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as - modular operads & GT
① Modular operads & category of graphs
Open problems/exercises
② A-modular ephod cperads
+ related objects
openads problems /exercises
③ GT + related groups
Lecture 1: Graphs and Modular
Operads
joint work : Philip Hackney/Donald Yau
Cyclic Operads
A C-coloured cyclic operad is an algebraic structure consisting of:
- an involutive set of colours C;
- for each c1, . . . , cn 2 C a ⌃n-set P(c1, . . . , cn);
- a family of equivariant, associative and unital composition operations
P(c1, . . . , cn)⇥P(d1, . . . , dm) �! P(c1, . . . , ci , . . . , d1, . . . dj , . . . , dm),
when ci = dj .
1
i :&→ e ie -- id
¥ ,
44"
oii
CzG
043¥ →¥0T,?dg
Modular Operads
A C-coloured modular operad is a cyclic operad which also has
- a family of equivariant contraction operations
P(c1, . . . , cn) �! P(c1, . . . , ci , . . . , cj , . . . cn),
when ci = cj .
satisfying some axioms.
2
¥a/ Icy→
"
✓
Example: A modular operad of surfaces
3
÷. ¥0 → 8¥18¥⇒* Eas
Eff - >4.a
{a. 4
Exercise: ⇤-autonomous categories
⇤-autonomous categories are closed symmetric monoidal categories with aglobal dualizing object so that (a†)† ⇠= a. Show that all strict⇤-autonomous categories are examples of cyclic operads. (See Example2.2 in D-C–H).
4
✗ Drummond - Cole -Hackney"
Dwyer - Kan". .
Goal for Today:
There is an equivalence of categories:
ModOp SetUop
Segal.N
⇠=
5
a-connected
undirected-
graphs
10%00-7edgesw/no
vertices
Graphs:
A graph G is a diagram of finite sets:
A D Vis t
- i is a free involution;
- s is a monomorphism.
6
✗vertices
half-
edges→~
Thialf - edges # →fthat touch
auertex t:A→V
I 10
v={ u ,w} D= {2.9. 3,415,6
2,97 }
A- ={ 1 , . . .jo }3
i (2) =L
i(3) =4v1
g
w 6
7
8
Graphs: Edges and Internal Edges
The involution on half-edges i : a 7! a† determines the edges of a graph.
- An edge is just an i-orbit [a, a†].
- An internal edge is an edge of the form [b, b†] where both b and b†
are in D.
7
e =[3,4]
f-- [5,6]
3
e 4g f6
Special Graphs, Boundaries and Neighbourhoods
Figure 1: The exceptional edge l and the 4-star ?4.
Definition
- The boundary of a graph: @(G ) = A \ D.
- The neighbourhood of v 2 V (G ): nb(v) = t�1(v) ✓ D.
8
at A={a ,a±} @1*+1={15%5,4}D= it
v=0 at,
--
,
,
@ (1) ={a .at} if It✓ 1
1 41 5 /
yt\'- --
'
A zt
A loop with 2 nodes and a nodeless loop
9
•(g)
" $A = { a.
at}
% :&v
i. ✗w
Exercise
Draw a graph G with A = {1, 2, 3, 4, 5, 6, 7, 8}, D = {1, 2, 3, 4, 5, 6, 7, 8},V = {v1, v2, v3, v4} and i(n) = n � 1 for n = 2, 4, 6, 8.
10
Hint @(G) =p
Exercise: The star of a vertex and star of a graph
Definition
- The star of a graph FG is the one-vertex graph with A = @(G ) t@(G )†, D = @(G )† and @(FG ) = A \ D = @(G ). In other words,FG = F@(G).
- If v is a vertex in G , the star of a vertex Fv is the one-vertex graphwith A = nb(v) t nb(v)† and @(Fv ) = nb(v)†.
Exercise
For the graph G drawn above, write down FG and Fv for eachv 2 V (G ).
11
-
•
Morphisms of Graphs
Graphs are diagrams in FinSet in the shape of
I := • • •is t
Definition
A natural transformation f : G ! G0 is called an embedding if
A D V
A0
D0
V0
i
f
s
f
t
f
i0 s
0t0
- the right-hand square is a pullback;
- V ! V0 a monomorphism.
12
OKU) = connectedgraphs .
c- Fin self
fr f.
There is a class of vertex embeddings
Fv ,! G
for every v 2 V (G ):
Example
Explicitly:nb(v) t nb(v)† nb(v) {v}
A D V .
s t
s0
t
13
*- ¥¥=o¥-
There is a class of vertex embeddings
Fv ,! G
for every v 2 V (G ):
Example
Explicitly:nb(v) t nb(v)† nb(v) {v}
A D V .
s t
s0
t
13
e e
Embeddings are not necessarily injective on half-edges.
There is a natural embedding
Fn ,! ⇠Fn
which is not injective on half-edges.
14
5 qt
,
,
-
1 vs Fijii :>2 112
,
'
-
g-+
zt It
It m3zt↳ I
Graphical Maps
Definition
A graphical map ' : G ! G0 consists of:
- a map of involutive sets '0 : A ! A0;
- a function '1 : V ! Emb(G 0) satisfying the following conditions:- The embeddings '1(v) have no overlapping vertices- For each v , the diagram:
nb(v) A
@('1(v)) A0
⇠=
i
'0
commutes.- If @(G ) = ;, then there exists a v in V so that '1(v) 6= l.
15
v1→ HuÉ G
'
I
otghff.ph
{
v0 dl Hutiii.
Graphical Maps
16
E) *¥
iii.q G
'
'
e. -
'
G- rj:, ,
-
-
-
÷
,
1 !LiHu¥1s ,
'
i-- -
-
-
-
-
either embeddings
graph substitution
Inner coface maps
An inner coface map dv : G ! G0 is a graphical map defined by
“blowing-up” a single vertex v in G by a graph (dv )1 which has preciselyone internal edge.
17
every graphical morphism isa composite of some elementary
maps
,
-
-
-. (d) =
V
'
n.
? I G€
Outer coface maps
An outer coface map is either:
- an embedding de : G ! G0 in which G
0 has precisely one moreinternal edge than G or
- an embedding l! Fn.
18
{ 1 , it }← 0 → 0
{ t.lt?..,n,nt3-Yi...n3-v
T ✓
.
"
w e
Codegeneracy maps
A codegeneracy map sv : G ! G0 is a graphical map defined by
“blowing-up” a vertex v in G by l.
19
0 -
③ ① * 0
The graphical category
The graphical category U is the category whose objects are connectedgraphs. The morphisms are the graphical maps.
20
The modular operad hG i generated by a graph G is the free modularoperad whose:
- set of colours is the set of half-edges A;
- a collection E (a1, . . . , an) =
({v} if (a1, . . . , an) = @(?v )
; otherwise.
- hG i = F (E )
21
A- ={a, .at . . . . }
ai-cai.at]
F- (at.az?ats.As,Au)a. Yas
as= { u}
anw 94
F- Cat ,aÉaI,a ;-) ={w}at
Proposition (Proposition 2.25 HRY2)
The assignment G 7! hG i defines a faithful functor U ! ModOp which isinjective on isomorphism classes of objects.
22
V1 V2 V
W,WZ W
(G) - < G'
>
vi. V2- ✓
nographicalmagno
'
Wl , Wz-W
Graphical Sets
The category of graphical sets is SetUop
.
- XG : evaluation of X at G 2 U.
- ' : G ! G0 ) '⇤ : XG 0 ! XG .
- The representable presheaf at G :
U[G ] := U(�;G )
U[G ]H := U(H,G )
for all graphs H.
23
✗ : u"'→ set
¥ .
C)A
-
PCC, , . .
.. ,a)XPCdi , →
dm)
→ P (c, , . . .Ñi , --Ñj - -
ci='
Segal Maps
Internal edges ) diagram of embeddings
Fv l Fw
in U.
Let
X1G= lim
Fv l!Fw
0
BBB@
XFvXFw
Xl
1
CCCA.
24
maps in set""
that modelmodular openedcomp .
Segal Maps
25
☒ 1-- e.e-
☒ D
vez
e, we s÷÷¥*,)
1- ✗*
✗ ✗*wk
Segal Maps
The embeddings Fv ,! G induce a Segal map
XG X1G✓
Qv2V (G)
XFv
which factors through X1G.
Exercise
In the case when Xl = ⇤, show that X 1G=
Qv2V (G)
XFv.
26
'
¥I
* ✗ ¥1.
Segal Maps
The embeddings Fv ,! G induce a Segal map
XG X1G✓
Qv2V (G)
XFv
which factors through X1G.
Exercise
In the case when Xl = ⇤, show that X 1G=
Qv2V (G)
XFv.
26
Segal graphical sets
A graphical set X 2 SetUop
is strictly Segal if the Segal map
XG X1G✓
Qv2V (G)
XFv
is a bijection for each G in U.
27
\o/*
1¥13← ¥. ✗ iT
compositionin modular opened
The Nerve
N : ModOp SetUop
NPG = ModOp(hG i ,P)
for any P 2 ModOp and any G 2 U.
NPG is “the set of P decorations of the graph G ”:
- NPl = ModOp(hli ,P) = C;
- NP?n= ModOp(h?ni ,P) = P(c1, . . . cn).
28
The Nerve
N : ModOp SetUop
NPG = ModOp(hG i ,P)
for any P 2 ModOp and any G 2 U.
NPG is “the set of P decorations of the graph G ”:
- NPl = ModOp(hli ,P) = C;
- NP?n= ModOp(h?ni ,P) = P(c1, . . . cn).
28
colors of P
?G:*:
The Nerve Theorem
Theorem (Theorem 3.6 HRY2)
The nerve functor is fully faithful. Moreover, the following statements areequivalent for X 2 SetU
op
.
- There exists a modular operad P and an isomorphism X ⇠= NP.
- X satisfies the strict Segal condition.
In other words:ModOp Set
Uop
Segal.N
⇠=
29
-for
some
modular
cperadp
* Gr Joyal - Kock
Sophie Raynor= has nerve
Webber nerve
The Nerve Theorem
Theorem (Theorem 3.6 HRY2)
The nerve functor is fully faithful. Moreover, the following statements areequivalent for X 2 SetU
op
.
- There exists a modular operad P and an isomorphism X ⇠= NP.
- X satisfies the strict Segal condition.
In other words:ModOp Set
Uop
Segal.N
⇠=
29
✗ relax this
weak Segal condition
Exercise: Graphs exhibit some strange behaviour
Given a graph G 2 U, we now have two ways to assign an object inSetU
op
to G :
- the representable presheaf U[G ],
- taking the nerve of the modular operad hG i, N hG i.
The representable U[G ] is a sub-object of N hG i (since J : U ! ModOpis faithful) but they nearly never coincide.
- Let G be the loop with one node and show U[G ] ⇢ N hG i.- Show that we have U[?0] = N h?0i.
30
U[G) µ = UCH ,G)
•
Further Directions
Earlier we defined the notion of (inner and outer) coface maps of U.Given a coface map � with codomain G , one can define the horn ⇤�[G ]
which is a sub-object of the representable object U[G ]. A strict inner
Kan graphical set is a presheaf X 2 SetsUop
such that every diagram
⇤�[G ] X
U[G ]
with � an inner coface map admits a unique filler.
31
Further Directions
Michelle Strumila shows in her PhD thesis that :
Theorem (Strumila)
The nerve functorN : ModOp SetU
op
is fully faithful. Moreover, the following statements are equivalent forX 2 SetU
op
.
There exists a modular operad P and an isomorphism X ⇠= NP.
X satisfies the strict Segal condition.
X is strict inner Kan.
If one relaxes the inner Kan condition you arrive at a model for quasi or1-modular operads. Following the example of dendroidal sets, onecould find a model category structure in which the weak inner Kangraphical sets are the fibrant objects. Because the graphical categoriesfor cyclic operads, wheeled properads, etc can all be derived from U thiswould simultaneously create models for many flavours of 1-“operads”. 32
÷
✗6' I ✗ (G)
oz"
I
f)e→G-
' set ⇒ ssetu"
YO l ' Tt
sSeth"
sset
9-Nod → *eat
Fyc* - autonomous categories ←usecompact closed categories Grace