6.4 Modular Operads
139
surfaces with labeled marked points on the boundary. Stability means that we
exclude surfaces of genus 0 with either one boundary component and less that three
marked points, or two boundary components and no marked point. We also assume
that all surfaces have at least one boundary component.
To define the operadic composition, it is convenient to replace each marked point
by an interval embedded in the boundary and then glue one edge of a short strip to
the interval. The edge of the strip opposite to the one glued to the interval is called
an open end (of an open string). The surface(s) can be glued along the open ends
where we allow only gluing resulting in orientable surfaces.
It is clear that the homeomorphism class of such a surface is determined by its
genus, by the number of boundary components, and by the cyclically ordered sets of
open ends at each boundary component. The modular operad QO therefore admits
a purely combinatorial description given in the rest of this example. We will need
the following notion.
Definition 6.19 A cycle in a set S is an equivalence class ( (x 1 , . . . , x n )) of an n-tuple
(x 1 , . . . , x n ) of several distinct elements of S under the equivalence
(x 1 , . . . , x n ) ∼ τ (x 1 , . . . , x n ),
where τ ∈ Σ n is the cyclic permutation given by τ (i) := i + 1 for 1 ≤ i ≤ n − 1,
and τ (n) := 1. In other words,
((x 1 , . . . , x n ) ) = · · · = ( (x n−i+1 , . . . , x n , x 1 , . . . , x n−i ) ) = · · · = ((x 2 , . . . , x n , x 1 )) .
We call n the length of the cycle.
We also admit the empty cycle (() ), which is a cycle in any set. For a bijection
ρ : S
∼ =
− → T and a cycle ( (x 1 , . . . , x n )) in S, define an induced cycle in T by
ρ ( (x 1 , . . . , x n )) := ((ρ(x 1 ), . . . , ρ(x n )) ) .
We are going to introduce a stable modular operad
QO =
QO(S; G) ∈ Set | (S, G) ∈ S
in the category of sets which will be the combinatorial model of the stable part of
the operad QO. 10 Since the operadic genus of elements of QO does not coincide
with the geometric genus of the surface it represents, we denoted it in this particular
example by the capital G instead of g that we used for the operadic genus before.
10 The set S of stable pairs was defined in (6.60).
139
surfaces with labeled marked points on the boundary. Stability means that we
exclude surfaces of genus 0 with either one boundary component and less that three
marked points, or two boundary components and no marked point. We also assume
that all surfaces have at least one boundary component.
To define the operadic composition, it is convenient to replace each marked point
by an interval embedded in the boundary and then glue one edge of a short strip to
the interval. The edge of the strip opposite to the one glued to the interval is called
an open end (of an open string). The surface(s) can be glued along the open ends
where we allow only gluing resulting in orientable surfaces.
It is clear that the homeomorphism class of such a surface is determined by its
genus, by the number of boundary components, and by the cyclically ordered sets of
open ends at each boundary component. The modular operad QO therefore admits
a purely combinatorial description given in the rest of this example. We will need
the following notion.
Definition 6.19 A cycle in a set S is an equivalence class ( (x 1 , . . . , x n )) of an n-tuple
(x 1 , . . . , x n ) of several distinct elements of S under the equivalence
(x 1 , . . . , x n ) ∼ τ (x 1 , . . . , x n ),
where τ ∈ Σ n is the cyclic permutation given by τ (i) := i + 1 for 1 ≤ i ≤ n − 1,
and τ (n) := 1. In other words,
((x 1 , . . . , x n ) ) = · · · = ( (x n−i+1 , . . . , x n , x 1 , . . . , x n−i ) ) = · · · = ((x 2 , . . . , x n , x 1 )) .
We call n the length of the cycle.
We also admit the empty cycle (() ), which is a cycle in any set. For a bijection
ρ : S
∼ =
− → T and a cycle ( (x 1 , . . . , x n )) in S, define an induced cycle in T by
ρ ( (x 1 , . . . , x n )) := ((ρ(x 1 ), . . . , ρ(x n )) ) .
We are going to introduce a stable modular operad
QO =
QO(S; G) ∈ Set | (S, G) ∈ S
in the category of sets which will be the combinatorial model of the stable part of
the operad QO. 10 Since the operadic genus of elements of QO does not coincide
with the geometric genus of the surface it represents, we denoted it in this particular
example by the capital G instead of g that we used for the operadic genus before.
10 The set S of stable pairs was defined in (6.60).