6.1 Cyclic Operads
119
Graphs form a category. Intuitively, a morphism f : Γ 0 → Γ 1 of graphs is given
by a permutation of vertices, followed by a contraction of some edges of the graph
Γ 0 , leaving the legs untouched. Translated into the language of Definition 6.8, this
means an injection f ∗ : Flag(Γ 1 ) → Flag(Γ 0 ) that commutes with the involutions.
Moreover, the involution σ 0 of Γ 0 must act freely on the complement of the image
of f ∗ in Flag(Γ 0 ) (i.e., the legs of the graphs are preserved by the map f ) and two
flags a and b in Γ 1 meet either if f ∗ (a) and f ∗ (b) meet in Γ 0 or there is a chain of
edges in Γ 0 from a to b.
A morphism f : Γ 0 → Γ 1 clearly defines a surjective cellular map |f | : |Γ 0 | →
|Γ 1 | of geometric realizations such that the induced map Leg(f ) : Leg(Γ 0 ) →
Leg(Γ 1 ) of legs is bijective. We will denote by Grp the category of graphs and their
morphisms.
Example 6.14 There is a special class of morphisms which are given by contracting
a subset of edges, without permuting the vertices. First of all, for any subset I
of Edg(Γ ), there is a unique graph Γ /I such that Flag(Γ /I ) is obtained from
Flag(Γ ) by deleting the flags constituting the edges in I and combining blocks
of the partition that contain flags connected by a chain in I . Then the inclusion
Flag(Γ /I ) )→ Flag(Γ ) is a morphism of graphs, which we denote by π I : Γ →
Γ /I. An important special case is when I consists of a single edge e. We then
simplify our notation by writing Γ /e instead of Γ /{e} and π e instead of π {e} .
The graph Γ /I introduced in Example 6.14 is called the contraction of Γ along
the set of edges I . Any morphism f : Γ 0 → Γ 1 of graphs is isomorphic to a
morphism of this form. This means that there exists a subset I ⊂ Edg(Γ 0 ) and
an isomorphism φ : Γ 0 /I → Γ 1 such that the following diagram of graph maps
commutes:
Γ 0
Γ 1
Γ 0 /I
π I
f
φ
Example 6.15 The category Grp of graphs has two important full subcategories.
The first one is the category of corollas, easily seen to be isomorphic to the category
Cor introduced at the beginning of this section. Another important subcategory is
the category Tre of trees consisting of graphs with simply connected geometric
realizations. Notice that the only automorphism of a tree T ∈ Tre fixing the legs is
the identity.
Definition 6.10 A cyclic module is a covariant functor E : Cor → Chain. A
morphism Ψ : E → F of cyclic modules is a natural transformation from the
functor E to the functor F .
119
Graphs form a category. Intuitively, a morphism f : Γ 0 → Γ 1 of graphs is given
by a permutation of vertices, followed by a contraction of some edges of the graph
Γ 0 , leaving the legs untouched. Translated into the language of Definition 6.8, this
means an injection f ∗ : Flag(Γ 1 ) → Flag(Γ 0 ) that commutes with the involutions.
Moreover, the involution σ 0 of Γ 0 must act freely on the complement of the image
of f ∗ in Flag(Γ 0 ) (i.e., the legs of the graphs are preserved by the map f ) and two
flags a and b in Γ 1 meet either if f ∗ (a) and f ∗ (b) meet in Γ 0 or there is a chain of
edges in Γ 0 from a to b.
A morphism f : Γ 0 → Γ 1 clearly defines a surjective cellular map |f | : |Γ 0 | →
|Γ 1 | of geometric realizations such that the induced map Leg(f ) : Leg(Γ 0 ) →
Leg(Γ 1 ) of legs is bijective. We will denote by Grp the category of graphs and their
morphisms.
Example 6.14 There is a special class of morphisms which are given by contracting
a subset of edges, without permuting the vertices. First of all, for any subset I
of Edg(Γ ), there is a unique graph Γ /I such that Flag(Γ /I ) is obtained from
Flag(Γ ) by deleting the flags constituting the edges in I and combining blocks
of the partition that contain flags connected by a chain in I . Then the inclusion
Flag(Γ /I ) )→ Flag(Γ ) is a morphism of graphs, which we denote by π I : Γ →
Γ /I. An important special case is when I consists of a single edge e. We then
simplify our notation by writing Γ /e instead of Γ /{e} and π e instead of π {e} .
The graph Γ /I introduced in Example 6.14 is called the contraction of Γ along
the set of edges I . Any morphism f : Γ 0 → Γ 1 of graphs is isomorphic to a
morphism of this form. This means that there exists a subset I ⊂ Edg(Γ 0 ) and
an isomorphism φ : Γ 0 /I → Γ 1 such that the following diagram of graph maps
commutes:
Γ 0
Γ 1
Γ 0 /I
π I
f
φ
Example 6.15 The category Grp of graphs has two important full subcategories.
The first one is the category of corollas, easily seen to be isomorphic to the category
Cor introduced at the beginning of this section. Another important subcategory is
the category Tre of trees consisting of graphs with simply connected geometric
realizations. Notice that the only automorphism of a tree T ∈ Tre fixing the legs is
the identity.
Definition 6.10 A cyclic module is a covariant functor E : Cor → Chain. A
morphism Ψ : E → F of cyclic modules is a natural transformation from the
functor E to the functor F .