Stacks CATEGORICAL AND RINGS THEIR MORPHISMS OF ETTORE ALDROVANDI FLORIDA UNIVERSITY STATE Octoberfest 2015 Annual Meeting Ottawa, October 31-November 1
- Categories Ann ftamoi ) ( Regular ) > Picard Categorical Rings a ( Strictly ) ( Presentations ) Bimodubs Crossed Butterflies ) ( spans c- Moipbisms form ? What do categorical rings Classification Taxonomy Non Regular Cat . Rings arxiv Xiv 1501.07592 or 1501.04664 , ' TAC 15 ) 3-0 (
Categorical in gwwpoids Ringo stack fp monphisms of fibered categories 1g L site a- / 8 8 × 8 Two idol structures to @ : - mono , = Picard ) ( like ( e +0,0 , ) is group . symmetric , distributive ( e 0 , ID +0,9 , , E ex @ @ - : with respect to Bimomoidal to 1,2 ,w : ( × *2)o(y*D Objects xi QZ = ( YOY ) x*(z*w)=(x*Do(x*w)
Categorical in gwwpoids Ringo stack fp monphisms of fibered categories 1g L site a- / f 8 × 8 Two momoiowl structures to @ : - , about More like ( e +0,0 , ) is group . symmetric , this in a moment distributive ( 9 × 0 , ID +0,9 , E ex @ @ - : with respect to Bimomoidal to 1,2 ,w : ( × *2)o(y*D Objects xi QZ = ( YOY ) x*(z*w)=(x*Do(x*w)
- like / Categorical About Group Groups 2- 6 C Co - admits apuseutatiom (9+0,9) , - module crossed Presha .f|g > Th 1 Ulm Cdu ) XGCU ) : • : ~ • [ COXC IG ] C ~- : Equivalence , p Tk Cas associated stock . T thin ( Folk B Naoki og ) E A - : . . . ,
⇐ - like / Categorical About Group Groups 2- Co. C admits apuseutatiom ( e ,o , g) , module Tfttemhftreet Presha .f|g > Th Ulm C.lu ) ,(U)1G(U ) : XC • : 1G]~±D gassed • [ : coxc Equivalence , associated stock " - symmetric anti C , { } Coxco : - • ,
⇐ - like / Categorical About Group Groups 2- Co. C admits apuseutatiom (9+0,9) , module Presha .f|g > Th Ulm C.lu ) XGCU )1G(U ) : • : Aroaossed ~ • [ COXC IG ] C ~- : Equivalence , associated stock ALTERNATING : { C , { } Coxco }°Dq= : - , • eq , ,
- like / Categorical about Group Groups Abelian Groups < - admits ateusentation ( e , ⇒ ,g , Presha .f|g §x÷x9sIu4 > Th Ulm C.lu ) XGCU )1G(U ) : • : ~ • [ 16 ] C ~- : Coxc Equivalence , " associated stock ALTERNATING : { C , { } Coxco }°Dq= : - , • eq , , STAwDNGAgguMPT1°h.
( Back to ) Categorical Rings Definition 2- bimoduce C is crossed if Co a , with ) =L ) ( i ) ring ( Sha 1 , usually Co is of a . , bimodule C ii ) C , Co . - ' E C± PFEIFFER ( Jc CWD Ycnci ( dci ) : , ) C c = , ,
( Back to ) Categorical Rings D_ef.tw#m_ 2- bimodule C is crossed if Co a , with ) =L ) ( i ) ring ( Sha 1 , usually Co is of a . , bimodule C ii ) C , Co . - ' c- C± PFEIFFER ( Jc CWD Ycnci ( dci ) : , ) C c = , , is ) - theorem ( C Pstarrstcatyovud Ea : of L Ring Presentation -8 F Gec . Cussed Bimodnleeff
⇒ Morphism Picard -2 F of strictly orphism :C categorical rings : morphism underlying F B b/t :C D - stocks Picard strictly � 2 � ¥ E ° × 8 conditions IF + ext to % on A 2 × 2 D - *
Morphism Picard 2 F of strictly orphism : C - categorical rings : morphism underlying F B b/t :C D - stocks Picard strictly By in ( sib ) ) auth Span Ch+( I E yv Co Bo , ND (DeCigme SGA 4 ,
Morphism Picard -2 F of strictly orphism :C categorical rings : morphism underlying F B b/t :C D - stocks Picard strictly l§utteflyimCh+( ' SI ) ¥ C B , , 1¥ E v y < Co Bo
Morphism Picard -2 F of strictly orphism :C categorical rings : morphism underlying F B b/t :C D - stocks Picard strictly l§utteflyimCh+( It ¥ SID c B colt JTB Hal , , E v v 0
⇒ Morphism Picard -2 F of strictly orphism :C categorical rings : morphism underlying F B b/t :C D - stocks Picard strictly � 2 � ¥ E ° × 8 conditions IF + ext to % on A 2 × 2 D - *
Monphisms between the we need wssedbimodT µ Definition - Cussed Bimoohdhf Extension f Ring , homo T co E D ring . C By , , Is bilateral ideal E � 2 � B ( ) | ¥ | B ,2=|o % @ , ingenue E ✓ v so momsimgular - > \ z P , ti B Co 0
Monphisms between the we need wssedbimodT µ Definition : ihnotexaot j•kzo\ Extension Ring Just : ← homo npsco E 0 ring . C B E dig � 1 � , , , Is bilateral ideal E � 2 � B ( ) | X ringbone | , % B ,2=|o @ , ingenue E ✓ v → so monsimyular z y P ti B Co 0
Monphisms between the we need aossedbimodT µ Definition - e) b.) )=i(j( cis eilb , i ( b jle ifbpe ) ) lid = , exact not still j•kzo\ Extension Ring Just : ← homo npsco E 0 ring . C B E dig � 1 � , , , Is bilateral ideal E � 2 � B ( ) | X ringbone | , % B ,2=|o @ , ingenue E ✓ v → so monsimyular z y P ti B Co 0
Monphisms between the we need aossedbim°dT µ Definition ÷ )=i( jlelb . ) ekk eilb ( iii ) , )=k( place ) , , )e=i( b .sk ) ) ( ii ifb in ) , nlc )e=k( QPKD , exact not still j•kao\ Extension Ring Just : ← homo Etc 0 ring . C 13 . # � 1 � E , , , Is bilateral ideal E � 2 � B ( ) | X ringbone | , % B ,2=|o @ , ingenue E ✓ v → so momsimgular z y P ti B Co 0
Morphisms F F :@ B Ho±( GB ) form Tae moyohisms - agrowpoid G :C T I ex T e e ex e e fF) ⇒ G % FXF ( ⇒ )G × G | txt = g & 2 × 2 - × oD Mok 2 × 2
Morphism F :@ B Ho±( GB ) form ophisms - agrowpoid #@B :C T I ex T e e ex e e fF) ⇒ G % / FxFf ⇒ fG × G ext = g & 2 × 2=02 Mok 2 × 2 do so butterflies S+(c : . ,BD C , t.ec#Bi/ ¥¥¥* .
Morphis heeled ( ) ) 31 ( 2015 Ea TAC , T . Rings C B Picard Cat : of Strictly , G - p C - . > presentations 2 B - - B. , of gwupoids There is on equivalence Home ( e , B) ( c . ,BD tsp
Morphisms Thereof ( Ea Tac 30-(2 × 51) , Picard 5 C B . Rings : Strictly cat of , 9 p - - co > presentations 2 B - B. - , Pw# equivalence of gwupoids There is on Hot ( QD ) . ,B tsspcc . ) B C , , ti " Bo o D - F
Morphisms Thereof ( Ea Tac 30-(2 × 51) , Picard T C B . Rings : Strictly cat of , G p - - co > presentations 2 B - B. - , Pw# equivalence of gwupoids There is on Hott , B) . ,B tsspcc . ) B C , , Bo El l Stack fiber product f- Co fit I%3h,l ( cost ,bo ) ) , . f- A D bo ) Icc t( in . ) F
Morphisms Thereof ( Ea Tac 30-(2 × 51) , Picard T C B . Rings : Strictly Cat of , G p - - c. > presentations 2 B - B. - , T.ro# . gwwps at Exact equivalence of gwupoids There is on Hot ( 50 . ,B tsspcc . ) B , ti , £ e Bo Co Coffin ) 't to . D 8 - F
Morphisms Thereof ( Ea 30-(20151 ) TAC , Picard 5 C B . Rings : Strictly cat of , 9 p - - co > presentations 2 B - B. - , T.to#_ . gwwps Exact cat equivalence of gwupoids There is on Exact K # Hom- ( QB ) ( c . ,B tsp . ) C , , , yB £ . Bo Co Coffin ) 't to . D 8 - F
Morphisms Thereof ( Ea Tac 30-(2 × 51) , Picard T C B . Rings : Strictly Cat of , G p - - c. > presentations 2 B - B. - , Pw# equivalence of gwupoids There is on Hott , B) . ,B tsspcc . ) B C , , i. Not exact , ⇒ of £¥k> Bo Co coxed 't to . D 8 - F
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