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