Higher dimensional algebras

Scott Morrison joint with Kevin Walker NCGOA May 32019

(Kevin Walker) understand Our goal is to

modules for higher categories

in the semisimple particularly pivotal / )unitary settings.

mathematical formulation of This is the studying

domain walls for topological phases : Low - dimensional cases

vector consists of • An algebra acting on a space

→ V A ( ) . an algebra map

A is V is in decomposable When semisimple he and , ,

there is a PEA =p projection , E

V= action is multiplication . so that Ap and the just by

- consists - on a I of • A ① acting category

→ M End . - E ( ) a ⑦

When e is a and M is in decomposable firegory . ,

- m : A -0A → A K there is a special Frobenius Aee , Lt algebra ,

"

- that A . ( theorem so ME emod Ostrik's ) ① =/ • V on a E A 3- category acting 2- category V → { e }

' e - se V - { 2- }

natural transformations} V - { pseudo

modifications Is - { }

" ' . V - E 3 V -1.3) , braided lie . , When V is a category

we is a ① - and e category,

2- functor e → e - only see the identity exactly natural transformations of the identity are - pseudo

the Drin feld centre : Zk objectsdatainternaltoV-ahiohdassih.esof D ) the What is such modules? " "

For n - e we define its ) completion E pivotal , (algebra any category with the following properties :

⇐ Idem • For n -1 E (e) ,

-0 - • C a For n=2 category , , Frobenius (the 2- category of algebras, E ⇐ Alyce ) b. modules and intertwines ) ,

a functor → Morita • There is b :C E a equivalence , inducing

and terminal such ( IE. c) is amongst pains )

Morita and ⇒ e and D are equivalent At E ⑤ are

functionally equivalent.

' ' • For E an sufficientlysemisimple equivalence E Mod (e) . of Ostrik 's theorem The inverse provides a generalisation all in dimensions . full ) stratification ? is . . ( What . I string diagram)

stratification - manifold W is A of an n a sequence

- - - = 2 W 2 2 W ,

is submanifold of W so WoWna dcodimension k . Wfwk+, stratification if n It is a string

W - -2 has a nbhd in n x E Whlwn every ,

of its nbhd in which is a product Wn stratification and the code over a string diagram n =3

- some n - D - of ( k sphere . * if : Equivalently .

- stratification • the ball with its it is O unique

• it is W x I for some W with an SD stratification

• where it is cone w W is an SD stratification of a sphere diagram( ) .

• is Where W have stratification it w , and Wz SD or , wwz , ,

stratification • it to itself has an SD is W along Y where W , glued ,

Y the same stratification and both restrictions to give .

' [ Sieben mann 72 ! ] It is a full stratification if

"

- of WWW is a In k - ball component ) , every

and such component 2W does so in exactly one any meeting

- - - ball . ( n k D Example : but④not if : Equivalently .

- stratification • the ball with its it is O unique

• where it is cone w W is a full stratification of a sphere o.itisw-IforsomewwithanSDstrahf.ca/noX( ) .

• is where W have full stratification it w , and Wz or , wwz , ,

stratification • it to itself has a full is W along Y where W , glued ,

Y the same stratification and both restrictions to give . - - - m dimensional E is an n E diagram is If category , string

"

- an SD stratified ball W with each - k ball in , ) www.i ,

k - labelled by some of E .

definition of a n - let Any reasonable (pivotal) category you

a e - to an m - evaluate stringanImdiagram morphism ,

same evaluation . E - have the so isotopic string diagrams

eti¥¥=

this evaluation . - axiomatises definition of an n just map ) Indeed a minimal category ( , - E The O of are the dgebrasine .

of : b a a . . . . . consists . . . in E ( . a , an ,b , An . .az algebra . , ,

a - in - choice of do O morphism E a .

- t a choice of a morphism E. ,

- a k - S with II string diagram for every sphere ,

stratification labelled by morphisms . for q jsk ,

a choice of a htt morphism with S an ,s , boundary

- - each n for ans ksn an morphism bns with boundary , ,

" - h - @k.sxBn-k-7UCau.sx B Y

' '

in a moment - satisfying the egalitarian axiom dobe specified ) Examples

: I - e consists of TAN in a category algebra

satisfying conditions Ao ( some ) , ,

consists of • An algebra in a 2- category

o

. . a ...... , a •az a . a , ④ ,

( conditions isatisfying some ) labelled bn B a an Say string diagram by . replete

• all the bn the stratification is full , if ignoring , full stratification • each codimension k cell in that

of : satisfies exactly one

in a codimension ktl cell - the cell meets the boundary single

- the all contains a single bn .

' ' The egalitarian axiom :

with the same boundary two replete string diagrams any

are equal . Examples . ¥9 : and a a do . nd a. .

to the same amorphism . and hence are replete, required In fact this relation generates everything by

the egalitarian axiom . valuatenr : i÷==⑤¥ of relations These two classes generate

everythinge.. Highers ' i - ball X la j ) m : E of shape A j - morphism

on 2X Im which is a string diagram has a boundary '

' a on : E e- . -9 labelled of . by jsj morphisms

where

- . . . . Mn . . ( min , bi , , The data of m is a tuple Mj , , with - ball Y - I , is a choice for each me , j I hell labelled the . for 'Ll 24 containing a string diagram by me ,

of Xxx me , :@ f )

- - " - t ' " e ' with x B - boundary ( me )ufmµxB an n , - is morphism ) bei,

the ' ' - all satisfying b.)same egalitarian axiom . ° a -1 Cao Cad ' .¥ n . b.) a :b : , an , , , ) Examples Say , @ .

. = :D t.ba = E' l = Me ca . a :b { ÷ / ⇒

theorem E E Idem (e) via

is an idempotent) recall ab , → Cao a. b.) ( ( , b.) , aaa ,

: morphism in Idem ( e ) Cthis is a m b ! ( m ) ↳ ,

= ¥9.9 ) =

X id ← X. ) , ) ( p ( , p

f- ← f to check that Proof the only interesting thing is E Cao a do d. b. ida is an isomorphism in . ,b ( , , . ) ,

" " the completion . E is a puffier version of usual; idempotent (special Frobenius algebras) Examples n -1 : EE Alge

= = . : → m A -0A . ( b. b.) → A A ao.a.dz , , ( image? , , * , toga O : A- → A -0A = ¥ ) to

the : can verify relations we = = µ ¥ . µ =L

. - &= b : ¥ . boy equivalenceME

be e a - - E - of morphism of using a k morphism k gives ,

- - x k iterated identities an for > . , Aj j

- is a bindle with a Monta equivalence END category isomorphisms

⇐ ⇐

. O . i

need to be able to rewrite E as - we diagrams e diagrams

example ' is $4 M " stratification refining the to be removed choose an SD I . in region

label the in the E and it with underlying morphisms C . diagram , ? are these isomorphisms

. → →

÷i a "

dose to the identity on the nose : is very stratification the e we're refined the . in region just with labelling all new strata identity morphisms

•

→

relies on the - which mixed E in e retracts Lenny In a e diagram region

to he can test equality boundary , you by converting to full everything e diagrams . The inclusion :C → is terminal L E . amongst D f- Er Morita : functors E :C → D inducing a equivalence §qµ

an we can construct algebra - x in D : a O morphism Sketch given , Morita : O - morphism of E.) the isomorphisms ( i.e . a using

a :@ = for some ④ .

' for some :C = a ④ = .

' = = for some ④ a¥. a :c

The handle cancellation laws for Morita isomorphisms

imply the egalitarian axiom . APM.ca#mmpsi.ebilagen → B) : n VTA µ MENE

- B = - mod = mod where µ A , , N

- in D A. B 3 algebras .

" : - G a - labelled Example D= ( soap films)

- µ - Vec kith trivial a action)

- ⑦ - with a N a category acting by ⑦ automorphisms

Vec N E N ? the salmon -0A, , equivariant

-

- - example V RepG

E - Vec the de equivariantisahon . GaN Nda .

salmon dimensional analogues of (d) equivariant ? • . . higher : - dimensional in Vec are 3 algebras fusion categories

→ Funtime )

HELI → * eel # → HH

) us a ANY #

- T # y

pentagon ea.hn I → of " splitting ehe-szuwt-%fa-sxvwl.ec#-.yzv→ D II - lditi.me f→ da ' ① " t.TT • ¥a¥x→D ¥=¥-elated braided 3- a tensor algebras in V , More generally, , category

- fusion enriched . are N categories