118
6 Operads
Fig. 6.3 The geometric
realization of the sputnik Σ
a
b
c
d
f
e
g
h
i
•
•
its geometric realization. A tree is a graph T whose geometric realization |T | is
simply connected. 3
Example 6.12 (Taken from [3]) Consider the graph Σ with {a, b, . . . , i} as the set
of flags, the involution σ = (df )(eg) and the partition {a, b, c, d, e} ∪ {f, g, h, i}.
Its geometric realization |Σ| is the “sputnik” in Fig. 6.3.
Example 6.13 A S-corolla is the graph S with one vertex and Leg( S ) = S. Its
geometric realization is indeed the “corolla”
with the spikes indexed by S; whence the name and notation.
Graphs can be glued (grafted) together via their legs. Let Γ 1 be a graph with
Leg(Γ 1 ) = S 1 {a} and Γ 2 a graph with Leg(Γ 2 ) = S 1 {b}. We define the graph
Γ 1 a ◦ b Γ 2 by
Flag(Γ 1 a ◦ b Γ 2 ) := Flag(Γ 1 ) Flag(Γ 2 ).
The partition of Flag(Γ 1 a ◦ b Γ 2 ) is the union of the partitions of Flag(Γ 1 ) resp.
Flag(Γ 2 ), and the involution σ on Flag(Γ 1 a ◦ b Γ 2 ) agrees with the involution σ 1 of
Flag(Γ 1 ) on Flag(Γ 1 ) \ {a}, with the involution σ 2 of Flag(Γ 2 ) on Flag(Γ 2 ) \ {b},
and σ (a) := b.
Definition 6.9 We call Γ 1 a ◦ b Γ 2 the gluing or grafting of the graphs Γ 1 and Γ 2 .
In human language, Γ 1 a ◦ b Γ 2 is obtained by gluing the free end of the leg a to
the free end of b creating a new edge, symbolically:
Γ 1
Γ 2
a b
3 Meaning that |T | has no loops.
6 Operads
Fig. 6.3 The geometric
realization of the sputnik Σ
a
b
c
d
f
e
g
h
i
•
•
its geometric realization. A tree is a graph T whose geometric realization |T | is
simply connected. 3
Example 6.12 (Taken from [3]) Consider the graph Σ with {a, b, . . . , i} as the set
of flags, the involution σ = (df )(eg) and the partition {a, b, c, d, e} ∪ {f, g, h, i}.
Its geometric realization |Σ| is the “sputnik” in Fig. 6.3.
Example 6.13 A S-corolla is the graph S with one vertex and Leg( S ) = S. Its
geometric realization is indeed the “corolla”
with the spikes indexed by S; whence the name and notation.
Graphs can be glued (grafted) together via their legs. Let Γ 1 be a graph with
Leg(Γ 1 ) = S 1 {a} and Γ 2 a graph with Leg(Γ 2 ) = S 1 {b}. We define the graph
Γ 1 a ◦ b Γ 2 by
Flag(Γ 1 a ◦ b Γ 2 ) := Flag(Γ 1 ) Flag(Γ 2 ).
The partition of Flag(Γ 1 a ◦ b Γ 2 ) is the union of the partitions of Flag(Γ 1 ) resp.
Flag(Γ 2 ), and the involution σ on Flag(Γ 1 a ◦ b Γ 2 ) agrees with the involution σ 1 of
Flag(Γ 1 ) on Flag(Γ 1 ) \ {a}, with the involution σ 2 of Flag(Γ 2 ) on Flag(Γ 2 ) \ {b},
and σ (a) := b.
Definition 6.9 We call Γ 1 a ◦ b Γ 2 the gluing or grafting of the graphs Γ 1 and Γ 2 .
In human language, Γ 1 a ◦ b Γ 2 is obtained by gluing the free end of the leg a to
the free end of b creating a new edge, symbolically:
Γ 1
Γ 2
a b
3 Meaning that |T | has no loops.