PREFACE ix

Conventions. Most of the algebraic objects will be considered over a fixed field

of characteristic zero, although some results remain true in arbitrary characteristic,

or even over the ring of integers. The symbol ⊗ will denote the tensor product over

and Lin(−, −) the space of -linear maps. By Span(S) we denote the -vector

space spanned by the set S. The arity of a multilinear map is the number of its

arguments. For instance, a bilinear map has arity two. We will denote by

X

or

simply by when X is understood, the identity endomorphism of an object X (set,

vector space, algebra, &c.). The symbol Sn will refer to the symmetric group on n

elements. We will observe the usual convention of calling a commutative associative

algebra simply a commutative algebra.

We denote by the ring of integers and by the set {1, 2, 3, . . .} of natural

numbers. The adjective “graded” will usually mean -graded, though we will also

consider non-negatively or non-positively graded objects; the actual meaning will

always be clear from the context. The degree of an homogeneous element a will be

denoted by deg(a) or by |a|.

The grading will sometimes be indicated by ∗ in sub- or superscript, the sim-

plicial and cosimplicial degrees by •. If we write v ∈ V

∗

for a graded vector space

V

∗,

we automatically assume that v is homogeneous, i.e. it belongs to a specific

component of V .

The abbreviation ‘dg’ will mean ‘differential graded.’ Since we decided to re-

quire that the Maurer-Cartan elements are placed in degree +1, our preferred degree

of differentials is +1. As a consequence, resolutions are non-positively graded.

An unpleasant feature of the graded word is the necessity to keep track of

complicated signs. We will always use the Koszul sign convention requiring that,

whenever we interchange two graded objects of degrees p and q, respectively, we

change the overall sign by

(−1)pq.

This rule however does not determine the signs

uniquely. For instance, in Remark 3.51 we explain that the signs in the definition

of strongly homotopy algebras depend on the preference for the inversion of the

tensor power of the suspension. We will use the sign convention determined by

requiring that

(i) all terms in the L∞-Maurer-Cartan equation come with the + sign,1 and

that

(ii) the intrinsic bracket (9.22) agrees, in the associative algebra case, with the

one of [Ger63].

Requirement (i) fixes the signs in L∞-algebras. Requirement (ii) introduces the

correction (−1)k+1 to formula (9.4) and affects the sign in (9.13). Since A∞-algebras

are Maurer-Cartan elements in the extended Gerstenhaber-Hochschild dg-Lie alge-

bra (3.24), (ii) in turn determines the convention for A∞-algebras. An unfortunate

but necessary consequence is the minus sign in the expression (9.15) for the curva-

ture and in the related formulas.

1This

convention is used in [Get09a].