological spaces. In the subsequent two decades, A_{∞}-structures found applications and developments [30] [3] [1] in homotopy theory; their use remained essentially confined to this subject (cf. however [34], [42]). This changed at the beginning of the nineties when the relevance of A_{∞}-structures in algebra, geometry and mathematical physics became more and more apparent (cf. e.g. [11], [48], [31]). Of special influence was M. Kontsevich’s talk [23] at the International Congress in 1994: Inspired by K. Fukaya’s preprint [8] Kontsevich gave a conjectural interpretation of mirror symmetry as the ‘shadow’ of an equivalence between two triangulated categories associated with A_{∞}-categories. His conjecture was proved in the case of elliptic curves by A. Polishchuk and E. Zaslow [37].

1.3. Scope and sources

In these notes, we aim at presenting some basic theorems on A_{∞}-algebras, their modules and their derived categories. The notes may serve as an introduction to a (very) small part of Kontsevich’s lectures [25]. Besides these, our main sources were [38], [46], [43], [15], [16], [18]. Some of the results we state are not proved in this form in the literature. Complete proofs will be given in [28].

1.4. Contents

In section 2, we motivate the introduction of A_{∞}-algebras and modules by two basic problems from homological algebra:

1. the reconstruction of a complex from its homology,

2. the reconstruction of the category of iterated selfextensions of a module from

its extension algebra. Then we briefly present the topological origin of A_{∞}-structures. Section 3 is devoted to A_{∞}-algebras and their morphisms. The central result is the theorem on the existence of minimal models. In sections 4 and 5, we introduce the derived category of an A_{∞}-algebra and we present the natural framework for the solution of problem 1. In section 6, we sketch the formalism of standard functors and arrive at the solution of problem 2. Section 7 presents the category of twisted objects, which is of importance because it is ‘computable’.

1.5. Omissions

The links with Morita theory for derived categories [39] or dg categories [21] have not been made explicit, cf. however [22].

The relevance of A_{∞}-algebras to boxes and matrix problems was discovered by S. Ovsienko [34] (cf. also [35]). We only give a hint of this important development in example 7.9.

The notions of A_{∞}-equivalence and of A_{∞}-enhanced triangulated categories, crucial for [25], will be treated in [28].

Acknowledgments. The author is grateful to N. Marmaridis and C. M. Ringel for the organization of the Euroconference in Ioannina, Greece, where the material of sections 2 to 6 was first presented. He expresses his sincere thanks to J. A. de la Peña and C. Geiss for the hospitality he enjoyed at the Instituto de Matemáticas, UNAM, México, where the material of section 7 was first presented and a first draft of these notes completed.

2. Motivation and topological origin

2.1. Motivation

Let k be a field. The following two problems will guide us in our exploration of the world of A_{∞}-algebras and modules.

Problem 1. Let A be an associative k-algebra with 1, let

M = (… → M^{p} →^{d^{p}} M^{p+1} → …)

be a complex of (unital right) A-modules and let

H^{∗}M = ker d^{∗}/ im d^{∗} , ∗ ∈ Z ,

be its homology. It is a naive but natural question to ask whether, up to quasiisomorphism, we can reconstruct the complex M from its homology H^{∗}M. This is of course impossible, except under very restrictive hypotheses, for example when A is an hereditary algebra (i.e. an algebra such that submodules of projective Amodules are projective, see [6]). So we ask what additional structure is needed if we want to reconstruct M from its homology. The answer is that H^{∗}M carries a unique (up to isomorphism) A_{∞}-module structure over A (viewed as an A_{∞}-algebra) which encodes exactly the additional information needed for this task.

Problem 2. Let B be an associative k-algebra with 1, let M_{1}, …, M_{n} be Bmodules and let filt(M_{i}) denote the full subcategory of the category of right Bmodules whose objects admit finite filtrations with subquotients among the M_{i}. In other words, filt(M_{i}) is the closure under extensions of the M_{i}. A natural question to ask is whether filt(M_{i}) is determined by the extension algebra

Ext_{B}^{∗} (M, M) , where M = ⨁ M_{i} ,

together with its idempotents corresponding to the M_{i}. Again, the answer is no. This time, it is no even if we assume that B is hereditary. Again, we ask what additional structure on the extension algebra is needed to reconstruct the category of iterated extensions. The answer is that the extension algebra admits an A_{∞}-structure which encodes the additional information needed for this task.

The aim of these lectures is to explain the answers to the two problems and to show how they fit into a general theory.

2.2. Topological origin

At the beginning of the sixties, J. Stasheff invented A_{∞}-spaces and A_{∞}-algebras [45], [46] as a tool in the study of ‘group-like’ spaces. Let us consider the basic example:

Let (X, ∗) be a topological space with a base point ∗ and let ΩX denote the space of based loops in X: a point of ΩX is thus a continuous map f : S^{1} → X taking the base point of the circle to the base point ∗. We have the composition map

m_{2} : ΩX × ΩX → ΩX

sending a pair of loops (f_{1}, f_{2}) to the loop f_{1} ∗ f_{2} = m_{2}(f_{1}, f_{2}) obtained by running through f_{1} on the first half of the circle and through f_{2} on the second half

f_{1} ∗ f_{2}

This composition is non associative: for three loops f_{1}, f_{2}, f_{3}, the composition (f_{1}∗f_{2})∗f_{3} runs through f_{1} on the first quarter of the circle whereas the composition f_{1} ∗ (f_{2} ∗ f_{3}) runs through f_{1} on the first half of the circle. We symbolize the two possibilities by the two binary trees with three leaves

(f_{1} ∗ f_{2}) ∗ f_{3}

f_{1} ∗ (f_{2} ∗ f_{3})

Clearly, there is an homotopy

m_{3} : [0, 1] × ΩX × ΩX → ΩX.

joining the two possibilities of composing three loops. When we want to compose 4 factors, there are 5 possibilities corresponding to the 5 binary trees with 4 leaves. Using m_{3}, we obtain two paths of homotopies joining the composition (f_{1}, f_{2}, f_{3}, f_{4}) ↦ ((f_{1} ∗ f_{2}) ∗ f_{3}) ∗ f_{4} to the composition (f_{1}, f_{2}, f_{3}, f_{4}) ↦ f_{1} ∗ (f_{2} ∗ (f_{3} ∗ f_{4})).

These two paths are clearly homotopic. Let us denote an homotopy by

m_{4} : K_{4} × (ΩX)^{4} → ΩX

where K_{4} denotes the pentagon bounded by the two paths. When we want to compose 5 factors, there are 14 possibilities corresponding to the 14 binary trees with 5 leaves. Using m_{4} and m_{3}, we obtain paths linking the compositions and faces linking the paths. The resulting sphere is the boundary of the polytope K_{5} depicted [48] below.

The pentagonal faces are copies of K_{4}.

More generally, Stasheff [45] defined polytopes K_{n} of dimension n−2 for all n ⩾ 2 (we have K_{2} = ∗, K_{3} = [0, 1]) and he defined an A_{∞}-space to be a topological space Y endowed with maps m_{n} : K_{n} × Y^{n} → Y, n ⩾ 2, satisfying suitable compatibility conditions and admitting a ‘strict unit’. The loop space ΩX is the prime example of such a space Y. Conversely [1, 2.2], a topological space that admits the structure of an A_{∞}-space and whose connected components form a group is homotopy equivalent to a loop space.

If Y is an A_{∞}-space, the singular chain complex of Y is the paradigmatic example of an A_{∞}-algebra [46].

3. A_{∞}-algebras and their morphisms

3.1. Definition of an A_{∞}-algebra

Let k be a field. An A_{∞}-algebra over k (also called a ‘strongly homotopy associative algebra’ or an ‘sha algebra’) is a Z-graded vector space

A = ⨁_{p∈Z} A^{p}

endowed with graded maps (=homogeneous k-linear maps)

m_{n} : A^{⊗n} → A , n ⩾ 1 ,

of degree 2 − n satisfying the following relations • We have m_{1} m_{1} = 0, i.e. (A, m_{1}) is a differential complex.

• We have

m_{1} m_{2} = m_{2} (m_{1} ⊗ 1 + 1 ⊗ m_{1})

as maps A^{⊗2} → A. Here 1 denotes the identity map of the space A. So m_{1} is a (graded) derivation with respect to the multiplication m_{2}.

• We have

m_{2}(1 ⊗ m_{2} − m_{2} ⊗ 1)
= m_{1}m_{3} + m_{3} (m_{1} ⊗ 1 ⊗ 1 + 1 ⊗ m_{1} ⊗ 1 + 1 ⊗ 1 ⊗ m_{1})

as maps A^{⊗3} → A. Note that the left hand side is the associator for m_{2} and that the right hand side may be viewed as the boundary of m_{3} in the morphism complex Hom^{•}_{k} (A^{⊗3}, A) (the definition of the morphism complex is recalled below). This implies that m_{2} is associative up to homotopy.

• More generally, for n ⩾ 1, we have

∑(−1)^{r+st} m_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = 0

where the sum runs over all decompositions n = r+s+t and we put u = r+1+t. We have adopted the sign conventions of Getzler-Jones [11]. Note that when these formulas are applied to elements, additional signs appear because of the Koszul sign rule:

(f ⊗ g)(x ⊗ y) = (−1)^{|g||x|}f(x) ⊗ g(y)

where f, g are graded maps, x,y homogeneous elements and the vertical bars denote the degree. For example, we have

(m_{1} ⊗ 1 + 1 ⊗ m_{1})(x ⊗ y) = m_{1}(x) ⊗ y + (−1)^{|x|}x ⊗ m_{1}(y)

so that m_{1} ⊗ 1 + 1 ⊗ m_{1} is the usual differential on the tensor product. The defining identities are pictorially represented [25] by

∑ ± = 0.

We note three immediate consequences of the definition:

(1) In general, an A_{∞}-algebra A is not associative but its homology H^{∗}A = H^{∗}(A, m_{1}) is an associative graded algebra for the multiplication induced by m_{2}.

(2) If A^{p} = 0 for all p ≠ 0, then A = A^{0} is an ordinary associative algebra. Indeed, since m_{n} is of degree 2 − n, all m_{n} other than m_{2} have to vanish.

(3) If m_{n} vanishes for all n ⩾ 3, then A is an associative dg (=differential Zgraded) algebra and conversely each dg algebra yields an A_{∞}-algebra with m_{n} = 0 for all n ⩾ 3.

Note that we have not defined unitality for A_{∞}-algebras. Cf. however 3.5 below.

3.2. Link with deformation theory

The following class of examples, due to Penkava-Schwarz [36], shows the close link between deformation theory and A_{∞}-algebras (cf. also [26]): Let B be an ordinary associative algebra, n ⩾ 1 an integer and ε an indeterminate of degree 2 − n and with ε^{2} = 0. An infinitesimal n-deformation of B is a k[ε]-multilinear A_{∞}-algebra structure on B ⊗ k[ε] which lifts the multiplication of B. Thus an infinitesimal 2-deformation is just an algebra deformation of B. It is well-known and easy to check that infinitesimal 2-deformations are in natural bijection with Hochschild 2cocycles c : B^{⊗2} → B. Analogously, infinitesimal n-deformations are in natural bijection with Hochschild n-cocycles. Thus, by working in the class of A_{∞}-algebras, we may interpret Hochschild cocycles of arbitrary degree ⩾ 1 as deformations.

3.3. Minimal models

Let us recall the dg algebra which is implicit in problem 2: Let B be an ordinary associative k-algebra with 1 and and M a B-module. Choose an injective resolution

0 → M → I^{0} → I^{1} → …

of M and denote by I the complex

0 → I^{0} → I^{1} → ….

Let Hom^{•}_{B} (I, I) be the morphism complex: recall that its nth component is the group of graded B-linear maps f : I → I of degree n (no compatibility with d_{I}) and its differential is defined by d(f) = d_{I} ∘ f − (−1)^{n}f ∘ d_{I}. Then A = Hom^{•}_{B} (I, I) endowed with its differential becomes a dg algebra for the natural composition of graded maps and its homology algebra is the extension algebra of M:

H^{∗}A = Ext_{B}^{∗} (M, M).

We claim that this has a canonical (up to equivalence) structure of A_{∞}-algebra with m_{1} = 0 and m_{2} the usual multiplication. Indeed, more generally, we have the

Theorem (Kadeishvili [15], see also [14], [43], [38], [12], [13], [32]). Let A be an A_{∞}-algebra. Then the homology H^{∗}A has an A_{∞}-algebra structure such that

1) we have m_{1} = 0 and m_{2} is induced by m_{2}^{A} and

2) there is a quasi-isomorphism of A_{∞}-algebras H^{∗}A → A lifting the identity of

H^{∗}A. Moreover, this structure is unique up to (non unique) isomorphism of A_{∞}-algebras.

The notion of morphism and quasi-isomorphism of A_{∞}-algebras will be defined below (3.4). The most explicit construction of the A_{∞}-structure on H^{∗}A can be found in Merkulov’s [32]. His construction is explained in terms of trees in [27, 6.4].

Let us introduce some terminology: An A_{∞}-algebra with vanishing m_{1} is called minimal. In the situation of the theorem, the A_{∞}-algebra H^{∗}A is called a minimal model for A. The A_{∞}-algebra A is called formal if its minimal model can be chosen to be an ordinary associative graded algebra, i.e. such that m_{n} = 0 for all n ⩾ 3.

A graded algebra B is intrinsically formal if each A_{∞}-algebra A whose homology is isomorphic to B is formal. A sufficient condition for B to be intrinsically formal in terms of its Hochschild homology is given in [19], cf. also [41, 4.7].

It was shown in [7] that the dg algebra of smooth real forms on a compact Kähler manifold is formal. In particular, this holds for smooth, complex projective manifolds.

In contrast to the statement of the above theorem, one can also show that each A_{∞}-algebra A admits an ‘anti-minimal’ model, i.e. there is a dg algebra C and a quasi-isomorphism of A_{∞}-algebras A → C. Thus, passing from dg algebras to A_{∞}- algebras does not yield new quasi-isomorphism classes. What it does yield is a new description of these classes by minimal models.

If, in the above example, we choose B to be a finite-dimensional algebra and M to be the sum of the simple B-modules, one can show that the dg algebra A is formal iff B is a Koszul algebra. The smallest examples where A is not formal is the algebra B defined by the quiver with relations (cf. [40], [9], [2])

• →^{α} • →^{β} • →^{γ} • , γ β α = 0.

Here the extension algebra H^{∗}A is given by the quiver with relations

, ab = 0 , bc = 0 ,

where the arrows a, b, c are of degree 1 and the arrow e is of degree 2. We have m_{n} = 0 for n ≠ 2, 3, m_{3}(a, b, c) = e and all other products vanish (the method for computing this is given at the end of section 3.4). This example illustrates the fact that the higher products of the A_{∞}-algebra H^{∗}A lift the Massey products [46] whenever these are defined.

3.4. Morphisms of A_{∞}-algebras A morphism of A_{∞}-algebras f : A → B is a family

f_{n} : A^{⊗n} → B

of graded maps of degree 1 − n such that • we have f_{1}m_{1} = m_{1}f_{1}, i.e. f_{1} is a morphism of complexes,

• we have

f_{1}m_{2} = m_{2} (f_{1} ⊗ f_{1}) + m_{1}f_{2} + f_{2} (m_{1} ⊗ 1 + 1 ⊗ m_{1}) ,

which means that f_{1} commutes with the multiplication m_{2} up to a homotopy given by f_{2},

• more generally, for n ⩾ 1, we have

∑(−1)^{r+st} f_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = ∑(−1)^{s} m_{r} (f_{i_{1}} ⊗ f_{i_{2}} ⊗ … ⊗ f_{i_{r}})

where the first sum runs over all decompositions n = r + s + t, we put u = r + 1 + t, and the second sum runs over all 1 ⩽ r ⩽ n and all decompositions n = i_{1} + ⋯ + i_{r}; the sign on the right hand side is given by

s = (r − 1)(i_{1} − 1) + (r − 2)(i_{2} − 1) + ⋯ + 2(i_{r−2} − 1) + (i_{r−1} − 1).

The morphism f is a quasi-isomorphism if f_{1} is a quasi-isomorphism. It is strict if f_{i} = 0 for all i ≠ 1. The identity morphism is the strict morphism f such that f_{1} is the identity of A. The composition of two morphisms f : B → C and g : A → B is given by

(f ∘ g)_{n} = ∑(−1)^{s}f_{r} ∘ (g_{i_{1}} ⊗ ⋯ ⊗ g_{i_{r}})

where the sum and the sign are as in the defining identity. Here is a (slightly stylized) pictorial representation of the defining identity for a morphism of A_{∞}-algebras

∑ ±  = ∑ ±  .

A_{∞}-algebras may be viewed as algebras over an operad (cf. e.g. [29]). However, only strict morphisms of A_{∞}-algebras may then be viewed as morphisms of algebras.

As an illustration of the notion of morphism, suppose that A is a dg algebra and let us construct the multiplication m_{3} on H^{∗}A and, at the same time, the first three terms of a quasi-isomorphism f : H^{∗}A → A lifting the identity of H^{∗}A. For this, we view H^{∗}A as a complex with zero differential and we choose a morphism of complexes f_{1} : H^{∗}A → A inducing the identity in homology. Then f_{1} commutes with the multiplication up to boundaries, i.e. there exists a graded map f_{2} : H^{∗}A ⊗ H^{∗}A → A degree −1 such that

f_{1} m_{2} = m_{2} (f_{1} ⊗ f_{1}) + m_{1} f_{2}.

Now we look for f_{3} and m_{3} such that

f_{1} ∘ m_{3} + f_{2} ∘ (m_{2} ⊗ 1 − 1 ⊗ m_{2})
+f_{3} ∘ (m_{1} ⊗ 1^{⊗2} + 1 ⊗ m_{1} ⊗ 1 + 1^{⊗2} ⊗ m_{1})
= m_{3} ∘ (f_{1} ⊗ f_{1} ⊗ f_{1}) + m_{2} ∘ (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) + m_{1}f_{3}.

In our situation, this simplifies into

f_{1} ∘ m_{3} = m_{2} ∘ (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) + m_{1}f_{3}.

Now it is easy to check that the map m_{2} (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) has its image in the cycles of A. So there is indeed a map f_{3} : (H^{∗}A)^{⊗3} → A of degree −2 and a map m_{3} : (H^{∗}A)^{⊗3} → H^{∗}A of degree −1 such that the identity holds.

3.5. Augmented A_{∞}-algebras

If an A_{∞}-algebra A has a unital homology algebra H^{∗}A, one can often simplify the computation of the minimal model for A by passing to the category of augmented A_{∞}-algebras. Here are the relevant definitions:

A strictly unital A_{∞}-algebra is an A_{∞}-algebra A endowed with an element 1_{A} of degree 0 such that we have m_{1}(1_{A}) = 0, m_{2}(1_{A}, a) = a = m_{2}(a, 1_{A}) for all a ∈ A and such that, for all i > 2, and all a_{1}, …, a_{i} in A, the product m_{i}(a_{1}, …, a_{i}) vanishes if one of the a_{j} equals 1_{A}. If A and B are strictly unital A_{∞}-algebras, a morphism of A_{∞}-algebras f : A → B is strictly unital if we have f_{1}(1_{A}) = 1_{B} and if for all i > 1 and all a_{1}, …, a_{i}, the element f_{i}(a_{1}, …, a_{i}) vanishes if one of the a_{j} equals 1. Each strictly unital A_{∞}-algebra is canonically endowed with a strict (hence strictly unital) morphism η : k → A mapping 1_{k} to 1_{A}. It is augmented if it is moreover endowed with a strictly unital morphism ε : A → k such that ε∘η = 1_{k}. A morphism of augmented A_{∞}-algebras is a strictly unital morphism f : A → B such that ε_{B} ∘ f = ε_{A}. It is then easy to check that the functor A ↦ ker ε is an equivalence from the category of augmented A_{∞}-algebras to the category of A_{∞}- algebras. The quasi-inverse functor sends an A_{∞}-algebra B to k ⊕ B endowed with the unique strictly unital structure extending the structure of B.

Let A be an augmented A_{∞}-algebra and let \overline{A} = ker ε. Clearly we obtain a minimal model for A by taking k ⊕ \overline{A}′, where \overline{A}′ is a minimal model for \overline{A} in the category of A_{∞}-algebras. The computation of the minimal model for \overline{A} is simpler than that of A, because we do not have to deal with the ‘loop’ 1.

The construction of minimal models and the above discussion extend to the case where the ground ring k is no longer a field but a semi-simple ring R, for example a product of fields. We then consider A_{∞}-algebras in the monoidal category of R-R-bimodules.

For example, suppose that A is an ordinary algebra augmented over a semi-simple ring R, say A = R ⊕ \overline{A}. Then the standard complex computing Ext_{A}(R, R) is of the form

0 → R →^{0} Hom_{R}(\overline{A}, \overline{A}) → Hom_{R}(\overline{A} ⊗_{R} \overline{A}, \overline{A}) → …

It has a canonical structure of dg algebra augmented over R. Thus the A_{∞}-algebra Ext_{A}(R, R) may be chosen to be augmented over R and, in particular, strictly unital.

Let us consider another example: Suppose that k is a field and that A is a unital dg k-algebra endowed with a decomposition

1_{A} = e_{1} + … + e_{N}

of 1_{A} into idempotents e_{i} = e_{i}^{2} such that the complexes e_{j}Ae_{i} are acyclic for i < j and that e_{i}Ae_{i} is quasi-isomorphic to k e_{i}. Clearly A is isomorphic to the matrix algebra

⨁_{i,j=1}^{N} e_{j}Ae_{i}

and this algebra is quasi-isomorphic to its subalgebra of upper triangular matrices

B = ke_{1} ⊕ … ⊕ ke_{N} ⊕ ⨁_{i<j} e_{j}Ae_{i}.

Clearly B is an augmented algebra over R = k × … × k (N factors). So we obtain a minimal model for A as the sum of R with a minimal model for

\overline{B} = ⨁_{i<j} e_{j}Ae_{i}.

viewed as an A_{∞}-algebra over R. Let \overline{B}′ be this minimal model. Then we have a decomposition

\overline{B}′ = ⨁_{i<j} H^{∗}(e_{j}Ae_{i}) = ⨁_{i<j} e_{j}H^{∗}(A)e_{i}

of R-R-bimodules and the tensor power

\overline{B}′ ⊗_{R} \overline{B}′ ⊗_{R} … ⊗_{R} \overline{B}′ (n factors)

is isomorphic to the sum of the

e_{i_{n+1}}H^{∗}(A)e_{i_{n}} ⊗_{k} e_{i_{n}}H^{∗}(A)e_{i_{n−1}} ⊗ … ⊗_{k} e_{i_{2}}H^{∗}(A)e_{i_{1}}

taken over all sequences i_{1} < i_{2} < … < i_{n+1}. These spaces vanish for n ⩾ N so that in the minimal model we automatically have m_{n} = 0 for n ⩾ N. For example, if N = 3, i.e. if the unit of A decomposes into 1 = e_{1}+e_{2}+e_{3}, then \overline{B}′ is automatically formal. Hence \overline{B} and A are also formal in this case.

The following lemma presents another case where the passage to the category of augmented A_{∞}-algebras is easy. A subalgebra of an A_{∞}-algebra A is a graded subspace B such that m_{i} maps B^{⊗i} to B for all i ⩾ 1.

Lemma. Let A be a strictly unital A_{∞}-algebra (for example a unital dg algebra). Suppose that H^{i}A = 0 for i < 0 and that H^{0}A = k. Then there is a subalgebra B ⊂ A whose inclusion is a quasi-isomorphism and which is augmented.

3.6. The bar construction

Following [46] and [16] we will restate the definitions of A_{∞}-algebras and their morphisms in a more efficient way. Let V be a Z-graded vector space and let

\overline{T} V = V ⊕ V^{⊗2} ⊕ ⋯

be the reduced tensor algebra (in the category of graded spaces). We make \overline{T} V into a graded coalgebra via the comultiplication

Δ : \overline{T} V → \overline{T} V ⊗ \overline{T} V

defined by

Δ(v_{1}, …, v_{n}) = ∑_{1⩽i⩽n}(v_{1}, …, v_{i}) ⊗ (v_{i+1}, …, v_{n}) .

Note that Δ(v) = 0 and Δ(v_{1}, v_{2}) = v_{1} ⊗ v_{2}. It is then easy to see that each graded map b : \overline{T} V → V of degree 1 uniquely lifts to a graded coderivation b : \overline{T} V → \overline{T} V of degree 1, where coderivation means that we have

Δb = (b ⊗ 1 + 1 ⊗ b) Δ.

To see that this is plausible, one should remember that algebra derivations \overline{T} V → \overline{T} V are in bijection with maps V → \overline{T} V. The component of b that maps V^{⊗n} to V^{⊗u} is

1^{⊗r} ⊗ b_{s} ⊗ 1^{⊗t}

where r + 1 + t = u. Thus we have a bijection between coderivations b : \overline{T} V → \overline{T} V of degree 1 and families of maps b_{n} : V^{⊗n} → V, n ⩾ 1, of degree 1.

Now let A be another Z-graded space and let SA be its suspension: (SA)^{p} = A^{p+1}. Define a bijection between the families of maps b_{n} : (SA)^{⊗n} → SA of degree 1 and maps m_{n} : A^{⊗n} → A, n ⩾ 1, by the commutative square

\begin{matrix}(SA)^{⊗n} & →^{b_{n}} & SA \\ s^{⊗n} ↑ & & ↑ s \\ A^{⊗n} & →^{m_{n}} & A \end{matrix} ,

where s : A → SA is the canonical map a ↦ a of degree −1. Note that m_{n} is of degree −n + 1 − (−1) = 2 − n. Then we have the following easy

Lemma. The following are equivalent

(i) The maps m_{n} : A^{⊗n} → A yield an A_{∞}-algebra structure on A.

(ii) The coderivation b : \overline{T} SA → \overline{T} SA satisfies b^{2} = 0.

(iii) For each n ⩾ 1, we have

∑ b_{u} (1^{⊗r} ⊗ b_{s} ⊗ 1^{⊗t}) = 0 ,

where the sum runs over all decompositions n = r+s+t and we put u = r+1+t.

Note that no signs appear in the last formula. The lemma yields a bijection between A_{∞}-algebra structures on A and coalgebra differentials on \overline{T} SA. It is also easy to check that if A and B are A_{∞}-algebras, then morphisms of A_{∞}-algebras f : A → B are in bijection with morphisms of dg coalgebras \overline{T} SA → \overline{T} SB.

3.7. Homotopy equivalences and quasi-isomorphisms

Let A and B be A_{∞}-algebras and f, g morphisms from A to B. Let F, G denote the corresponding morphisms of dg coalgebras \overline{T} SA → \overline{T} SB. One defines f and g to be homotopic if there is a homogeneous map H : \overline{T} SA → \overline{T} SB of degree −1 such that

ΔH = F ⊗ H + H ⊗ G and F − G = b ∘ H + H ∘ b.

It is easy to translate this into the existence of a family of maps h_{n} : A^{⊗n} → B satisfying suitable conditions.

Theorem (Prouté [38], cf. also [18]). (a) Homotopy is an equivalence relation on the set of morphisms of A_{∞}-algebras A → B.

(b) A morphism of A_{∞}-algebras s : A → B is a quasi-isomorphism iff it is an homotopy equivalence (i.e. it becomes an isomorphism in the category obtained by dividing the category of A_{∞}-algebras by the homotopy relation).

4. A_{∞}-modules and the derived category

4.1. Reminder on the derived category

We refer to [20, Ch. I] for a concise presentation of derived categories and to [51] and [10] for more leisurely introductions. We refer to [44] for foundational material on resolutions of unbounded complexes.

Let A be an associative k-algebra with 1 and denote by Mod A the category of unital right A-modules. The derived category DA is classically constructed [49] [50] in three steps:

1st step: We define the category CA to be the category of complexes

… → M^{p} → M^{p+1} → …

with morphisms of complexes as morphisms.

2nd step: We define the homotopy category HA. Its objects are the same as those of CA. The group of morphisms M → N in HA is the group of morphisms of complexes divided by the subgroup of null homotopic morphisms.

3rd step: We define the derived category DA to be the localization of the homotopy category HA with respect to all quasi-isomorphisms. Thus the objects of DA are the same as those of HA and its morphisms are obtained from morphisms in the homotopy category by formally inverting all quasi-isomorphisms.

To get an idea of what the morphisms in the derived category are it is helpful to consider the following special case: Let M and N be two A-modules. Identify M with the complex

… → 0 → M → 0 → …

concentrated in degree 0. For any complex K and any n ∈ Z, denote by K[n] the complex K shifted by n degrees to the left: K[n]^{p} = K^{n+p} and d_{K[n]} = (−1)^{n}d_{K}. Then there is a canonical quasi-isomorphism

Hom_{DA} (M, N[n]) = Ext_{A}^{n} (M, N)

valid for any n ∈ Z if we take the right hand side to vanish for negative n.

4.2. Generalization to A_{∞}-algebras Let A be an A_{∞}-algebra. We generalize the above step by step.

1st step: An A_{∞}-module over A (cf. [17]) is a Z-graded space M endowed with maps

m_{n}^{M} : M ⊗ A^{⊗n−1} → M , n ⩾ 1 ,

of degree 2 − n such that an identity of the same form as the one in the definition (3.1) of an A_{∞}-algebra holds. However, the term

m_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t})

has to be interpreted as

m_{u}^{M} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t})

if r > 0 and as

m_{u}^{M} (m_{s}^{M} ⊗ 1^{⊗t})

if r = 0. Equivalently, the datum of an A_{∞}-structure on a graded space M is the datum of a differential m_{1}^{M} on M and of a morphism of A_{∞}-algebras from A to the opposite of the dg algebra Hom^{•}_{k}(M, M) (cf. 3.3). A morphism of A_{∞}-modules f : L → M is a sequence of graded morphisms

f_{n} : L ⊗ A^{⊗n−1} → M  (4.1)

of degree 1 − n such that for each n ⩾ 1, we have

∑(−1)^{r+st} f_{u} ∘ (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = ∑(−1)^{(r+1)s} m_{u} ∘ (f_{r} ⊗ 1^{⊗s}) ,  (4.2)

where the left hand sum is taken over all decompositions n = r+s+t, r, t ⩾ 0, s ⩾ 1 and we put u = r + 1 + t; and the right hand sum is taken over all decompositions n = r + s, r ⩾ 1, s ⩾ 0 and we put u = 1 + s. The morphism f is a quasiisomorphism if f_{1} is a quasi-isomorphism. The identical morphism f : M → M is given by f_{1} = 1_{M}, f_{i} = 0, for all i > 0. The composition fg is given by

(fg)_{n} = ∑(−1)^{(r−1) s}f_{u}(g_{r} ⊗ 1^{⊗s})

where the sum runs over all decompositions n = r + s and we put u = 1 + s.

In analogy with (3.6), one can easily check that A_{∞}-structures on a graded space M are in bijection with comodule differentials on SM ⊗ T SA, where T SA is the coaugmented tensor coalgebra on SA :

T SA = k ⊕ \overline{T} SA.

The comultiplication on T SA is given by

Δ(x) = 1 ⊗ x + Δ_{\overline{T} SA}(x) + x ⊗ 1.

The differential of T SA is induced by b on \overline{T} SA and maps k to zero. The comodule structure on SM ⊗T SA is induced from the coalgebra structure on T SA. Morphisms of A_{∞}-modules then correspond bijectively to morphisms of dg comodules.

We define C_{∞}A to be the category of A_{∞}-modules. If A is an ordinary associative algebra, we have a faithful functor

CA → C_{∞}A

but C_{∞}A has more objects and more morphisms than CA. A typical example of an A_{∞}-module over A is obtained as follows: Suppose that A is the endomorphism algebra of a module M′ over an ordinary algebra B and let M be a projective resolution of M′ over B. Then it is well-known that each element of A lifts to an endomorphism of the complex M and this endomorphism is unique up to homotopy. The resulting ‘action up to homotopy’ of A on M may be shown to arise from a structure of A_{∞}-module, cf. [22, 6.1].

2nd step: By definition, a morphism f : L → M of A_{∞}-modules is nullhomotopic if there is a family of graded maps

h_{n} : L ⊗ A^{⊗n−1} → M , n ⩾ 1 ,

homogeneous of degree −n such that

f_{n} = ∑(−1)^{rs} m_{1+s} ∘ (h_{r} ⊗ 1^{⊗s})
+ ∑(−1)^{r+st} h_{u} ∘ (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) ,

where the first sum runs over all decompositions n = r + s, r ⩾ 1, s ⩾ 0 and the second sum over all decompositions n = r + s + t, r, t ⩾ 0, s ⩾ 1 and we put u = r + 1 + t.

The homotopy category H_{∞}A has the same objects as C_{∞}A and morphisms from L to M are morphisms of A_{∞}-modules modulo nullhomotopic morphisms.

3rd step: To define the derived category, we should formally invert all quasiisomorphisms. This turns out not to be necessary:

Theorem. Each quasi-isomorphism of A_{∞}-modules is an homotopy equivalence.

We therefore define the derived category D_{∞}A to be equal to the homotopy category H_{∞}A. If k is only assumed to be a commutative ring, the assertion of the theorem is no longer true in general and the derived category is different from the homotopy category.

4.3. Solution of problem 1 Theorem. Let A be an A_{∞}-algebra.

a) If A is an ordinary associative algebra with 1, the canonical functor

DA → D_{∞}A

is an equivalence onto the full subcategory of homologically unital A_{∞}-modules, i.e. A_{∞}-modules M such that 1 ∈ A acts by the identity on H^{∗}M.

b) For each A_{∞}-module M, the graded space H^{∗}M admits an A_{∞}-module structure such that m_{1} = 0 and M is isomorphic to H^{∗}M in D_{∞}A. It is unique up to (non unique, non strict) isomorphism of A_{∞}-modules.

The A_{∞}-module H^{∗}M of b) is called the minimal model of M. The theorem explains the answer to problem 1: Indeed, if M is a complex over an algebra A as in (2.1), then we may view it as an A_{∞}-module, or, in other words, consider its image in D_{∞}A. By b), the image is isomorphic to H^{∗}M endowed with its canonical A_{∞}- structure m_{1} = 0, m_{2}, m_{3}, . . .. The full faithfulness of the functor DA → D_{∞}A then shows that (H^{∗}M, m_{1} = 0, m_{2}, …) determines the complex M up to isomorphism in the derived category DA.

In fact the theorem tells us more: If we also use the essential surjectivity of the functor DA → D_{∞}A onto its image, we see that if V is a graded vector space, then we have a bijection between complexes of A-modules M with H^{∗}M →^{∼} V (as graded vector spaces) up to isomorphism in DA, and homologically unital A_{∞}- structures m_{1} = 0, m_{2}, m_{3}, m_{4}, … on V up to isomorphism of such structures in C_{∞}A. Note that the latter set is naturally the set of orbits under the action of a group, namely the group of families f_{i}, i ⩾ 1, as in (4.1) acting on the set of homologically unital module structures by a conjugation action deduced from (4.2). If the total dimensions of V and A are finite, we even obtain an algebraic group acting on an algebraic variety.

5. Triangulated structure

5.1. Towards problem 2

Consider problem 2 of (2.1) for the special case n = 1: We then have an ordinary associative algebra B with 1 and a B-module M and we wish to reconstruct the category filt M ⊂ Mod B of iterated extensions of M by itself from the extension algebra

E = Ext_{B}^{∗} (M, M)

with its canonical A_{∞}-structure. The basic idea is to realize filt M as a subcategory of the derived category of E. In fact, this subcategory will simply be the subcategory of iterated extensions of the free E-module of rank one by itself. Now the derived category is not abelian in general so that a priori it makes no sense to speak of extensions. However, like the derived category of an ordinary associative algebra, the derived category of an A_{∞}-algebra does admit a triangulated structure and this suffices for our purposes.

5.2. Triangulated structure on D_{∞}A

Recall that a triangulated category is an additive category T endowed with an additive endofunctor S : T → T called its suspension (or shift) and with a class of sequences of the form

X → Y → Z → SX (∗)

called triangles (or distinguished triangles). These have to satisfy a number of axioms (see the references at the beginning of 4.1). If Y occurs in a triangle (∗), it is called an extension of Z by X.

For example, if A is an ordinary associative algebra with 1, the derived category DA is triangulated: The suspension of a complex K is defined by

(SK)^{p} = K^{p+1} , d_{SK} = −d_{K}.

Each short exact sequence of complexes

0 → K → L → M → 0

gives rise to a canonical triangle

K → L → M → SK

and (up to isomorphism) all triangles are obtained in this way. This generalizes to A_{∞}-algebras as follows Proposition. Let A be an A_{∞}-algebra. Then the derived category D_{∞}A admits a triangulated structure with suspension functor S defined by

(SM)^{p} = M^{p+1} , m_{n}^{SM} = (−1)^{n} m_{n}^{M}

and such that each short exact sequence

0 → K →^{i} L →^{p} M → 0

of A_{∞}-modules with strict morphisms i and p gives rise to a canonical triangle. Up to isomorphism in D_{∞}A, all triangles are obtained in this way.

6. Standard functors

6.1. More on problem 2

If T is a triangulated category and M an object of T , we denote by filt M = filt_{T} M the full subcategory of T whose objects are iterated extensions of M by itself. We denote by tria M the closure of M under suspension, desuspension and forming extensions. Thus tria M is the smallest triangulated subcategory of T containing M and filt M is a full subcategory of tria M.

If S and T are triangulated categories a triangle functor S → T is an additive functor F : S → T together with an isomorphism of functors φ : F S → SF such that for each triangle (X, u, Y, v, Zw) of S, the sequence

F X →^{F u} F Y →^{F v} F Z ⟶^{(φX)(F w)} SF X

is a triangle of T . If M is an object of T and F a triangle functor then clearly F tria M ⊂ tria F M and similarly for filt M.

In the situation of (5.1), consider the subcategory filt E ⊂ D_{∞}E obtained by taking the closure under extensions of the free E-module of rank one (also denoted by E). We would like to show that the subcategory filt E is equivalent to filt_{Mod B} M. For this, we first observe that the inclusion

Mod B → DB

induces an equivalence

filt_{Mod B} M →^{∼} filt_{DB} M.

It will thus suffice to construct a triangle functor DB → D_{∞}E inducing an equivalence from filt M onto filt E. The following diagram summarizes the setup:

Mod B ⟶  DB  ⟶ D_{∞}E
↑  ↑  ↑
filt M →^{∼} filt_{DB} M →^{∼} filt E

The triangle functor DB → D_{∞}E is constructed as a composition of standard functors.

6.2. Restriction

Let f : A → B be a morphism of A_{∞}-algebras and let M be a module over B. Define

m_{n}^{A} : M ⊗ A^{⊗n−1} → M , n ⩾ 1 ,

by

m_{n}^{A} = ∑(−1)^{s} m_{r+1}(1 ⊗ f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{r}})

where the sum runs over all 1 ⩽ r ⩽ n−1 and all decompositions n−1 = i_{1} +⋯+i_{r} and the sign is like in (3.4). Then it is not hard to check that the m_{n}^{A} define an A_{∞}-module f^{∗}M over A. The assignment M → f^{∗}M becomes a functor

C_{∞}B → C_{∞}A

and induces a triangle functor

D_{∞}B → D_{∞}A

which will still be denoted by f^{∗}.

Proposition. The functor f^{∗} : D_{∞}A → D_{∞}B is an equivalence if f is a quasiisomorphism.

6.3. Hom and Tensor

Let B be an A_{∞}-algebra and X and M two B-modules. For n ∈ Z, let Hom^{∞}_{B}^{n}(X, M) be the space of graded comodule maps f : SX ⊗ T SB → SM ⊗ T SB of degree n and define

d(f) = b ∘ f − (−1)^{n}f ∘ b.

Then Hom^{∞}_{B}^{•}(X, M) is a complex which is bifunctorial in X and M. If B is an ordinary algebra and X, M are ordinary modules, then Hom^{∞}_{B}^{•}(X, M) computes Ext_{B}^{∗} (X, M).

Now suppose that A is another A_{∞}-algebra and that the B-module structure on X comes from an A-B-bimodule structure, i.e. X is endowed with graded maps

m_{t,u} : A^{⊗t−1} ⊗ X ⊗ B^{⊗u−1} → X , t, u ⩾ 1 ,

of degree 3 − t − u such that we have the associativity identity of (3.1), where the terms 1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t} have to be interpreted according to the type of their arguments. Then Hom^{∞}_{B}^{•}(X, M) becomes a (right) A_{∞}-module over A and we have the Hom-functor

Hom^{∞}_{B}^{•}(X,?) : C_{∞}B → C_{∞}A

Proposition. The Hom-functor admits a left adjoint

? ⊗^{∞}_{A} X : C_{∞}A → C_{∞}B.

We have an induced pair of adjoint triangle functors

\begin{aligned}& D_{∞}A \\ ? ⊗^{∞}_{A} X & ↓↑ & Hom^{∞}_{B}^{•}(X,?) \\ & D_{∞}B \end{aligned} .

It is not hard to make L ⊗^{∞}_{A} X explicit. Its underlying graded space is L ⊗_{k} T SA ⊗_{k} X. If A is an ordinary associative algebra with 1 and L, X are ordinary modules, then L ⊗^{∞}_{A} X is a complex computing Tor_{∗}^{A} (L, X).

6.4. Application to problem 2 Let B be an A_{∞}-algebra and M a B-module. Then

A = Hom^{∞}_{B}^{•} (M, M)

is a dg algebra and in particular an A_{∞}-algebra. The module M becomes an A-B-bimodule.

Proposition. The functor

F = Hom^{∞}_{B}^{•} (M,?) : D_{∞}B → D_{∞}A

induces a triangle equivalence

tria M →^{∼} tria A

taking M to A and thus an equivalence filt M →^{∼} filt A.

Now consider the situation of (5.1). In this case, A is a dg algebra whose homology is the extension algebra of M. Let us simply write

E = Ext_{B}^{∗} (M, M)

for the extension algebra endowed with its canonical structure of A_{∞}-algebra. Then we know by Theorem 3.3 that we have a quasi-isomorphism E → A. This yields the restriction functor

res : D_{∞}A → D_{∞}E

which is an equivalence by Proposition 6.2. Consider the square

D_{∞}B ⟶^{{∞ \atop Hom}_{B}^{•} (M,?)} D_{∞}A
can ↑  ↓ res
DB  ⟶  D_{∞}E

The left vertical arrow induces an equivalence onto its image and maps M to M. By the above proposition, the top horizontal arrow induces an equivalence filt M →^{∼} filt A. The right vertical arrow is an equivalence mapping the free module A to the free module E and hence induces an equivalence filt A →^{∼} filt E. It follows that the bottom arrow induces an equivalence filt_{DB} M →^{∼} filt E. Finally, we know that filt_{DB} M is equivalent to filt M ⊂ Mod B so that we do obtain the required equivalence between filt E ⊂ D_{∞}E and filt M ⊂ Mod B.

6.5. Solution of the original problem 2

In order to treat problem 2 as it is stated in (2.1) with n modules M_{1}, …, M_{n}, let e_{i} be the idempotent of E corresponding to the direct factor M_{i} of M = ⨁ M_{i}. There is a canonical isomorphism

E^{0} →^{∼} Hom_{D_{∞}E} (E, E)

and thus e_{i} yields an idempotent endomorphism of E in the derived category. Now the derived category is a triangulated category with infinite sums so that by a wellknown trick [4], the idempotent e_{i} splits in D_{∞}E. Let X_{i} denote its image. We have

E →^{∼} ⨁ X_{i}

in D_{∞}E. The functor constructed in 6.4 sends X_{i} to M_{i} and thus induces an equivalence filt X_{i} →^{∼} filt M_{i}.

7. Twisted objects

7.1. Motivation

Let A be an associative algebra with 1 and tria A the triangulated subcategory of DA generated by the free A-module of rank one (as in 6.1). Up to isomorphism, the objects of tria A are the bounded complexes of finitely generated free A-modules and the morphisms between two such complexes are in bijection with the homotopy classes of morphisms of complexes. In other words, tria A is triangle equivalent to the bounded homotopy category of finitely generated free A-modules. The advantage of this description of tria A is that morphisms and objects are readily computable. The aim of this section is to generalize this description to any A_{∞}-algebra A with a strict unit (cf. below). The generalization will be of the form

tria A →^{∼} H^{0} tw A

where tw A is the A_{∞}-category of twisted objects over A. An A_{∞}-category is an ‘A_{∞}-algebra with several objects’ analogous to a ‘ring with several objects’ [33]. The above equivalence illustrates the general fact that working with several objects is natural and useful. From the outset, we will therefore replace the A_{∞}-algebra A by an A_{∞}-category with strict identities A. We will then have more generally

tria A →^{∼} H^{0} tw A.

The material in this section is adapted from [23], [25]. The special case of dg categories goes back to Bondal-Kapranov’s article [5].

7.2. A_{∞}-categories Let k be a field. An A_{∞}-category A is the datum of

• a class of objects obj(A),

• for all A, B ∈ A (we write A ∈ A instead of A ∈ obj(A)), a Z-graded vector space Hom_{A}(A, B), often denoted by (A, B),

• for all n ⩾ 1 and all A_{0}, …, A_{n} ∈ A, a graded map

m_{n} : (A_{n−1}, A_{n}) ⊗ (A_{n−2}, A_{n−1}) ⊗ ⋯ ⊗ (A_{0}, A_{1}) → (A_{0}, A_{n})

of degree 2 − n such that for each n ⩾ 1 and all A_{0}, …, A_{n} ∈ A, we have the identity

∑(−1)^{r+st} m_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = 0

of maps

(A_{n−1}, A_{n}) ⊗ (A_{n−2}, A_{n−1}) ⊗ ⋯ ⊗ (A_{0}, A_{1}) → (A_{0}, A_{n}) ,

where the sum runs over all decompositions n = r + s + t and we put u = r + 1 + t.

For example, the datum of an A_{∞}-category A with one object ∗ is equivalent to the datum of the endomorphism A_{∞}-algebra Hom_{A} (∗, ∗). Another class of examples is provided by dg categories : these are exactly the A_{∞}-categories whose composition maps m_{n} vanish for all n ⩾ 3.

Note that for two reasons, an A_{∞}-category is not a category in general : firstly, the composition m_{2} may not be associative; secondly, there may not be identity morphisms.

Let A be an A_{∞}-category and A ∈ A. A morphism e_{A} ∈ Hom_{A}^{0} (A, A) is a strict identity if we have

m_{2}(f, e_{A}) = f , m_{2}(e_{A}, g) = g

whenever these make sense and

m_{n}(…, e_{A}, …) = 0

for all n ≠ 2. In particular, e_{A} is then a cycle of the complex (Hom_{A} (A, A), m_{1}). Clearly, if e_{A} exists, it is unique. The A_{∞}-category A has strict identities if there is a strict identity e_{A} for each object A ∈ A.

7.3. A_{∞}-functors Let A, B be two A_{∞}-categories. An A_{∞}-functor F : A → B is the datum of

• a map F : obj(A) → obj(B),

• for all n ⩾ 1 and all A_{0}, …, A_{n} ∈ A a graded map

F_{n} : (A_{n−1}, A_{n}) ⊗ (A_{n−2}, A_{n−1}) ⊗ ⋯ ⊗ (A_{0}, A_{1}) → Hom_{B} (F A_{0}, F A_{n})

of degree 1 − n

such that conditions analogous to those of (3.4) hold. The A_{∞}-functor F is strict if we have F_{n} = 0 for all n ⩾ 2. This implies that F_{1} ‘commutes with all compositions’.

7.4. Modules

We leave it as an exercise to the reader to generalize (4.2) to A_{∞}-categories. In particular, if A is an A_{∞}-category, he may

• define the category C_{∞}A of A-modules

• define the derived category D_{∞}A

• define a structure of dg category

Hom^{∞} (L, M) , L, M ∈ C_{∞}A

on C_{∞}A as in 6.3

• define the Yoneda functor

Y : A → C_{∞}A , A ↦ A (?, A).

Here we view C_{∞}A as a dg category (and hence an A_{∞}-category) and the Yoneda functor is to be an A_{∞}-functor.

7.5. Factorization of the Yoneda functor

Let A be an A_{∞}-category and let tria A denote the triangulated subcategory of D_{∞}A generated by the image of the Yoneda functor.

Theorem. Suppose that A has strict identities. Then there is an A_{∞}-category tw A and a factorization

[figure: A; Y; Y_{1}; tw A; Y_{2}; C_{∞}A]

of the Yoneda functor Y such that Y_{1} is a strict fully faithful A_{∞}-functor and Y_{2} is an A_{∞}-functor inducing an equivalence

H^{0} tw A →^{∼} tria A ⊂ D_{∞}A.

The theorem may be enhanced so as to characterize the category tw A of twisted objects by a universal property.

7.6. Explicit description of tw A

Let A be an A_{∞}-category with strict identities. We will explicitly construct the category of twisted objects tw A in two steps. First, we construct the A_{∞}-category ZA, which is the ‘closure under shifts’ of A: the objects of ZA are the pairs (A, n) of an object A ∈ A and an integer n ∈ Z. The morphisms from (A, n) to (A′, n′) are in bijection with the elements of the graded space

Hom_{A}(A, A′)[n′ − n].

We will identify A with the full sub-A_{∞}-category of ZA on the objects (A, 0). The compositions of ZA are obtained by shifting the compositions of A. The reader is invited to write down an explicit formula with correct signs.

In a second step, we construct tw A as the ‘closure under extensions’ of ZA: Objects of tw A are pairs (B, δ), where B is a sequence (A_{1}, …, A_{r}) of objects of ZA, and δ = (δ_{ij}) a matrix of morphisms of degree 1

δ_{ij} ∈ Hom_{ZA}^{1} (A_{j}, A_{i})

such that we have δ_{ij} = 0 for all i ⩾ j and

∑_{t=1}^{∞}(−1)^{\frac{t(t−1)}{2}} m_{t} (δ, δ, …, δ) = 0.  (7.1)

Here we denote by m_{n} the extension of m_{n}^{ZA} to matrices. Note that since δ is strictly upper triangular, the sum contains only finitely many non zero terms. The strange sign in the formula disappears in the b_{n}-language (3.6).

Alternatively, we can view an object (B, δ) as the datum of an object B of the additive hull of ZA endowed with a finite split filtration and with an endomorphism δ of degree 1 which strictly decreases the filtration.

The space of morphisms from (B, δ) to (B′, δ′) is defined as

⨁_{i,j} Hom_{ZA} (A_{i}, A′_{j}).

Let n ⩾ 1 and let (B_{0}, δ_{0}), …, (B_{n}, δ_{n}) be objects of tw A. We will define the composition map (where the δ_{i} are suppressed from the notation)

m_{n} : (B_{n−1}, B_{n}) ⊗ ⋯ ⊗ (B_{0}, B_{1}) → (B_{0}, B_{n}).

In the following formula, we write δ for all δ_{i}

m_{n}^{tw A} = ∑_{t=0}^{∞} ∑ ±m_{n+t}^{ZA} ∘ (1^{⊗i_{1}} ⊗ δ^{⊗j_{1}} ⊗ 1^{⊗i_{2}} ⊗ δ^{⊗j_{2}} ⊗ … ⊗ 1^{⊗i_{r}} ⊗ δ^{⊗j_{r}}).

Here the terms of the second sum are in bijection with the non-commutative monomials

X^{i_{1}}Y^{j_{1}} … X^{i_{r}}Y^{j_{r}}

of degree n in X and degree t in Y and the sign is given by the identity

(SX)^{i_{1}}(SY)^{j_{1}} … (SX)^{i_{r}}(SY)^{j_{r}} = ±S^{n+t}X^{i_{1}}Y^{j_{1}} … X^{i_{r}}Y^{j_{r}}

in the algebra k < X, Y, S > /(SX − XS, SY + Y S). The pictorial version of the formula is

m_{n}^{tw A} = ∑ ± .

Here the fat edges symbolize tensor powers of δ’s to be inserted in all possible places and in any number. The concatenation of the thin edges represents a given tensor f_{n} ⊗ f_{n−1} ⊗ ⋯ ⊗ f_{1}.

It is worthwile to make the formula explicit for m_{1}. If f is a morphism of even degree from (B, δ) to (B′, δ′), we have

m_{1}^{tw A}(f) = m_{1}(f) − m_{2}(δ′, f) + m_{2}(f, δ)

− m_{3}(δ′, δ′, f) + m_{3}(δ′, f, δ) − m_{3}(f, δ, δ)

+ m_{4}(δ′, δ′, δ′, f) − m_{4}(δ′, δ′, f, δ) + m_{4}(δ′, f, δ, δ) − m_{4}(f, δ, δ, δ) + …

If f is of odd degree, we have

m_{1}^{tw A}(f) = m_{1}(f) − m_{2}(δ′, f) − m_{2}(f, δ)

− m_{3}(δ′, δ′, f) − m_{3}(δ′, f, δ) − m_{3}(f, δ, δ)

+ m_{4}(δ′, δ′, δ′, f) + m_{4}(δ′, δ′, f, δ) + m_{4}(δ′, f, δ, δ) + m_{4}(f, δ, δ, δ) + … In the sequel, we identify A with the subcategory of the objects (A, 0) of ZA. With this convention, the functor Y_{1} of (7.5) sends an object A ∈ A to (A, δ = 0). The functor Y_{2} sends B = (A_{1}, …, A_{n}) to Y A_{1} ⊕ ⋯ ⊕ Y A_{n} (where Y denotes the canonical extension of the Yoneda functor to ZA) endowed with the multiplications

m_{n} = ∑_{t=0}^{∞}(−1)^{\frac{t(t−1)}{2}} m_{n+t} ∘ (δ^{⊗t} ⊗ 1^{⊗n}) , n ⩾ 1 ,

where the m_{n+t} on the right hand side denote the multiplications of the direct sum of the Y A_{i}.

7.7. Description of filtered modules by twisted objects

Let B be an associative k-algebra with 1, let M_{1}, …, M_{n} be B-modules and let filt(M_{i}) denote the full subcategory of the category of right B-modules whose objects admit finite filtrations with subquotients among the M_{i}. Let A be the A_{∞}-category with objects 1, …, n and with morphism spaces

A (i, j) = Ext_{B}^{∗} (M_{i}, M_{j}).

The composition maps of A are defined as follows: The differential m_{1} vanishes, the composition m_{2} is the Yoneda composition, and the m_{i}, i > 3, are the canonical higher compositions constructed as in (3.3, 3.4). We assume that these may be chosen so as to make A strictly unital. For example, this holds if Ext_{B}^{∗} (M_{i}, M_{i}) is one-dimensional and Ext_{B}^{∗} (M_{i}, M_{j}) = 0 for i > j (cf. 3.5) or if the M_{i} are the simple modules over a finite-dimensional basic algebra B. Indeed, in the latter case, both B and the dg algebra computing the extensions are augmented over a product of fields (cf. 3.5).

Let filt A denote the full subcategory of H^{0} tw A whose objects are the (B, δ) such that B is a filtered direct sum of (non shifted) copies of objects of A (we use the alternative description of tw A and identify A with a subcategory of ZA).

Theorem. There is an equivalence of categories

filt A →^{∼} filt(M_{i}).

Although the category filt A ⊂ H^{0} tw A looks complicated, it is often quite easy to describe in practice. We illustrate this in two examples below.

The theorem is a consequence of (7.5) and (6.5). Indeed, by its construction, the category A is the A_{∞}-Ext-category of the B-modules M_{1}, …, M_{n} in the sense of problem 2 of (2.1). So by (6.4) and (6.5), there is a triangle functor

F : D_{∞}A → DB

inducing an equivalence

tria(Y 1, …, Y n) →^{∼} tria(M_{1}, …, M_{n}).

By (7.5), the Yoneda functor induces an equivalence

H^{0} tw A → tria(Y 1, …, Y n).

Thus we get an equivalence

H^{0} tw A → tria(M_{1}, …, M_{n})

which takes the object i to the corresponding module M_{i} and therefore induces an equivalence

filt A →^{∼} filt(M_{i}).

7.8. Example: Reconstruction from simple modules

Let B be a finite-dimensional basic algebra. Theorem 7.7 allows us in particular to reconstruct the category mod B of finite-dimensional B-modules from the A_{∞}- Ext-category of the simple B-modules.

We illustrate this by an example. Let B be the algebra given by the quiver with relations

x →^{α} y →^{β} z →^{γ} t , γ β α = 0

and let M_{1} = S_{x}, …, M_{4} = S_{t} be the four simple B-modules. The A_{∞}-category A of (7.7) is then given by the quiver with relations

where the arrows a, b, c are of degree 1, the arrow e is of degree 2, and all compositions other than m_{3}(a, b, c) = e are defined to be zero (this example has already appeared in 3.3). We would like to determine the subcategory filt A of H^{0} tw A whose objects are the (A, δ) where A is a filtered sum of objects of A (we use the alternative description and identify A with a subcategory of ZA). By an obvious abuse of notation, we have

A = V_{x} ⊗ x ⊕ … ⊕ V_{t} ⊗ t

where V_{x}, …, V_{t} are finite-dimensional vector spaces and the datum of

δ ∈ Hom_{ZA}^{1} (A, A)

corresponds to the datum of three linear maps V_{a}, V_{b}, V_{c} :

V_{x} ⟵^{V_{a}} V_{y} ⟵^{V_{b}} V_{z} ⟵^{V_{c}} V_{t}.

The condition

m_{1}(δ) + m_{2}(δ, δ) − m_{3}(δ, δ, δ) − … = 0

translates into V_{a}V_{b}V_{c} = 0. By definition, morphisms from (A, δ) to (A′, δ′) are in bijection with

H^{0} Hom_{tw A} ((A, δ), (A′, δ′)).

Now Hom_{A}^{−1} and hence

Hom_{tw A}^{−1} ((A, δ), (A′, δ′))

vanishes so that the homology classes in degree zero are exactly the zero cycles. Now it is easy to see that the elements f of Hom_{tw A}^{0} ((A, δ), (A′, δ′)) correspond exactly to the quadruples of linear maps

f_{x} : V_{x} → V′_{x} , …, f_{t} : V_{t} → V′_{t}

and the condition

m_{1}(f) − m_{2}(δ′, f) + m_{2}(f, δ) − m_{3}(…) + … = 0

translates into

f_{x}V_{a} = V′_{a}f_{y} , …, f_{z}V_{c} = V′_{c}f_{t}.

Hence, as expected, the category filt A is equivalent to the category of contravariant representations of the quiver with relations

x →^{α} y →^{β} z →^{γ} t , γ β α = 0 ,

i.e. to the category mod B of finite-dimensional right modules over the algebra B. This confirms theorem 7.7 in this example.

7.9. Example: Classifying filtered modules

This example arose from an email exchange with T. Brüstle. Consider the following quiver Q of type D_{4}

Denote by S_{i} its simple (covariant) representation associated with the point i, 1 ⩽ i ⩽ 4. Let Δ_{1} = S_{1}, Δ_{2} = S_{2}, Δ_{4} = S_{4} and let Δ_{3} be the following indecomposable representation

.

We are interested in the category filt(Δ_{1}, …, Δ_{4}) of representations of Q admitting a filtration with subquotients among the Δ_{i}. The A_{∞}-Ext-category A of the Δ_{i} is given by the quiver

where a, b, c are of degree 1, f is of degree 0 and all multiplications other than m_{3}(b, f, a) = c are defined to be zero. As in example 7.8, the subcategory filt A of H^{0} tw A can be described as follows: Its objects are the finite-dimensional (covariant) representations V of the quiver

2 ←^{a} 1 →^{c} 4 ←^{b} 3

of type A_{4}. Its morphisms f : V → V′ are the quintuples of linear maps f_{i} : V_{i} → V′_{i} and φ : V_{2} → V′_{3}

[figure: V_{2}; f_{2}; φ; V_{1}; V_{a}; f_{1}; V_{c}; V_{4}; f_{4}; V_{3}; f_{3}; V_{b}; V′_{2}; V′_{1}; V′_{a}; V′_{c}; V′_{4}; V′_{3}; V′_{b}]

satisfying

0 = V′_{a} f_{1} − f_{2}V_{a} , 0 = V′_{b}f_{3} − f_{4}V_{b}
0 = V′_{c}f_{1} − f_{4}V_{c} + V′_{b} φ V_{a} (∗)

Thanks to theorem 7.7, we have an equivalence

filt A →^{∼} filt(Δ_{1}, …, Δ_{4}).

The term V′_{b} φ V_{a} in (∗) is essential: If it is omitted, the problem is that of classifying the representations of the quiver of type A_{4} above. Then we find 10 isoclasses of indecomposables instead of the 9 isoclasses present on the right hand side.

8. On the proofs

8.1. The minimality theorem 3.3

The article [13] gives a detailed account of various approaches to this theorem (for the case of connected dg algebras A). For the case where A is a Z-graded dg algebra, it is proved in [32]. Here one also finds new explicit formulas for the higher multiplication maps on H^{∗}A.

In the general case, one may proceed as follows [28]: Let A and B be two A_{∞}- algebras and (f_{1}, …, f_{s}) an A_{s}-morphism (i.e. the defining equation from 3.4 holds for n ⩽ s). One can then define a morphism of complexes c = c(f_{1}, …, f_{s}) : A^{⊗s+1} → B such that (f_{1}, …, f_{s}) extends to an A_{s+1}-morphism (f_{1}, …, f_{s+1}) iff c is null homotopic and in this case, the homotopy is given by f_{s+1}. Here, A and B are endowed with the differentials given by m_{1}. It follows that if (B, m_{1}) is contractible, each A_{s}-morphism with target B extends to an A_{∞}-morphism.

Now suppose that A is an A_{∞}-algebra and A = A′ ⊕ N a decomposition of the complex (A, m_{1}) with contractible N. We wish to make A′ into an A_{∞}-algebra quasi-isomorphic to A. For this, we make N into an A_{∞}-algebra by taking all m_{n}, n ⩾ 2, to vanish. Since N is contractible, the morphism of complexes A → N extends to an A_{∞}-morphism. We thus obtain a dg coalgebra morphism p : \overline{T} SA → \overline{T} SN (cf. 3.6). It is easy to check that its kernel in the category of graded coalgebras is isomorphic to \overline{T} SA′. Thanks to 3.6, this yields an A_{∞}-algebra structure on A′ and an A_{∞}-morphism A′ → A extending the inclusion.

It is easy to see that an A_{∞}-morphism f between minimal A_{∞}-algebras is invertible iff f_{1} is invertible. This yields the uniqueness assertion of the theorem.

8.2. The theorem on homotopies 3.7 The theorem results from the facts, proved in [28], that

1) the bar construction functor (3.6) is an equivalence from the category of A_{∞}- algebras onto the full subcategory of fibrant-cofibrant objects of a closed model category of coalgebras and

2) the intrinsic homotopy relation of this closed model category induces the explicit relation given in 3.7.

Alternatively, one can prove b) by the following argument taken from [24, 4.5]: One shows that the quasi-isomorphism i_{A} : H^{∗}A → A constructed above is actually an homotopy equivalence. The given morphism s : A → B thus fits into a diagram

[figure: A; s; B; H^{∗}A; i_{A}; s′; H^{∗}B; i_{B}]

of A_{∞}-algebras which is commutative up to homotopy. The first component s′_{1} identifies with H^{∗}s. So H^{∗}s is invertible iff s′ is an isomorhism iff s is a homotopy equivalence.

8.3. A_{∞}-modules (section 4)

Theorem 4.2 on A_{∞}-modules has an analogous (but simpler) proof as the corresponding theorem 3.7 on A_{∞}-algebras, cf. [28].

Part a) of theorem 6.5 is proved in [22, section 7]. The existence of the minimal model in b) is shown in [12]. Uniqueness is easy (as in the case of A_{∞}-algebras).

8.4. Triangulated structure (section 5)

The following argument is taken from [28]: The category of A_{∞}-modules over an A_{∞}-algebra A is equivalent to a full subcategory C′ of the category C of differential counital comodules over the counital dg coalgebra T SA (cf. 4.2). We endow the category C with the class of exact sequences whose underlying sequences of graded comodules split. As in [21, 2.2], one shows that C is a Frobenius category. The associated stable category is thus triangulated. This category equals the homotopy category H of the category of dg counital comodules. It is easy to check that the essential image H′ of C′ in H is a triangulated subcategory. By theorem 4.2, H′ is equivalent to the derived category D_{∞}A. Exact sequences of strict morphisms of A_{∞}-modules give rise to exact sequences of coinduced graded T SA-comodules. Such sequences split in the graded category (coinduced modules are injective). Thus each exact sequence of strict morphisms of A_{∞}-modules gives rise to a canonical triangle. When we have an arbitrary triangle (K, L, M) of H′, we can form a mapping cylinder over the morphism M → SK to obtain an isomorphic triangle K → L′ → M → SK. By the construction of the mapping cylinder, the sequence of graded comodules K → L′ → M identifies with the canonical sequence K → K ⊕ M → M. So the sequence K → L′ → M does come from a sequence of strict A_{∞}-morphisms.

8.5. Standard functors (section 6)

The assertions in this section are based on the following description [28] of the derived category D_{∞}A : The functor M ↦ M ⊗ T SA is an equivalence of the category of A_{∞}-modules onto the full subcategory of fibrant-cofibrant objects of a closed model category of counital dg comodules over T SA. Thus it induces an equivalence of D_{∞}A onto the corresponding homotopy category.

8.6. Twisted objects (section 7)

We follow [28]. Below, we will prove that tw A is an A_{∞}-category. The proof that Y_{2} is an A_{∞}-functor follows the same lines. It remains to be proved that Y_{2} induces an equivalence of H^{0} tw A onto its image in the homotopy category (=derived category) of A_{∞}-modules. For this, let B_{1} and B_{2} be two twisted objects and consider the map

H^{∗} tw A(B_{1}, B_{2}) → H^{∗}Hom^{∞}_{A}(Y_{2}B_{1}, Y_{2}B_{2})

induced by Y_{2}. We have to show it is an isomorphism. Using the canonical filtrations of B_{1} and B_{2}, one easily reduces this to the special case where the B_{i} are objects A_{i} of A (and thus the corresponding twisting elements δ vanish). Then one has to show that the map

H^{∗}A(A_{1}, A_{2}) → H^{∗}Hom^{∞}_{A}(Y A_{1}, Y A_{2})

is bijective. Now using the strict identity of A_{1}, it is not hard to construct an explicit inverse of this map.

Let us show that tw A is an A_{∞}-category. Let B be the datum of

• the same class of objects as tw A,

• the same graded morphism spaces as tw A,

• the natural composition maps m′_{n} obtained by extending the maps m_{n}^{ZA} to

matrices. It is clear that B is an A_{∞}-category. For each sequence B_{0}, …, B_{n} of objects of B, let b′_{n} be the map

S(B_{n−1}, B_{n}) ⊗ S(B_{n−2}, B_{n−1}) ⊗ … ⊗ S(B_{0}, B_{1}) → S(B_{0}, B_{n})

deduced from m′_{n} as in the bar construction (3.6). Here, S(B_{i−1}, B_{i}) denotes the suspension of the morphism space (B_{i−1}, B_{i}). For each 0 ⩽ i ⩽ n, let α : k → S(B_{i}, B_{i}) be the composition of the degree 1 map

k → (B_{i}, B_{i}) , 1 ↦ δ

with the canonical morphism s : (B_{i}, B_{i}) → S(B_{i}, B_{i}) of degree −1. Note that α is of degree 0. A sign check shows that the identity 7.1 translates into

0 = ∑_{i⩾1} b′_{i} ∘ α^{⊗i} .

Now for each t ⩾ 0, let φ_{n,t} be the sum of the maps

1^{⊗i_{1}} ⊗ α^{⊗j_{1}} ⊗ … ⊗ 1^{⊗i_{r}} ⊗ α^{⊗j_{r}}

defined on

S(B_{n−1}, B_{n}) ⊗ … ⊗ S(B_{0}, B_{1})

where i_{1}+⋯+i_{r} = n and j_{1}+⋯+j_{r} = t as in 7.6. Note that for each decomposition n = n_{1} + n_{2}, we have

φ_{n,t} = ∑_{t=t_{1}+t_{2}} φ_{n_{1},t_{1}} ⊗ φ_{n_{2},t_{2}}.  (8.1)

Define

b_{n} = ∑_{t⩾0} b′_{n+t} ∘ φ_{n,t}.

A sign check shows that b_{n} corresponds to the twisted composition m_{n}^{tw A} defined in 7.6. We have to show that

∑ b_{l} (1^{⊗i} ⊗ b_{j} ⊗ 1^{⊗k′}) = 0 ,

where the sum runs over the decompositions n = i+j +k′ and we put l = i+1+k′. Now we have

∑ b_{l} (1^{⊗i} ⊗ b_{j} ⊗ 1^{⊗k′}) = ∑ b′_{l+s} φ_{l,s} (1^{⊗i} ⊗ (b′_{j+t} φ_{j,t}) ⊗ 1^{⊗k′})
= ∑ b′_{l+s} (φ_{i,s_{1}} ⊗ (b′_{j+t} φ_{j,t}) ⊗ φ_{k′,s_{2}})

the indices run trough the integers ⩾ 0 such that s = s_{1} + s_{2}. Here, we have used the decomposition formula 8.1 and the fact that α is of degree 0. Let Σ_{1} denote the last sum. Graphically, the terms of Σ_{1} correspond to diagrams

,

where the fat edges symbolize α’s to be inserted in any number and in all positions such that the arc above the upper b′ is not entirely fat (it must always contain a thin piece symbolizing the j ⩾ 1 arguments of b_{j}). To show that Σ_{1} vanishes, we compare it to

∑_{r⩾0} ∑ b′_{L} (1^{⊗I} ⊗ b′_{J}⊗ , 1^{⊗K}) φ_{n,r}  (8.2)

where the inner sum runs over all decompositions of N = n + r into N = I + J + K and we put L = I + 1 + K. Since B is an A_{∞}-category, the inner sum vanishes for each r. On the other hand, the whole expression equals

∑ b′_{L} (φ_{I′,r_{1}} ⊗ (b′_{J′+r_{2}} φ_{J′,r_{2}}) ⊗ φ_{K′,r_{3}}) ,

where the sum runs over the non negative integers r, r_{1}, r_{2}, r_{3}, I′, J′, K′ satisfying r = r_{1} + r_{2} + r_{3}, I′ + J′ + K′ = n, and we put L = (I′ + r_{1}) + 1 + (K′ + r_{3}). Let us denote the last sum by Σ_{2}. Each term of Σ_{1} appears exactly once in Σ_{2} but not all terms of Σ_{2} appear in Σ_{1}. Those that do not appear are precisely the terms where J′ = 0, i.e. the terms

b′_{L} (φ_{I′,r_{1}} ⊗ (b′_{r_{2}} φ_{0,r_{2}}) ⊗ φ_{K′,r_{3}}).

Graphically, they correspond to diagrams

.

Now we have b′_{r_{2}} φ_{0,r_{2}} = b′_{r_{2}} ∘ α^{⊗r_{2}} so that the sum over all r_{2} ⩾ 0 of these terms vanishes. Thus we have Σ_{1} = Σ_{2} = 0.

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