ggeqt.FI IItaeA Imt Ker h add Ae prg A.TT B B B pmjAkes EAe3 - - PDF document

SLIDE 1

lk1k field

Htt

comod cat

cat of f cog

mod

cat

cat of f g mod

right comod

1k cat

a

add

1hlinear category

Ican.my

AEA

proj

A f d.lkalg

Short

exact sequence

• f

1k cat's

Vasserotivaragaolo

• B

B

G

B

O

Bi

i findary 1k cat's

F fully faithful

Gf

t

dense

ggeqt.FI IItaeA

Ker

h

Imt

B

B

B addAe prgA.TT

EAe3

pmjAkes

Recollement of abelian cat's

Fuses of ab.cat

g

l

c c

A

A

A

7

y

2

I

3

p r s t

Ai

abelian

q i p

L e r

adjoint triples

Is

are

fully faithful

Ker

e

Im

i

SLIDE 2

Fact

see for

example

Psaroudakis Vitoria

If

A

mod A

A fad only

then

recollement

is

equivalent to

Ake

AeAe

µ

dake

moat

man

e

standard

Hamal

Ake T

Homme eti

where

e

et EA

idem

in A

Fat

Rego i

Indian K cat

q l e

g

nod

Akey

mont

modele

c

page Ae pngA pnj

Ake

we will

dualiu

mod

Comal

Upgrade

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pro

j Mj

Thyle

Mtemiett

finitelymanyobj t Han's are finitary calls

e

nice

2

category

e HE ftp.III

f7nte.ota.afegjd.cat

s.es

• f

ztrepresentation

• f e

comaldDE corrodeI C

I comodelE

• 7MjE

Mj C

mjD e

co

algebra

F coalgebm object in the abelianisation ofthe morphism 1 off's

SLIDE 3

Note

Classical setting

is

when

E

Kmod

horizontalcompo

Th

vertical compo

compo of linearmaps

Intrinsic

description of

Dd E relative

to

C

Note

Unlike 1k

co algebra setting

there is

no

correspondence

between

idempotent and injective C comodules

ALGEBRA

COALGEBRA

From

tansy

la

c

eAeT

endomorphism ab

HaelQ

g

it

e D

Hai Yi

a

ftp.msn.subcoaiaota

To

Soc

Q ED

To

Amon I

comodC

We

can

also start from the

• ther side

E

ii

ice

102

I

4D

YD Ae

to

I

Y's.it'S

ete Hm_ QQ

SLIDE 4

Remarks

The

functors

are

C E

e

HeemLD 3

Heel

HELQ.CI 3

A

cHEl

corrode D corroded

cnn.detunCQ

ED

D Q

Hutton

iQ

Wh

Hom es

27cal

1M

2 representation

• ver

e

w

i internal Hom 7

no

bifunctural

nom

b

Homme

M FN EHomeci.pltnENiMJ F

Non trivial e.g

• f

nice

2 cat

su e.g

NOT

single obj

case

tensor category

e.g

Cat of

Soergel

bonodules

Su

e g Elias Williamson

2 art of

finite truncationof

upper low halfof

categoritoed

quantum gp Khorana Lauda

Rougnier