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].
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