Unital A∞-categories
Ми доводимо, що три означення унiтальностi для A∞-категорiй запропонованi Любашенком, Концевичем i Сойбельманом, та Фукая є еквiвалентними.
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| Цитувати: | Unital A∞-categories/ V. Lyubashenko, O. Manzyuk // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 235-268. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-62902010-02-23T12:00:50Z Unital A∞-categories Lyubashenko, V. Manzyuk, O. Геометрія, топологія та їх застосування Ми доводимо, що три означення унiтальностi для A∞-категорiй запропонованi Любашенком, Концевичем i Сойбельманом, та Фукая є еквiвалентними. We prove that three definitions of unitality for A∞-categories suggested by Lyubashenko, by Kontsevich and Soibelman, and by Fukaya are equivalent. 2006 Article Unital A∞-categories/ V. Lyubashenko, O. Manzyuk // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 235-268. — Бібліогр.: 11 назв. — англ. 1815-2910 http://dspace.nbuv.gov.ua/handle/123456789/6290 en Інститут математики НАН України |
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Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування |
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Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування Lyubashenko, V. Manzyuk, O. Unital A∞-categories |
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Ми доводимо, що три означення унiтальностi для A∞-категорiй запропонованi Любашенком, Концевичем i Сойбельманом, та Фукая є еквiвалентними. |
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Lyubashenko, V. Manzyuk, O. |
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| title |
Unital A∞-categories |
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Unital A∞-categories |
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Unital A∞-categories |
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Unital A∞-categories |
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Unital A∞-categories |
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unital a∞-categories |
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Інститут математики НАН України |
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2006 |
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Геометрія, топологія та їх застосування |
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Unital A∞-categories/ V. Lyubashenko, O. Manzyuk // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 235-268. — Бібліогр.: 11 назв. — англ. |
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Збiрник праць
Iн-ту математики НАН України
2006, т.3, №3, 235-268
Volodymyr Lyubashenko
Institute of Mathematics, NAS of Ukraine, Kyiv
E-mail: lub@imath.kiev.ua
Oleksandr Manzyuk
Institute of Mathematics, NAS of Ukraine, Kyiv
E-mail: manzyuk@mathematik.uni-kl.de
Unital A∞-categories
Ми доводимо, що три означення унiтальностi для A∞-категорiй за-
пропонованi Любашенком, Концевичем i Сойбельманом, та Фукая є
еквiвалентними.
We prove that three definitions of unitality for A∞-categories suggested by
Lyubashenko, by Kontsevich and Soibelman, and by Fukaya are equivalent.
Keywords: A∞-category, unital A∞-category, weak unit
1. Introduction
Over the past decade, A∞-categories have experienced a
resurgence of interest due to applications in symplectic geom-
etry, deformation theory, non-commutative geometry, homo-
logical algebra, and physics.
The notion of A∞-category is a generalization of Stasheff’s
notion of A∞-algebra [11]. On the other hand, A∞-categories
generalize differential graded categories. In contrast to differ-
ential graded categories, composition in A∞-categories is asso-
ciative only up to homotopy that satisfies certain equation up
to another homotopy, and so on. The notion of A∞-category
appeared in the work of Fukaya on Floer homology [1] and
c© Volodymyr Lyubashenko, Oleksandr Manzyuk, 2006
236 V. Lyubashenko, O.Manzyuk
was related to mirror symmetry by Kontsevich [5]. Basic con-
cepts of the theory of A∞-categories have been developed by
Fukaya [2], Keller [4], Lefèvre-Hasegawa [7], Lyubashenko [8],
Soibelman [10].
The definition of A∞-category does not assume the exis-
tence of identity morphisms. The use of A∞-categories with-
out identities requires caution: for example, there is no a sen-
sible notion of isomorphic objects, the notion of equivalence
does not make sense, etc. In order to develop a comprehen-
sive theory of A∞-categories, a notion of unital A∞-category,
i.e., A∞-category with identity morphisms (also called units),
is necessary. The obvious notion of strictly unital A∞-cate-
gory, despite its technical advantages, is not quite satisfac-
tory: it is not homotopy invariant, meaning that it does
not translate along homotopy equivalences. Different defi-
nitions of (weakly) unital A∞-category have been suggested
by Lyubashenko [8, Definition 7.3], by Kontsevich and Soibel-
man [6, Definition 4.2.3], and by Fukaya [2, Definition 5.11].
We prove that these definitions are equivalent. The main in-
gredient of the proofs is the Yoneda Lemma for unital (in the
sense of Lyubashenko) A∞-categories proven in [9, Appen-
dix A].
2. Preliminaries
We follow the notation and conventions of [8], sometimes
without explicit mentioning. Some of the conventions are re-
called here.
Throughout, k is a commutative ground ring. A graded
k-module always means a Z-graded k-module.
A graded quiver A consists of a set ObA of objects and a
graded k-module A(X, Y ), for each X, Y ∈ ObA. A mor-
phism of graded quivers f : A → B of degree n consists of
Unital A∞-categories 237
a function Obf : ObA → ObB, X 7→ Xf , and a k-linear
map f = fX,Y : A(X, Y ) → B(Xf, Y f) of degree n, for each
X, Y ∈ ObA.
For a set S, there is a category Q/S defined as follows.
Its objects are graded quivers whose set of objects is S. A
morphism f : A → B in Q/S is a morphism of graded quiv-
ers of degree 0 such that Obf = idS. The category Q/S is
monoidal. The tensor product of graded quivers A and B is
a graded quiver A ⊗ B such that
(A ⊗ B)(X, Z) =
⊕
Y ∈S
A(X, Y ) ⊗ B(Y, Z), X, Z ∈ S.
The unit object is the discrete quiver kS with ObkS = S and
(kS)(X, Y ) =
{
k if X = Y ,
0 if X 6= Y ,
X, Y ∈ S.
Note that a map of sets f : S → R gives rise to a morphism of
graded quivers kf : kS → kR with Obkf = f and (kf)X,Y =
idk is X = Y and (kf)X,Y = 0 if X 6= Y , X, Y ∈ S.
An augmented graded cocategory is a graded quiver C equip-
ped with the structure of on augmented counital coassociative
coalgebra in the monoidal category Q/ObC. Thus, C comes
with a comultiplication ∆ : C → C⊗C, a counit ε : C → kObC,
and an augmentation η : kObC → C, which are morphisms
in Q/ObC satisfying the usual axioms. A morphism of aug-
mented graded cocategories f : C → D is a morphism of graded
quivers of degree 0 that preserves the comultiplication, counit,
and augmentation.
The main example of an augmented graded cocategory is
the following. Let A be a graded quiver. Denote by TA
the direct sum of graded quivers T nA, where T nA = A⊗n
is the n-fold tensor product of A in Q/ObA; in particular,
238 V. Lyubashenko, O.Manzyuk
T 0A = kObA, T 1A = A, T 2A = A ⊗ A, etc. The graded
quiver TA is an augmented graded cocategory in which the
comultiplication is the so called ‘cut’ comultiplication ∆0 :
TA → TA ⊗ TA given by
f1 ⊗ · · · ⊗ fn 7→
n∑
k=0
f1 ⊗ · · · ⊗ fk
⊗
fk+1 ⊗ · · · ⊗ fn,
the counit is given by the projection pr0 : TA → T 0A =
kObA, and the augmentation is given by the inclusion in0 :
kObA = T 0A →֒ TA.
The graded quiver TA admits also the structure of a graded
category, i.e., the structure of a unital associative algebra in
the monoidal category Q/ObA. The multiplication µ : TA⊗
TA → TA removes brackets in tensors of the form (f1 ⊗· · ·⊗
fm)
⊗
(g1 ⊗ · · · ⊗ gn). The unit η : kObA → TA is given by
the inclusion in0 : kObA = T 0A →֒ TA.
For a graded quiver A, denote by sA its suspension, the
graded quiver given by ObsA = ObA and (sA(X, Y ))n =
A(X, Y )n+1, for each n ∈ Z and X, Y ∈ ObA. An A∞-cat-
egory is a graded quiver A equipped with a differential b :
TsA → TsA of degree 1 such that (TsA, ∆0, pr0, in0, b) is an
augmented differential graded cocategory. In other terms, the
equations
b2 = 0, b∆0 = ∆0(b ⊗ 1 + 1 ⊗ b), bpr0 = 0, in0b = 0
hold true. Denote by
bmn
def
=
[
TmsA
inm−−→ TsA
b
−→ TsA
prn−−→ T nsA
]
matrix coefficients of b, for m, n > 0. Matrix coefficients bm1
are called components of b and abbreviated by bm. The above
equations imply that b0 = 0 and that b is unambiguously
Unital A∞-categories 239
determined by its components via the formula
bmn =
∑
p+k+q=m
p+1+q=n
1⊗p ⊗ bk ⊗ 1⊗q : TmsA → T nsA, m, n > 0.
The equation b2 = 0 is equivalent to the system of equations
∑
p+k+q=m
(1⊗p ⊗ bk ⊗ 1⊗q)bp+1+q = 0 : TmsA → sA, m > 1.
For A∞-categories A and B, an A∞-functor f : A → B is a
morphism of augmented differential graded cocategories f :
TsA → TsB. In other terms, f is a morphism of augmented
graded cocategories and preserves the differential, meaning
that fb = bf . Denote by
fmn
def
=
[
TmsA
inm−−→ TsA
f
−→ TsB
prn−−→ T nsB
]
matrix coefficients of f , for m, n > 0. Matrix coefficients fm1
are called components of f and abbreviated by fm. The con-
dition that f is a morphism of augmented graded cocategories
implies that f0 = 0 and that f is unambiguously determined
by its components via the formula
fmn =
∑
i1+···+in=m
fi1 ⊗ · · · ⊗ fin : TmsA → T nsB, m, n > 0.
The equation fb = bf is equivalent to the system of equations
∑
i1+···+in=m
(fi1 ⊗ · · · ⊗ fin)bn
=
∑
p+k+q=m
(1⊗p ⊗ bk ⊗ 1⊗q)fp+1+q : TmsA → sB,
for m > 1. An A∞-functor f is called strict if fn = 0 for
n > 1.
240 V. Lyubashenko, O.Manzyuk
3. Definitions
3.1. Definition (cf. [2,4]). An A∞-category A is strictly uni-
tal if, for each X ∈ ObA, there is a k-linear map X iA0 :
k → (sA)−1(X, X), called a strict unit, such that the fol-
lowing conditions are satisfied: XiA0 b1 = 0, the chain maps
(1 ⊗ Y iA0 )b2,−(X iA0 ⊗ 1)b2 : sA(X, Y ) → sA(X, Y ) are equal
to the identity map, for each X, Y ∈ ObA, and (· · · ⊗ iA0 ⊗
· · · )bn = 0 if n > 3.
For example, differential graded categories are strictly uni-
tal.
3.2. Definition (Lyubashenko [8, Definition 7.3]). An A∞-ca-
tegory A is unital if, for each X ∈ ObA, there is a k-linear
map XiA0 : k → (sA)−1(X, X), called a unit, such that the
following conditions hold: X iA0 b1 = 0 and the chain maps
(1 ⊗ Y iA0 )b2,−(X iA0 ⊗ 1)b2 : sA(X, Y ) → sA(X, Y ) are homo-
topic to the identity map, for each X, Y ∈ ObA. An arbi-
trary homotopy between (1 ⊗ Y iA0 )b2 and the identity map is
called a right unit homotopy. Similarly, an arbitrary homo-
topy between −(X iA0 ⊗ 1)b2 and the identity map is called a
left unit homotopy. An A∞-functor f : A → B between uni-
tal A∞-categories is unital if the cycles XiA0 f1 and Xf i
B
0 are
cohomologous, i.e., differ by a boundary, for each X ∈ ObA.
Clearly, a strictly unital A∞-category is unital.
With an arbitrary A∞-category A a strictly unital A∞-cat-
egory Asu with the same set of objects is associated. For each
X, Y ∈ ObA, the graded k-module sAsu(X, Y ) is given by
sAsu(X, Y ) =
{
sA(X, Y ) if X 6= Y ,
sA(X, X) ⊕ kX iA
su
0 if X = Y ,
Unital A∞-categories 241
where XiA
su
0 is a new generator of degree −1. The element
XiA
su
0 is a strict unit by definition, and the natural embedding
e : A →֒ Asu is a strict A∞-functor.
3.3. Definition (Kontsevich–Soibelman [6, Definition 4.2.3]).
A weak unit of an A∞-category A is an A∞-functor U : Asu →
A such that [
A
e
−֒→ A
su U
−→ A
]
= idA.
3.4. Proposition. Suppose that an A∞-category A admits a
weak unit. Then the A∞-category A is unital.
Proof. Let U : Asu → A be a weak unit of A. For each
X ∈ ObA, denote by XiA0 the element XiA
su
0 U1 ∈ sA(X, X)
of degree −1. It follows from the equation U1b1 = b1U1 that
XiA0 b1 = 0. Let us prove that X iA0 are unit elements of A.
For each X, Y ∈ ObA, there is a k-linear map
h = (1 ⊗ Y i0)U2 : sA(X, Y ) → sA(X, Y )
of degree −1. The equation
(3.1) (1 ⊗ b1 + b1 ⊗ 1)U2 + b2U1 = U2b1 + (U1 ⊗ U1)b2
implies that
−b1h + 1 = hb1 + (1 ⊗ Y iA0 )b2 : sA(X, Y ) → sA(X, Y ),
thus h is a right unit homotopy for A. For each X, Y ∈ ObA,
there is a k-linear map
h′ = −(X i0 ⊗ 1)U2 : sA(X, Y ) → sA(X, Y )
of degree −1. Equation (3.1) implies that
b1h
′ − 1 = −h′b1 + (XiA0 ⊗ 1)b2 : sA(X, Y ) → sA(X, Y ),
thus h′ is a left unit homotopy for A. Therefore, A is unital.
�
242 V. Lyubashenko, O.Manzyuk
3.5. Definition (Fukaya [2, Definition 5.11]). An A∞-cate-
gory C is called homotopy unital if the graded quiver
C
+ = C ⊕ kC ⊕ skC
(with ObC+ = ObC) admits an A∞-structure b+ of the follow-
ing kind. Denote the generators of the second and the third
direct summands of the graded quiver sC+ = sC⊕skC⊕s2
kC
by XiC
su
0 = 1s and jCX = 1s2 of degree respectively −1 and −2,
for each X ∈ ObC. The conditions on b+ are:
(1) for each X ∈ ObC, the element XiC0
def
= XiC
su
0 − jCXb+
1 is
contained in sC(X, X);
(2) C+ is a strictly unital A∞-category with strict units
X iC
su
0 , X ∈ ObC;
(3) the embedding C →֒ C+ is a strict A∞-functor;
(4) (sC ⊕ s2
kC)⊗nb+
n ⊂ sC, for each n > 1.
In particular, C+ contains the strictly unital A∞-category
Csu = C ⊕ kC. A version of this definition suitable for filtered
A∞-algebras (and filtered A∞-categories) is given by Fukaya,
Oh, Ohta and Ono in their book [3, Definition 8.2].
Let D be a strictly unital A∞-category with strict units iD0 .
Then it has a canonical homotopy unital structure (D+, b+).
Namely, jDXb+
1 = XiD
su
0 − X iD0 , and b+
n vanishes for each n > 1
on each summand of (sD ⊕ s2
kD)⊗n except on sD⊗n, where
it coincides with bD
n . Verification of the equation (b+)2 = 0 is
a straightforward computation.
3.6. Proposition. An arbitrary homotopy unital A∞-cate-
gory is unital.
Proof. Let C ⊂ C+ be a homotopy unital category. We claim
that the distinguished cycles XiC0 ∈ C(X, X)[1]−1, X ∈ ObC,
turn C into a unital A∞-category. Indeed, the identity
(1 ⊗ b+
1 + b+
1 ⊗ 1)b+
2 + b+
2 b+
1 = 0
Unital A∞-categories 243
applied to sC ⊗ jC or to jC ⊗ sC implies
(1 ⊗ iC0 )bC
2 = 1 + (1 ⊗ jC)b+
2 bC
1 + bC
1 (1 ⊗ jC)b+
2 : sC → sC,
(iC0 ⊗ 1)bC
2 = −1 + (jC ⊗ 1)b+
2 bC
1 + bC
1 (jC ⊗ 1)b+
2 : sC → sC.
Thus, (1⊗ jC)b+
2 : sC → sC and (jC⊗ 1)b+
2 : sC → sC are unit
homotopies. Therefore, the A∞-category C is unital. �
The converse of Proposition 3.6 holds true as well.
3.7. Theorem. An arbitrary unital A∞-category C with unit
elements iC0 admits a homotopy unital structure (C+, b+) with
jCb+
1 = iC
su
0 − iC0 .
Proof. By [9, Corollary A.12], there exists a differential graded
category D and an A∞-equivalence φ : C → D. By [9, Re-
mark A.13], we may choose D and φ such that ObD = ObC
and Obφ = idObC. Being strictly unital D admits a canonical
homotopy unital structure (D+, b+). In the sequel, we may
assume that D is a strictly unital A∞-category equivalent to
C via φ with the mentioned properties. Let us construct si-
multaneously an A∞-structure b+ on C+ and an A∞-functor
φ+ : C+ → D+ that will turn out to be an equivalence.
Let us extend the homotopy isomorphism φ1 : sC → sD to
a chain quiver map φ+
1 : sC+ → sD+. The A∞-equivalence
φ : C → D is a unital A∞-functor, i.e., for each X ∈ ObC,
there exists vX ∈ D(X, X)[1]−2 such that XiD0 −XiC0φ1 = vXb1.
In order that φ+ be strictly unital, we define XiC
su
0 φ+
1 = XiD
su
0 .
We should have
jCXφ+
1 b+
1 = jCXb+
1 φ+
1 = XiC
su
0 φ+
1 − XiC0φ1
= X iD
su
0 − XiD0 + X iD0 − XiC0φ1 = (jCX + vX)b+
1 ,
so we define jCXφ+
1 = jDX + vX .
244 V. Lyubashenko, O.Manzyuk
We claim that there is a homotopy unital structure (C+, b+)
of C satisfying the four conditions of Definition 3.5 and an
A∞-functor φ+ : C+ → D+ satisfying four parallel conditions:
(1) the first component of φ+ is the quiver morphism φ+
1
constructed above;
(2) the A∞-functor φ+ is strictly unital;
(3) the restriction of φ+ to C gives φ;
(4) (sC ⊕ s2
kC)⊗nφ+
n ⊂ sD, for each n > 1.
Notice that in the presence of conditions (2) and (3) the first
condition reduces to jCX(φ+)1 = jDX + vX , for each X ∈ ObC.
Components of the (1,1)-coderivation b+ : TsC+ → TsC+ of
degree 1 and of the augmented graded cocategory morphism
φ+ : TsC+ → TsD+ are constructed by induction. We already
know components b+
1 and φ+
1 . Given an integer n > 2, assume
that we have already found components b+
m, φ+
m of the sought
b+ and φ+ for m < n such that the equations
((b+)2)m = 0 : TmsC+(X, Y ) → sC+(X, Y ),(3.2)
(φ+b+)m = (b+φ+)m: TmsC+(X, Y ) → sD+(Xf, Y f)(3.3)
are satisfied for all m < n. Define b+
n , φ+
n on direct summands
of T nsC+ which contain a factor iC
su
0 by the requirement of
strict unitality of C+ and φ+. Then equations (3.2), (3.3)
hold true for m = n on such summands. Define b+
n , φ+
n on the
direct summand T nsC ⊂ T nsC+ as bC
n and φn. Then equations
(3.2), (3.3) hold true for m = n on the summand T nsC. It
remains to construct those components of b+ and φ+ which
have jC as one of their arguments.
Extend b1 : sC → sC to b′1 : sC+ → sC+ by iC
su
0 b′1 = 0 and
jCb′1 = 0. Define b−1 = b+
1 −b′1 : sC+ → sC+. Thus, b−1
∣∣
sCsu
= 0,
jCb−1 = iC
su
0 − iC0 and b+
1 = b′1 + b−1 . Introduce for 0 6 k 6 n
Unital A∞-categories 245
the graded subquiver Nk ⊂ T n(sC ⊕ s2
kC) by
Nk =
⊕
p0+p1+···+pk+k=n
T p0sC ⊗ jC ⊗ T p1sC ⊗ · · · ⊗ jC ⊗ T pksC
stable under the differential dNk =
∑
p+1+q=n 1⊗p ⊗ b′1 ⊗ 1⊗q,
and the graded subquiver Pl ⊂ T nsC+ by
Pl =
⊕
p0+p1+···+pl+l=n
T p0sCsu⊗ jC⊗T p1sCsu⊗· · ·⊗ jC⊗T plsCsu.
There is also the subquiver
Qk =
k⊕
l=0
Pl ⊂ T nsC+
and its complement
Q
⊥
k =
n⊕
l=k+1
Pl ⊂ T nsC+.
Notice that the subquiver Qk is stable under the differential
dQk =
∑
p+1+q=n 1⊗p ⊗ b+
1 ⊗ 1⊗q, and Q⊥
k is stable under the
differential dQ⊥
k =
∑
p+1+q=n 1⊗p ⊗ b′1 ⊗ 1⊗q. Furthermore, the
image of 1⊗a ⊗ b−1 ⊗ 1⊗c : Nk → T nsC+ is contained in Qk−1
for all a, c > 0 such that a + 1 + c = n.
Firstly, the components b+
n , φ+
n are defined on the graded
subquivers N0 = T nsC and Q0 = T nsCsu. Assume for an
integer 0 < k 6 n that restrictions of b+
n , φ+
n to Nl are already
found for all l < k. In other terms, we are given b+
n : Qk−1 →
sC+, φ+
n : Qk−1 → sD such that equations (3.2), (3.3) hold
on Qk−1. Let us construct the restrictions b+
n : Nk → sC,
φ+
n : Nk → sD, performing the induction step.
Introduce a (1,1)-coderivation b̃ : TsC+ → TsC+ of degree
1 by its components (0, b+
1 , . . . , b+
n−1, prQk−1
· b+
n |Qk−1
, 0, . . . ).
Introduce also a morphism of augmented graded cocategories
246 V. Lyubashenko, O.Manzyuk
φ̃ : TsC+ → TsD+ with Obφ̃ = Obφ by its components
(φ+
1 , . . . , φ+
n−1, prQk−1
·φ+
n |Qk−1
, 0, . . . ). Here prQk−1
: T nsC+ →
Qk−1 is the natural projection, vanishing on Q⊥
k−1. Then λ
def
=
b̃2 : TsC+ → TsC+ is a (1,1)-coderivation of degree 2 and
ν
def
= −φ̃b+ + b̃φ̃ : TsC+ → TsD+ is a (φ̃, φ̃)-coderivation of
degree 1. Equations (3.2), (3.3) imply that λm = 0, νm = 0 for
m < n. Moreover, λn, νn vanish on Qk−1. On the complement
the n-th components equal
λn =
1<r<n∑
a+r+c=n
(1⊗a ⊗ b+
r ⊗ 1⊗c)b+
a+1+c
+
∑
a+1+c=n
(1⊗a ⊗ b−1 ⊗ 1⊗c)b̃n : Q
⊥
k−1 → sC+,
νn = −
1<r6n∑
i1+···+ir=n
(φ+
i1
⊗ · · · ⊗ φ+
ir
)b+
r
+
1<r<n∑
a+r+c=n
(1⊗a ⊗ b+
r ⊗ 1⊗c)φ+
a+1+c
+
∑
a+1+c=n
(1⊗a ⊗ b−1 ⊗ 1⊗c)φ̃n : Q
⊥
k−1 → sD.
The restriction λn|Nk
takes values in sC. Indeed, for the first
sum in the expression for λn this follows by the induction
assumption since r > 1 and a+1+ c > 1. For the second sum
this follows by the induction assumption and strict unitality if
n > 2. In the case of n = 2, k = 1 this is also straightforward.
The only case which requires computation is n = 2, k = 2:
(jC⊗ jC)(1⊗b−1 +b−1 ⊗1)b̃2 = jC− (jC⊗ iC0 )b+
2 − jC− (iC0 ⊗ jC)b+
2 ,
which belongs to sC by the induction assumption.
Unital A∞-categories 247
Equations (3.2), (3.3) for m = n take the form
−b+
n b1 −
∑
a+1+c=n
(1⊗a ⊗ b′1 ⊗ 1⊗c)b+
n = λn : Nk → sC,(3.4)
φ+
n b1 −
∑
a+1+c=n
(1⊗a ⊗ b′1 ⊗ 1⊗c)φ+
n − b+
n φ1 = νn : Nk → sD.
(3.5)
For arbitrary objects X, Y of C, equip the graded k-module
Nk(X, Y ) with the differential dNk =
∑
p+1+q=n 1⊗p ⊗ b′1 ⊗1⊗q
and denote by u the chain map
C
k
(Nk(X, Y ), sC(X, Y )) → C
k
(Nk(X, Y ), sD(Xφ, Y φ)),
λ 7→ λφ1.
Since φ1 is homotopy invertible, the map u is homotopy invert-
ible as well. Therefore, the complex Cone(u) is contractible,
e.g. by [8, Lemma B.1], in particular, acyclic. Equations (3.4)
and (3.5) have the form −b+
n d = λn, φ+
n d + b+
n u = νn, that is,
the element (λn, νn) of
C
2
k
(Nk(X, Y ), sC(X, Y )) ⊕ C
1
k
(Nk(X, Y ), sD(Xφ, Y φ))
= Cone1(u)
has to be the boundary of the sought element (b+
n , φ+
n ) of
C
1
k
(Nk(X, Y ), sC(X, Y )) ⊕ C
0
k
(Nk(X, Y ), sD(Xφ, Y φ))
= Cone0(u).
These equations are solvable because (λn, νn) is a cycle in
Cone1(u). Indeed, the equations to verify −λnd = 0, νnd +
248 V. Lyubashenko, O.Manzyuk
λnu = 0 take the form
−λnb1 +
∑
p+1+q=n
(1⊗p ⊗ b′1 ⊗ 1⊗q)λn = 0 : Nk → sC,
νnb1 +
∑
p+1+q=n
(1⊗p ⊗ b′1 ⊗ 1⊗q)νn − λnφ1 = 0 : Nk → sD.
Composing the identity −λb̃ + b̃λ = 0 : T nsC+ → TsC+ with
the projection pr1 : TsC+ → sC+ yields the first equation.
The second equation follows by composing the identity νb+ +
b̃ν − λφ̃ = 0 : T nsC+ → TsD+ with pr1 : TsD+ → sD+.
Thus, the required restrictions of b+
n , φ+
n to Nk (and to
Qk) exist and satisfy the required equations. We proceed by
induction increasing k from 0 to n and determining b+
n , φ+
n
on the whole Qn = T nsC+. Then we replace n with n + 1
and start again from T n+1sC. Thus the induction on n goes
through. �
3.8. Remark. Let (C+, b+) be a homotopy unital structure
of an A∞-category C. Then the embedding A∞-functor ι :
C → C+ is an equivalence. Indeed, it is bijective on objects.
By [8, Theorem 8.8] it suffices to prove that ι1 : sC → sC+
is homotopy invertible. And indeed, the chain quiver map
π1 : sC+ → sC, π1|sC = id, X iC
su
0 π1 = X iC0 , jCXπ1 = 0, is
homotopy inverse to ι1. Namely, the homotopy h : sC+ →
sC+, h|sC = 0, XiC
su
0 h = jCX , jCXh = 0, satisfies the equation
idsC+ − π1 · ι1 = hb+
1 + b+
1 h.
The equation between A∞-functors
[
C
ιC
−→ C
+ φ+
−→ D
+
]
=
[
C
φ
−→ D
ιD
−→ D
+
]
obtained in the proof of Theorem 3.7 implies that φ+ is an
A∞-equivalence as well. In particular, φ+
1 is homotopy invert-
ible.
Unital A∞-categories 249
The converse of Proposition 3.4 holds true as well, how-
ever its proof requires more preliminaries. It is deferred until
Section 5.
4. Double coderivations
4.1. Definition. For A∞-functors f, g : A → B, a double
(f, g)-coderivation of degree d is a system of k-linear maps
r : (TsA ⊗ TsA)(X, Y ) → TsB(Xf, Y g), X, Y ∈ ObA,
of degree d such that the equation
(4.1) r∆0 = (∆0 ⊗ 1)(f ⊗ r) + (1 ⊗ ∆0)(r ⊗ g)
holds true.
Equation (4.1) implies that r is determined by a system of
k-linear maps rpr1 : TsA ⊗ TsA → sB with components of
degree d
rn,m : sA(X0, X1) ⊗ · · · ⊗ sA(Xn+m−1, Xn+m)
→ sB(X0f, Xn+mg),
for n, m > 0, via the formula
rn,m;k = (r|T nsA⊗T msA)prk : T nsA ⊗ TmsA → T ksB,
rn,m;k =
p+1+q=k∑
i1+···+ip+i=n,
j1+···+jq+j=m
fi1 ⊗ · · · ⊗ fip ⊗ ri,j ⊗ gj1 ⊗ · · · ⊗ gjq
.
(4.2)
This follows from the equation
(4.3) r∆
(l)
0 =
∑
p+1+q=l
(∆
(p+1)
0 ⊗ ∆
(q+1)
0 )(f⊗p ⊗ r ⊗ g⊗q) :
TsA ⊗ TsA → (TsB)⊗l,
250 V. Lyubashenko, O.Manzyuk
which holds true for each l > 0. Here ∆
(0)
0 = ε, ∆
(1)
0 = id,
∆
(2)
0 = ∆0 and ∆
(l)
0 means the cut comultiplication iterated
l − 1 times.
Double (f, g)-coderivations form a chain complex, which we
are going to denote by (D(A, B)(f, g), B1). For each d ∈ Z,
the component D(A, B)(f, g)d consists of double (f, g)-coderi-
vations of degree d. The differential B1 of degree 1 is given
by
rB1
def
= rb − (−)d(1 ⊗ b + b ⊗ 1)r,
for each r ∈ D(A, B)(f, g)d. The component [rB1]n,m of rB1
is given by
(4.4) ∑
i1+···+ip+i=n,
j1+···+jq+j=m
(fi1 ⊗ · · · ⊗ fip ⊗ rij ⊗ gj1 ⊗ · · · ⊗ gjq
)bp+1+q
− (−)r
∑
a+k+c=n
(1⊗a ⊗ bk ⊗ 1⊗c+m)ra+1+c,m
− (−)r
∑
u+t+v=m
(1⊗n+u ⊗ bt ⊗ 1⊗v)rn,u+1+v,
for each n, m > 0. An A∞-functor h : B → C gives rise to a
chain map
D(A, B)(f, g) → D(A, C)(fh, gh), r 7→ rh.
The component [rh]n,m of rh is given by
(4.5)
∑
i1+···+ip+i=n,
j1+···+jq+j=m
(fi1 ⊗· · ·⊗fip ⊗ri,j ⊗gj1 ⊗· · ·⊗gjq
)hp+1+q,
for each n, m > 0. Similarly, an A∞-functor k : D → A gives
rise to a chain map
D(A, B)(f, g) → D(D, B)(kf, kg), r 7→ (k ⊗ k)r.
Unital A∞-categories 251
The component [(k ⊗ k)r]n,m of (k ⊗ k)r is given by
(4.6)
∑
i1+···+ip=n
j1+···+jq=m
(ki1 ⊗ · · · ⊗ kip ⊗ kj1 ⊗ · · · ⊗ kjq
)rp,q,
for each n, m > 0. Proofs of these facts are elementary and
are left to the reader.
Let C be an A∞-category. For each n > 0, introduce a
morphism
νn =
n∑
i=0
(−)n−i(1⊗i ⊗ ε ⊗ 1⊗n−i) : (TsC)⊗n+1 → (TsC)⊗n,
in Q/ObC. In particular, ν0 = ε : TsC → kObC. Denote
ν = ν1 = (1⊗ ε)− (ε⊗ 1) : TsC⊗ TsC → TsC for the sake of
brevity.
4.2. Lemma. The map ν : TsC ⊗ TsC → TsC is a double
(1, 1)-coderivation of degree 0 and νB1 = 0.
Proof. We have:
(∆0 ⊗ 1)(1 ⊗ ν) + (1 ⊗ ∆0)(ν ⊗ 1)
= (∆0 ⊗ 1)(1 ⊗ 1 ⊗ ε) − (∆0 ⊗ 1)(1 ⊗ ε ⊗ 1)
+ (1 ⊗ ∆0)(1 ⊗ ε ⊗ 1) − (1 ⊗ ∆0)(ε ⊗ 1 ⊗ 1)
= (∆0 ⊗ ε)− (ε⊗∆0) = ((1⊗ ε)− (ε⊗ 1))∆0 = ν∆0,
due to the identities
(∆0 ⊗ 1)(1 ⊗ ε ⊗ 1) = 1 ⊗ 1 = (1 ⊗ ∆0)(1 ⊗ ε ⊗ 1) :
TsC ⊗ TsC → TsC ⊗ TsC.
This computation shows that ν : TsC ⊗ TsC → TsC is a
double (1, 1)-coderivation. Its only non-vanishing components
are X,Y ν1,0 = 1 : sC(X, Y ) → sC(X, Y ) and X,Y ν0,1 = 1 :
sC(X, Y ) → sC(X, Y ), X, Y ∈ ObC.
252 V. Lyubashenko, O.Manzyuk
Since νB1 is a double (1, 1)-coderivation of degree 1, the
equation νB1 = 0 is equivalent to its particular case νB1pr1 =
0, i.e., for each n, m > 0
∑
06i6n,
06j6m
(1⊗n−i ⊗ νi,j ⊗ 1⊗m−j)bn−i+1+m−j
−
∑
a+k+c=n
(1⊗a ⊗ bk ⊗ 1⊗c+m)νa+1+c,m
−
∑
u+t+v=m
(1⊗n+u ⊗ bt ⊗ 1⊗v)νn,u+1+v = 0 :
T nsC ⊗ TmsC → sC.
It reduces to the identity
χ(n > 0)bn+m − χ(m > 0)bn+m
− χ(m = 0)bn + χ(n = 0)bm = 0,
where χ(P ) = 1 if a condition P holds and χ(P ) = 0 if P
does not hold. �
Let C be a strictly unital A∞-category. The strict unit iC0
is viewed as a morphism of graded quivers iC0 : kObC → sC of
degree −1, identity on objects. For each n > 0, introduce a
morphism of graded quivers
ξn =
[
(TsC)⊗n+1 1⊗iC0⊗1⊗···⊗iC0⊗1
−−−−−−−−−−→
TsC ⊗ sC ⊗ TsC ⊗ · · · ⊗ sC ⊗ TsC
µ(2n+1)
−−−−→ TsC
]
,
of degree −n, identity on objects. Here µ(2n+1) denotes com-
position of 2n + 1 composable arrows in the graded cate-
gory TsC. In particular, ξ0 = 1 : TsC → TsC. Denote
ξ = ξ1 = (1 ⊗ iC0 ⊗ 1)µ(3) : TsC ⊗ TsC → TsC for the sake of
brevity.
Unital A∞-categories 253
4.3. Lemma. The map ξ : TsC ⊗ TsC → TsC is a double
(1, 1)-coderivation of degree −1 and ξB1 = ν.
Proof. The following identity follows directly from the defini-
tions of µ and ∆0:
µ∆0 = (∆0 ⊗ 1)(1 ⊗ µ) + (1 ⊗ ∆0)(µ ⊗ 1) − 1 :
TsC ⊗ TsC → TsC ⊗ TsC.
It implies
(4.7)
µ(3)∆0 = (∆0 ⊗ 1 ⊗ 1)(1 ⊗ µ(3)) + (1 ⊗ 1 ⊗ ∆0)(µ
(3) ⊗ 1)
+ (1 ⊗ ∆0 ⊗ 1)(µ ⊗ µ) − (1 ⊗ µ) − (µ ⊗ 1) :
TsC ⊗ TsC ⊗ TsC → TsC ⊗ TsC.
Since iC0∆0 = iC0 ⊗ η + η ⊗ iC0 : kObC → TsC⊗ TsC, it follows
that
(1⊗ iC0∆0⊗1)(µ⊗µ)− (1⊗ (iC0 ⊗1)µ)− ((1⊗ iC0)µ⊗1) = 0 :
TsC ⊗ TsC → TsC ⊗ TsC.
Equation (4.7) yields
(1 ⊗ iC0 ⊗ 1)µ(3)∆0
= (∆0⊗1)(1⊗(1⊗iC0 ⊗1)µ(3))+(1⊗∆0)((1⊗iC0 ⊗1)µ(3)⊗1),
i.e., ξ = (1 ⊗ iC0 ⊗ 1)µ(3) : TsC ⊗ TsC → TsC is a double
(1, 1)-coderivation. Its the only non-vanishing components
are Xξ0,0 = XiC0 ∈ sC(X, X), X ∈ ObC.
Since both ξB1 and ν are double (1, 1)-coderivations of de-
gree 0, the equation ξB1 = ν is equivalent to its particular
254 V. Lyubashenko, O.Manzyuk
case ξB1pr1 = νpr1, i.e., for each n, m > 0
∑
06p6n
06q6m
(1⊗n−p ⊗ ξp,q ⊗ 1⊗m−q)bn−p+1+m−q
+
∑
a+k+c=n
(1⊗a ⊗ bk ⊗ 1⊗c+m)ξa+1+c,m
+
∑
u+t+v=m
(1⊗n+u ⊗ bt ⊗ 1⊗v)ξn,u+1+v = νn,m :
T nsC ⊗ TmsC → sC.
It reduces to the the equation
(1⊗n ⊗ iC0 ⊗ 1⊗m)bn+1+m = νn,m : T nsC ⊗ TmsC → sC,
which holds true, since iC0 is a strict unit. �
Note that the maps νn, ξn obey the following relations:
(4.8)
ξn = (ξn−1 ⊗ 1)ξ, νn = (1⊗n ⊗ ε)− (νn−1 ⊗ 1), n > 1.
In particular, ξnε = 0 : (TsC)⊗n+1 → kObC, for each n > 1,
as ξε = 0 by equation (4.3).
4.4. Lemma. The following equations hold true:
ξn∆0 =
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(ξi ⊗ ξn−i), n > 0,(4.9)
ξnb − (−)n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)ξn = νnξn−1, n > 1.(4.10)
Proof. Let us prove (4.9). The proof is by induction on n. The
case n = 0 is trivial. Let n > 1. By (4.8) and Lemma 4.3,
ξn∆0 = (ξn−1⊗1)ξ∆0 = (ξn−1∆0⊗1)(1⊗ξ)+(ξn−1⊗∆0)(ξ⊗1).
Unital A∞-categories 255
By induction hypothesis,
ξn−1∆0 =
n−1∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−1−i)(ξi ⊗ ξn−1−i),
therefore
ξn∆0 =
n−1∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(ξi ⊗ ξn−1−i ⊗ 1)(1 ⊗ ξ)
+ (1⊗n ⊗ ∆0)((ξn−1 ⊗ 1)ξ ⊗ 1)
=
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(ξi ⊗ ξn−i),
since (ξn−1−i ⊗ 1)ξ = ξn−i if 0 6 i 6 n − 1.
Let us prove (4.10). The proof is by induction on n. The
case n = 1 follows from Lemma 4.3. Let n > 2. By (4.8) and
Lemma 4.3,
ξnb − (−)n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)ξn
= (ξn−1 ⊗ 1)ξb − (−)n
n−1∑
i=0
((1⊗i ⊗ b ⊗ 1⊗n−1−i)ξn−1 ⊗ 1)ξ
− (−)n(1⊗n ⊗ b)(ξn−1 ⊗ 1)ξ
= −(ξn−1b ⊗ 1)ξ − (ξn−1 ⊗ b)ξ + (ξn−1 ⊗ 1)ν
+ (−)n−1
n−1∑
i=0
((1⊗i ⊗ b ⊗ 1⊗n−1−i)ξn−1 ⊗ 1)ξ + (ξn−1 ⊗ b)ξ
= (ξn−1 ⊗ 1)ν
−
([
ξn−1b − (−)n−1
n−1∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−1−i)ξn−1
]
⊗ 1
)
ξ.
256 V. Lyubashenko, O.Manzyuk
By induction hypothesis
ξn−1b − (−)n−1
n−1∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−1−i)ξn−1 = νn−1ξn−2,
therefore
ξnb−(−)n
n∑
i=0
(1⊗i⊗b⊗1⊗n−i)ξn = (ξn−1⊗1)ν−(νn−1ξn−2⊗1)ξ.
Since by (4.8),
(ξn−1 ⊗ 1)ν − (νn−1ξn−2 ⊗ 1)ξ
= (ξn−1 ⊗ ε) − (ξn−1ε ⊗ 1) − (νn−1 ⊗ 1)ξn−1
= (1⊗n ⊗ ε)ξn−1 − (νn−1 ⊗ 1)ξn−1 = νnξn−1,
equation (4.10) is proven. �
5. An augmented differential graded cocategory
Let now C = Asu, where A is an A∞-category. There is an
isomorphism of graded k-quivers, identity on objects:
ζ :
⊕
n>0
(TsA)⊗n+1[n] → TsAsu.
The morphism ζ is the sum of morphisms
(5.1) ζn =
[
(TsA)⊗n+1[n]
s−n
−−→ (TsA)⊗n+1
e⊗n+1
−֒−−→ (TsAsu)⊗n+1 ξn
−→ TsAsu
]
,
where e : A →֒ Asu is the natural embedding. The graded
quiver
E
def
=
⊕
n>0
(TsA)⊗n+1[n]
Unital A∞-categories 257
admits a unique structure of an augmented differential graded
cocategory such that ζ becomes an isomorphism of augmented
differential graded cocategories. The comultiplication ∆̃ :
E → E ⊗ E is found from the equation
[
E
ζ
−→ TsAsu
∆0−→ TsAsu ⊗ TsAsu
]
=
[
E
∆̃
−→ E ⊗ E
ζ⊗ζ
−−→ TsAsu ⊗ TsAsu
]
.
Restricting the left hand side of the equation to the summand
(TsA)⊗n+1[n] of E, we obtain
ζn∆0 = s−ne⊗n+1ξn∆0
= s−n
n∑
i=0
(e⊗i ⊗ e∆0 ⊗ e⊗n−i)(ξi ⊗ ξn−i) :
(TsA)⊗n+1[n] → TsAsu ⊗ TsAsu,
by equation (4.9). Since e is a morphism of augmented graded
cocategories, it follows that
ζn∆0 = s−n
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(e⊗i+1ξi ⊗ e⊗n−i+1ξn−i)
= s−n
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(si ⊗ sn−i)(ζi ⊗ ζn−i) :
(TsA)⊗n+1[n] → TsAsu ⊗ TsAsu.
258 V. Lyubashenko, O.Manzyuk
This implies the following formula for ∆̃:
(5.2) ∆̃|(TsA)⊗n+1[n] = s−n
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(si ⊗ sn−i) :
(TsA)⊗n+1[n] →
n⊕
i=0
(TsA)⊗i+1[i]
⊗
(TsA)⊗n−i+1[n − i].
The counit of E is ε̃ = [E
pr0−−→ TsA
ε
−→ kObA = kObE]. The
augmentation of E is η̃ = [kObE = kObA
η
−→ TsA
in0−→ E]. The
differential b̃ : E → E is found from the following equation:
[
E
ζ
−→ TsAsu b
−→ TsAsu
]
=
[
E
b̃
−→ E
ζ
−→ TsAsu
]
.
Let b̃n,m : (TsA)⊗n+1[n] → (TsA)⊗m+1[m], n, m > 0, denote
the matrix coefficients of b̃. Restricting the left hand side of
the above equation to the summand (TsA)⊗n+1[n] of E, we
obtain
ζnb = s−ne⊗n+1ξnb
= s−ne⊗n+1νnξn−1 + (−)ns−n
n∑
i=0
(e⊗i ⊗ eb ⊗ e⊗n−i)ξn :
(TsA)⊗n+1[n] → TsAsu,
by equation (4.10). Since e preserves the counit, it follows
that
e⊗n+1νn = νne⊗n : (TsA)⊗n+1 → (TsAsu)⊗n.
Unital A∞-categories 259
Furthermore, e commutes with the differential b, therefore
ζnb = s−nνns
n−1(s−(n−1)e⊗nξn−1)
+ (−)ns−n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)sn(s−ne⊗n+1ξn)
= s−nνns
n−1ζn−1 + (−)ns−n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)snζn :
(TsA)⊗n+1[n] → TsAsu.
We conclude that
(5.3) b̃n,n = (−)ns−n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)sn :
(TsA)⊗n+1[n] → (TsA)⊗n+1[n],
for n > 0, and
(5.4) b̃n,n−1 = s−nνns
n−1 : (TsA)⊗n+1[n] → (TsA)⊗n[n − 1],
for n > 1, are the only non-vanishing matrix coefficients of b̃.
Let g : E → TsB be a morphism of augmented differential
graded cocategories, and let gn : (TsA)⊗n+1[n] → TsB be its
components. By formula (5.2), the equation g∆0 = ∆̃(g ⊗ g)
is equivalent to the system of equations
gn∆0 = s−n
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(sigi ⊗ sn−ign−i) :
(TsA)⊗n+1[n] → TsB ⊗ TsB, n > 0.
The equation gε = ε̃(kObg) is equivalent to the equations
g0ε = ε(kObg0), gnε = 0, n > 1. The equation η̃g = (kObg)η
is equivalent to the equation ηg0 = (kObg0)η. By formu-
las (5.3) and (5.4), the equation gb = b̃g is equivalent to
260 V. Lyubashenko, O.Manzyuk
g0b = bg0 : TsA → TsB and
gnb = (−)ns−n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)sngn + s−nνnsn−1gn−1 :
(TsA)⊗n+1[n] → TsB, n > 1.
Introduce k-linear maps φn = sngn : (TsA)⊗n+1(X, Y ) →
TsB(Xg, Y g) of degree −n, X, Y ∈ ObA, n > 0. The above
equations take the following form:
(5.5) φn∆0 =
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(φi ⊗ φn−i) :
(TsA)⊗n+1 → TsB ⊗ TsB,
for n > 1;
(5.6) φnb = (−)n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)φn + νnφn−1 :
(TsA)⊗n+1 → TsB,
for n > 1;
φ0∆0 = ∆0(φ0 ⊗ φ0), φ0ε = ε, φ0b = bφ0,(5.7)
φnε = 0, n > 1.(5.8)
Summing up, we conclude that morphisms of augmented dif-
ferential graded cocategories E → TsB are in bijection with
collections consisting of a morphism of augmented differen-
tial graded cocategories φ0 : TsA → TsB and of k-linear
maps φn : (TsA)⊗n+1(X, Y ) → TsB(Xφ0, Y φ0) of degree
−n, X, Y ∈ ObA, n > 1, such that equations (5.5), (5.6),
and (5.8) hold true.
In particular, A∞-functors f : Asu → B, which are aug-
mented differential graded cocategory morphisms TsAsu →
Unital A∞-categories 261
TsB, are in bijection with morphisms g = ζf : E → TsB of
augmented differential graded cocategories. With the above
notation, we may say that to give an A∞-functor f : Asu → B
is the same as to give an A∞-functor φ0 : A → B and a system
of k-linear maps φn : (TsA)⊗n+1(X, Y ) → TsB(Xφ0, Y φ0) of
degree −n, X, Y ∈ ObA, n > 1, such that equations (5.5),
(5.6) and (5.8) hold true.
5.1. Proposition. The following conditions are equivalent.
(a) There exists an A∞-functor U : Asu → A such that
[
A
e
−֒→ A
su
U
−→ A
]
= idA.
(b) There exists a double (1, 1)-coderivation φ : TsA ⊗
TsA → TsA of degree −1 such that φB1 = ν.
Proof. (a)⇒(b) Let U : Asu → A be an A∞-functor such that
eU = idA, in particular ObU = id : ObAsu = ObA → ObA.
It gives rise to the family of k-linear maps φn = snζnU :
(TsA)⊗n+1(X, Y ) → TsB(X, Y ) of degree −n, X, Y ∈ ObA,
n > 0, that satisfy equations (5.5), (5.6) and (5.8). In par-
ticular, φ0 = eU = idA. Equations (5.5) and (5.6) for n = 1
read as follows:
φ1∆0 = (∆0 ⊗ 1)(φ0 ⊗ φ1) + (1 ⊗ ∆0)(φ1 ⊗ φ0)
= (∆0 ⊗ 1)(1 ⊗ φ1) + (1 ⊗ ∆0)(φ1 ⊗ 1),
φ1b = (1 ⊗ b + b ⊗ 1)φ1 + ν1φ0 = (1 ⊗ b + b ⊗ 1)φ1 + ν.
In other words, φ1 is a double (1, 1)-coderivation of degree −1
and φ1B1 = ν.
(b)⇒(a) Let φ : TsA ⊗ TsA → TsA be a double (1, 1)-
coderivation of degree −1 such that φB1 = ν. Define k-linear
maps
φn : (TsA)⊗n+1(X, Y ) → TsA(X, Y ), X, Y ∈ ObA,
262 V. Lyubashenko, O.Manzyuk
of degree −n, n > 0, recursively via φ0 = idA and φn =
(φn−1⊗1)φ, n > 1. Let us show that φn satisfy equations (5.5),
(5.6) and (5.8). Equation (5.8) is obvious: φnε = (φn−1 ⊗
1)φε = 0 as φε = 0 by (4.3). Let us prove equation (5.5) by
induction. It holds for n = 1 by assumption, since φ1 = φ is
a double (1, 1)-coderivation. Let n > 2. We have:
φn∆0 = (φn−1 ⊗ 1)φ1∆0
= (φn−1 ⊗ 1)((∆0 ⊗ 1)(1 ⊗ φ1) + (1 ⊗ ∆0)(φ1 ⊗ 1))
= (φn−1∆0 ⊗ 1)(1 ⊗ φ1)
+ (1⊗n ⊗ ∆0)((φn−1 ⊗ 1)φ1 ⊗ 1).
By induction hypothesis,
φn−1∆0 =
n−1∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−1−i)(φi ⊗ φn−1−i),
so that
φn∆0 =
n−1∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(φi ⊗ φn−1−i ⊗ 1)(1 ⊗ φ1)
+ (1⊗n ⊗ ∆0)((φn−1 ⊗ 1)φ1 ⊗ 1)
=
n∑
i=0
(1⊗i ⊗ ∆0 ⊗ 1⊗n−i)(φi ⊗ φn−i),
since (φn−1−i ⊗ 1)φ1 = φn−i, 0 6 i 6 n − 1.
Unital A∞-categories 263
Let us prove equation (5.6) by induction. For n = 1 it is
equivalent to the equation φB1 = ν, which holds by assump-
tion. Let n > 2. We have:
φnb − (−)n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)φn
= (φn−1 ⊗ 1)φb − (−)n
n−1∑
i=0
((1⊗i ⊗ b ⊗ 1⊗n−1−i)φn−1 ⊗ 1)φ
− (−)n(1⊗n ⊗ b)(φn−1 ⊗ 1)φ
= −(φn−1b ⊗ 1)φ − (φn−1 ⊗ b)φ + (φn−1 ⊗ 1)ν
+ (−)n−1
n−1∑
i=0
((1⊗i ⊗ b ⊗ 1⊗n−1−i)φn−1 ⊗ 1)φ + (φn−1 ⊗ b)φ
= (φn−1 ⊗ 1)ν
−
([
φn−1b − (−)n−1
n−1∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−1−i)φn−1
]
⊗ 1
)
φ.
By induction hypothesis,
φn−1b − (−)n−1
n−1∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−1−i)φn−1 = νn−1φn−2,
therefore
φnb − (−)n
n∑
i=0
(1⊗i ⊗ b ⊗ 1⊗n−i)φn
= (φn−1 ⊗ 1)ν − (νn−1φn−2 ⊗ 1)φ.
264 V. Lyubashenko, O.Manzyuk
Since by (4.8)
(φn−1 ⊗ 1)ν − (νn−1φn−2 ⊗ 1)φ
= (φn−1 ⊗ ε) − (φn−1ε ⊗ 1) − (νn−1 ⊗ 1)φn−1
= (1⊗n ⊗ ε)φn−1 − (νn−1 ⊗ 1)φn−1 = νnφn−1,
and equation (5.6) is proven.
The system of maps φn, n > 0, corresponds to an A∞-func-
tor U : Asu → A such that φn = snζnU , n > 0. In particular,
eU = φ0 = idA. �
5.2. Proposition. Let A be a unital A∞-category. There ex-
ists a double (1, 1)-coderivation h : TsA ⊗ TsA → TsA of
degree −1 such that hB1 = ν.
Proof. Let A be a unital A∞-category. By [9, Corollary A.12],
there exist a differential graded category D and an A∞-equiv-
alence f : A → D. The functor f is unital by [8, Corol-
lary 8.9]. This means that, for every object X of A, there
exists a k-linear map Xv0 : k → (sD)−2(Xf, Xf) such that
XiA0 f1 = Xf i
D
0 + Xv0b1. Here Xf i
D
0 denotes the strict unit of
the differential graded category D.
By Lemma 4.3, ξ = (1 ⊗ iD0 ⊗ 1)µ(3) : TsD ⊗ TsD → TsD
is a (1, 1)-coderivation of degree −1. Let ι denote the double
(f, f)-coderivation (f ⊗ f)ξ of degree −1. By Lemma 4.3,
ιB1 = (f ⊗ f)(ξB1) = (f ⊗ f)ν = νf.
By Lemma 4.2, the equation νB1 = 0 holds true. We conclude
that the double coderivations ν ∈ D(A, A)(idA, idA)0 and ι ∈
D(A, D)(f, f)−1 satisfy the following equations:
νB1 = 0,(5.9)
ιB1 − νf = 0.(5.10)
Unital A∞-categories 265
We are going to prove that there exist double coderivations
h ∈ D(A, A)(idA, idA)−1 and k ∈ D(A, D)(f, f)−2 such that
the following equations hold true:
hB1 = ν,
hf = ι + kB1.
Let us put Xh0,0 = X iA0 , Xk0,0 = Xv0, and construct the other
components of h and k by induction. Given an integer t > 0,
assume that we have already found components hp,q, kp,q of
the sought h, k, for all pairs (p, q) with p + q < t, such that
the equations
(5.11) (hB1 − ν)p,q = 0 :
sA(X0, X1) ⊗ · · · ⊗ sA(Xp+q−1, Xp+q) → sA(X0, Xp+q),
(5.12) (kB1 + ι − hf)p,q = 0 :
sA(X0, X1) ⊗ · · · ⊗ sA(Xp+q−1, Xp+q) → sD(X0f, Xp+qf)
are satisfied for all pairs (p, q) with p+q < t. Introduce double
coderivations h̃ ∈ D(A, A)(idA, idA) and k̃ ∈ D(A, D)(f, f)
of degree −1 resp. −2 by their components: h̃p,q = hp,q,
k̃p,q = kp,q for p + q < t, all the other components vanish.
Define a double (1, 1)-coderivation λ = h̃B1 − ν of degree 0
and a double (f, f)-coderivation κ = k̃B1 + ι − h̃f of degree
−1. Then λp,q = 0, κp,q = 0 for all p+q < t. Let non-negative
integers n, m satisfy n+m = t. The identity λB1 = 0 implies
that
λn,mb1 −
n+m∑
l=1
(1⊗l−1 ⊗ b1 ⊗ 1⊗n+m−l)λn,m = 0.
266 V. Lyubashenko, O.Manzyuk
The (n, m)-component of the identity κB1 + λf = 0 gives
κn,mb1 +
n+m∑
l=1
(1⊗l−1 ⊗ b1 ⊗ 1⊗n+m−l)κn,m + λn,mf1 = 0.
The chain map f1 : A(X0, Xn+m) → sD(X0f, Xn+mf) is ho-
motopy invertible as f is an A∞-equivalence. Hence, the chain
map Φ given by
C
•
k
(N, sA(X0, Xn+m)) → C
•
k
(N, sD(X0f, Xn+mf)),
λ 7→ λf1,
is homotopy invertible for each complex of k-modules N , in
particular, for N = sA(X0, X1) ⊗ · · · ⊗ sA(Xn+m−1, Xn+m).
Therefore, the complex Cone(Φ) is contractible, e.g. by [8,
Lemma B.1]. Consider the element (λn,m, κn,m) of
C
0
k
(N, sA(X0, Xn+m)) ⊕ C
−1
k
(N, D(X0f, Xn+mf)).
The above direct sum coincides with Cone−1(Φ). The equa-
tions −λn,md = 0, κn,md+λn,mΦ = 0 imply that (λn,m, κn,m) is
a cycle in the complex Cone(Φ). Due to acyclicity of Cone(Φ),
(λn,m, κn,m) is a boundary of some element (hn,m,−kn,m) of
Cone−2(Φ), i.e., of
C
−1
k
(N, sA(X0, Xn+m)) ⊕ C
−2
k
(N, D(X0f, Xn+mf)).
Unital A∞-categories 267
Thus, −kn,md + hn,mf1 = κn,m, −hn,md = λn,m. These equa-
tions can be written as follows:
− hn,mb1 −
∑
u+1+v=n+m
(1⊗u ⊗ b1 ⊗ 1⊗v)hn,m
= (h̃B1 − ν)n,m,
− kn,mb1 +
∑
u+1+v=n+m
(1⊗u ⊗ b1 ⊗ 1⊗v)kn,m + hn,mf1
= (k̃B1 + ι − h̃f)n,m.
Thus, if we introduce double coderivations h and k by their
components: hp,q = hp,q, kp,q = kp,q for p + q 6 t (using
just found maps if p + q = t) and 0 otherwise, then these
coderivations satisfy equations (5.11) and (5.12) for each p, q
such that p+q 6 t. Induction on t proves the proposition. �
5.3. Theorem. Every unital A∞-category admits a weak unit.
Proof. The proof follows from Propositions 5.1 and 5.2. �
6. Summary
We have proved that the definitions of unital A∞-category
given by Lyubashenko, by Kontsevich and Soibelman, and by
Fukaya are equivalent.
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