Octoberfest 2015 Annual Meeting Ottawa, October 31-November 1 - - - PowerPoint PPT Presentation

SLIDE 1

Octoberfest 2015 Annual Meeting Ottawa, October 31-November 1

Stacks OF CATEGORICAL RINGS AND THEIR MORPHISMS ETTORE ALDROVANDI FLORIDA STATE UNIVERSITY

SLIDE 2

(Strictly )

Picard Categorical Rings a > ( Regular) Ann

• Categories

ftamoi) Crossed Bimodubs

( Presentations ) Moipbisms

( spans

c-

Butterflies) What

do categorical rings

form ?

Taxonomy

Classification

Non Regular Cat . Rings arxiv 1501.07592 ,

• r

Xiv 1501.04664 TAC 3-0 ( ' 15 )

SLIDE 3 Categorical

Ringo

stack in gwwpoids

fp

monphisms

• f

L a- site fibered categories 1g

/

Two mono idol structures to , @ : 8×8

• 8

( e , +0,0 , ) is group . like symmetric

(

= Picard )

( e

, ,

+0,9

, ID distributive @ : ex @

• E

Bimomoidal with respect to to Objects xi 1,2 ,w : ( YOY ) QZ = ( ×*2)o(y*D x*(z*w)=(x*Do(x*w)

SLIDE 4 Categorical

Ringo

stack in gwwpoids

fp

monphisms

• f

L a- site fibered categories 1g

/

Two momoiowl structures to , @ : 8×8

• f

( e , +0,0 , ) is group . like symmetric More about this in a

( 9×0

,

+0,9

, ID distributive moment @ : ex @

• E

Bimomoidal with respect to to Objects xi 1,2 ,w : ( YOY ) QZ = ( ×*2)o(y*D x*(z*w)=(x*Do(x*w)

SLIDE 5 About Group

• like / Categorical

Groups

(9+0,9)

admits

apuseutatiom

C , 2- Co

• 6
• crossed

module

• Presha .f|g

: Ulm > Th : Cdu ) XGCU ) 1

Cas

~

• Equivalence

:

[ COXC

, IG ] ~- C associated stock p Tk . T

( Folk

thin , E . A .

• B

. Naoki :

• g )

SLIDE 6 About Group

• like / Categorical

Groups

( e ,o

, g) admits

apuseutatiom

C , 2- Co.

⇐

gassed

module Tfttemhftreet

• Presha .f|g

: Ulm > Th : C.lu ) XC ,(U)1G(U )

• Equivalence

:

[

coxc ,

1G]~±D

associated stock

• {

, } : Coxco

• C ,

anti

• symmetric

"

SLIDE 7 About Group

• like / Categorical

Groups

(9+0,9)

admits

apuseutatiom

C , 2- Co.

⇐

Aroaossed

module

• Presha .f|g

: Ulm > Th : C.lu ) XGCU )1G(U ) ~

• Equivalence

:

[ COXC

, IG ] ~- C associated stock

• {

, } : Coxco

• C ,

ALTERNATING : { , }°Dq= eq ,

SLIDE 8 about Group

• like / Categorical

Groups Abelian Groups <

• ( e

,⇒,g , admits

ateusentation

§x÷x9sIu4

• Presha .f|g

: Ulm > Th : C.lu ) XGCU )1G(U ) ~

• Equivalence

:

[

Coxc , 16 ] ~- C associated stock "

• {

, } : Coxco

• C ,

ALTERNATING : { , }°Dq= eq , STAwDNGAgguMPT1°h.

SLIDE 9

( Back

to) Categorical Rings Definition C , 2- Co is a crossed bimoduce if ( i ) Co is a ring (

• f

Sha ) =L ) , with 1 , usually . C ii ) C ,

• Co

. bimodule CWD PFEIFFER : Ycnci E C± ( Jc , ) C , ' = c , ( dci )

SLIDE 10

( Back

to) Categorical Rings D_ef.tw#m_ C , 2- Co is a crossed bimodule if ( i ) Co is a ring (

• f

Sha ) =L ) , with 1 , usually . C ii ) C ,

• Co

. bimodule CWD PFEIFFER : Ycnci c- C± ( Jc , ) C , ' = c , ( dci ) theorem ( Ea is ) C :

Pstarrstcatyovud

Ring

• f

L

• F

Presentation

Gec

.

• 8

Cussed Bimodnleeff

SLIDE 11

Morphism

• rphism

F :C

• 2
• f strictly

Picard categorical rings D F :C

• B

: morphism

b/t

underlying strictly Picard stocks 2 E × 8 ¥ ° conditions ext to

%

IF

+

• n

A 2×2

• D

⇒

*

SLIDE 12

Morphism

• rphism

F : C

• 2
• f strictly

Picard categorical rings D F :C

• B

: morphism

b/t

underlying strictly Picard stocks

auth

By

Span (

in Ch+( sib ))

I

E

yv

Co

Bo

(DeCigme , SGA 4 , ND

SLIDE 13

Morphism

• rphism

F :C

• 2
• f strictly

Picard categorical rings D F :C

• B

: morphism

b/t

underlying strictly Picard stocks C ,

¥

' B ,

l§utteflyimCh+(

SI )

1¥

< E y v Co

Bo

SLIDE 14

Morphism

• rphism

F :C

• 2
• f strictly

Picard categorical rings D F :C

• B

: morphism

b/t

underlying strictly Picard stocks c ,

It ¥

B ,

l§utteflyimCh+( SID

Hal

v E v

colt JTB

SLIDE 15

Morphism

• rphism

F :C

• 2
• f strictly

Picard categorical rings D F :C

• B

: morphism

b/t

underlying strictly Picard stocks 2 E × 8 ¥ ° conditions ext to

%

IF

+

• n

A 2×2

• D

⇒

*

SLIDE 16 Monphisms we need the µ between wssedbimodT

Definition

• Cussed

f

Bimoohdhf

, Ring Extension C ,

By

D E T co ring homo . 2 B , Is E ( bilateral ) ideal | ¥

%

| @ B ,2=|o , ingenue v z E \ , ✓

• >

so momsimgular Co P ti B

SLIDE 17 Monphisms we need the µ between wssedbimodT

Definition

:

ihnotexaot Just :

j•kzo\

← Ring Extension 1 E dig C , B , E npsco ring homo . 2 B , Is E ( bilateral ) ideal ringbone | X ,% | @ B ,2=|o , ingenue v z E y ✓ → so monsimyular Co P ti B

SLIDE 18 Monphisms we need the µ between aossedbimodT

Definition

• cis

eilb , )=i(j( e) b.) lid ifbpe = i ( b , jle )) still not exact Just :

j•kzo\

← Ring Extension 1 E dig C , B , E npsco ring homo . 2 B , Is E ( bilateral ) ideal ringbone | X ,% | @ B ,2=|o , ingenue v z E y ✓ → so monsimyular Co P ti B

SLIDE 19 Monphisms we need the µ between aossedbim°dT

Definition

÷

eilb , )=i( jlelb . )

( iii )

ekk , )=k( place ) ( ii ifb , )e=i( b , .sk ))

in)

nlc ,

)e=k(

QPKD

still not exact Just :

j•kao\

← Ring Extension 1 E

#

C , 13 , Etc . ring homo . 2 B , Is E ( bilateral ) ideal ringbone | X ,% | @ B ,2=|o , ingenue v z E y ✓ → so momsimgular Co P ti B

SLIDE 20 Morphisms F moyohisms F :@

• B

form agrowpoid Ho±( GB )

:C

Tae

G ex e T e ex e T e txt |

% fF)⇒G

= FXF ( ⇒ )G×G

I

g 2×2-×oD 2×2

Mok

&

SLIDE 21 Morphism

• phisms

F :@

• B

form agrowpoid Ho±( GB )

:C

#@B

ex e T e ex e T e ext /

% fF)⇒G

=

FxFf⇒fG×G

I

g

2×2=02

2×2

Mok

&

so do

butterflies

:

S+(c

. ,BD C ,t.ec#Bi/

¥¥¥*

.

SLIDE 22

Morphis

heeled (

Ea , TAC 31 ( 2015 ) ) C , B :

Strictly

Picard Cat . Rings

• f

T

G

• C

.

• p

B ,

• B.
• 2

>

presentations

There is

• n

equivalence

• f gwupoids

Home ( e , B)

tsp

( c

. ,BD

SLIDE 23 Morphisms Thereof ( Ea , Tac 30-(2×51) C , B : Strictly Picard cat . Rings

• f

5 9

• co
• p

B ,

• B.
• 2

> presentations Pw# There is

• n

equivalence of gwupoids C , B , Hot ( QD ) tsspcc . ,B .)

"

• Bo

ti

• D

F

SLIDE 24 Morphisms Thereof ( Ea , Tac 30-(2×51) C , B : Strictly Picard cat . Rings

• f

T G

• co
• p

B ,

• B.
• 2

> presentations Pw# There is

• n

equivalence of gwupoids C , B , Hott , B) tsspcc . ,B .)

l

I%3h,l

Stack fiber product Co Bo El

)

fit

. ( cost ,bo ) ,

f-

A Icc . ) f- t( bo ) in D F

SLIDE 25 Morphisms Thereof ( Ea , Tac 30-(2×51) C , B : Strictly Picard Cat . Rings

• f

T G

• c.
• p

B ,

• B.
• 2

> presentations T.ro# Exact at . gwwps There is

• n

equivalence of gwupoids

ti

B , Hot ( 50 tsspcc . ,B .)

e

Coffin

, Co Bo

£

)

to 't . 8

• D

F

SLIDE 26 Morphisms Thereof ( Ea , TAC 30-(20151 ) C , B : Strictly Picard cat . Rings

• f

5 9

• co
• p

B ,

• B.
• 2

> presentations T.to#_ Exact cat . gwwps There is

• n

equivalence of gwupoids K

#

Exact C ,

yB

, Hom- ( QB ) tsp ( c . ,B .)

.

Coffin

, Co Bo

£

)

to 't . 8

• D

F

SLIDE 27 Morphisms Thereof ( Ea , Tac 30-(2×51) C , B : Strictly Picard Cat . Rings

• f

T G

• c.
• p

B ,

• B.
• 2

> presentations Pw# There is

• n

equivalence of gwupoids C , B , Hott , B) tsspcc . ,B .) i. Not exact ⇒of

coxed

, Co Bo

£¥k>

to 't . 8

• D

F

SLIDE 28

llpg newsprint

Bicatcgoy

XM¥l(

s ) * Objects : C ,

• C

. crossed bimodules * ( Groupoids ) Moyshisms : $ ( C . ,B . ) with composition Sp_ ( D . ,C . )×Sp_

( C

. ,B . )

• s#(D

. ,B . )

Q

Wm

Dog

SLIDE 29

llpg newsprint

Bicatcgoy

Xtdfs

) * Objects : C ,

• C

. crossed bimodules * ( Groupoids ) Moyshisms : $ ( C . ,B . ) with composition Sp_ ( D . ,C . )×Sp_

( C

. ,B . )

• s#(D

. ,B . )

M

Wi→D§

E ' E C , E 't

SLIDE 30 ftp.w#Gespomo6neBicatcgoYXM=d(D 2- Category Pica ) : Stuitlypicardstaokyg has @ :P ,c(I ) xp ,c( f) → pica ) (Deligme ,sGa4 , Xvi " )

2RmgG )

: Imomoid

• bjects

in Prt )

SLIDE 31

llpg radiophone

Bicatcgoy

XII

) 2- Category Pica ) : Striotlypicardstaokyg has @ :P ,c(I ) xp ,c( f) → pica ) (Deligme ,sGa4 , Xvi " )

2RmgG )

: Lmomoid

• bjects

in Prt ) = Ourstricteypiaerd Categorical rings

SLIDE 32

llpg needs pomo6=

Bicatcgoy

XM¥l(S

) 2- Category Pic (s ) : Strictly Picard stocky has @ :P , c (f) xp , a (f) → Pic G) (Deligme , saa 4 , Xvi " ) 2 Ring (f ) : Imomoid

• bjects

in Pic ( S ) = Our Strictly Picard Categorical rings

Theory (

EA , ibid .) There is a bieqnivalemu 11M¥ (f) A 2 Rings (8) C ,

• co

17 [ Cox C , →→Co] "

SLIDE 33 Shukla ,BarrBeck , Andrei . Quillen Back to the Crossed bimoolule C , } Crossed extension

O#MsC

, → G→A→0 tltbimodule

• →M
• C

,

• Co
• A→o

up Equivalence : 11 E 11

• n
• →M→d

,

• to
• n

SLIDE 34 Shukla , Barr . Beck , Andrei

• Quillen

Back to the Crossed bimodule C , Isc . Crossed extension

O#MsC

,

• G→A→0

th . bimodule

• →M
• C

,

• Co

→ A→o is a Equivalence : H E 11

• n
• →M→d

,

• to
• n
• Wellknowm

:

XE×t(^,n)±SH3C^,M

)

~££j :c

,Aq

Depending

• nly
• n

@=[

GXC ,=G]~

SLIDE 35 Non strict Picard = Non regular cat . rings C , 2- c.

• e

⇐

Isle crossed module Q : f × 8

• P

momoiohd for to in both variables

thereof (

Ea ' is )

(

Gay ,qIe ) equivalent to a bi extension C , × Cn

*

cote

.FI

Co

SLIDE 36 Non strict Picard = Non regular cat . rings

• →

M

• C

, 2- Co

• +

kbi

module ring Q : Ex 8

• P

momoiohd for to in both variables

theory

(

Ea ' is )

(

Gay ,qIe ) equivalent to a bi extension C , ×

Ce

{

c }E*T§ 4×0×1n%alI×E*nE*

SLIDE 37 Non strict Picard = Non regular cat . rings

• →

M

• C

, 2- Co

• +

Xbi

module ring Q : 8×8

• P

momoiohd for to in both variables

theory

(

Ea ' is )

(

Gay ,qIe ) equivalent to a bi extension

Coe

• !c}E*T§

4=0×4 n %=lI×E*h%

N

[ Eo ] E HMP ( ^ ,M ) Mac Lane

SLIDE 38

THANK

You

!