II Mathematical Interpretation
93
differentials and the operations satisfy
d
α = (−1)
|α| α d
,
where d (resp. d ) is the induced differential on the product V
1 ⊗ · · · ⊗ V
s (resp.
on V
1 ⊗ · · · ⊗ V
t ).
For a graded vector space V =
p V p let ↑ V (resp. ↓ V ) denote the suspension
(resp. the desuspension) of V , i.e. the graded vector space defined by (↑ V ) p =
V p−1 (resp. (↓ V ) p = V p+1 ). We have the obvious natural maps ↑: V →↑ V and
↓: V →↓ V .
Notation for Categories
Set
The cartesian monoidal category of sets
Cor
The category of sets finite sets and their isomorphisms
Cor
The category of sets finite cyclically ordered sets and their isomorphisms
CycOp
The category of cyclic operads
CycMod The category of cyclic modules
CycOp
The category of non-Σ cyclic operads
CycMod The category of non-Σ cyclic modules
ModOp
The category of modular operads
ModMod The category of modular modules
dgVec
The category of dg-vector spaces
Chain
The category of dg-vector spaces and morphisms of degree 0
Grp
The category of graphs
Tre
The category of trees
Notation for Functors
Des : CycOp → CycOp
The desymmetrization
Sym : CycOp → CycOp
The symmetrization
: CycOp −→ CycMod
The forgetful functor
Mod : CycMod −→ CycOp The free cyclic operad functor
: ModOp −→ ModMod
The forgetful functor
Mod : ModMod −→ ModOp The free modular operad functor
: ModOp −→ CycOp
The forgetful functor
Mod : CycOp −→ ModOp
The modular completion functor
Reference
1. Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol.
5, 2nd edn. Springer, New York (1998)