Extention theory and the calculus of butterflies Alan Cigoli - PowerPoint PPT Presentation

Internal crossed modules Butterflies Extensions Extention theory and the calculus of butterflies Alan Cigoli Universit` a degli Studi di Milano (joint work with G. Metere) Categorical Methods in Algebra and Topology workshop in honour of Manuela Sobral on the occasion of her 70th birthday Coimbra, January 26, 2014 Alan Cigoli Extention theory and the calculus of butterflies

Internal crossed modules Butterflies Extensions Internal crossed modules Alan Cigoli Extention theory and the calculus of butterflies

Internal crossed modules Butterflies Extensions Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute). Alan Cigoli Extention theory and the calculus of butterflies

Internal crossed modules Butterflies Extensions Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute). An internal crossed module ( ∂, ξ ) in C is a morphism ∂ together with an action ξ ξ ∂ � G � G 0 G 0 ♭ G Alan Cigoli Extention theory and the calculus of butterflies

� � Internal crossed modules Butterflies Extensions Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute). An internal crossed module ( ∂, ξ ) in C is a morphism ∂ together with an action ξ ξ ∂ � G � G 0 G 0 ♭ G such that the following squares commute: χ G � G ♭ G G ∂♭ 1 � ξ G 0 ♭ G G 1 ♭∂ � ∂ � G 0 G 0 ♭ G 0 χ G 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � Internal crossed modules Butterflies Extensions A morphism of crossed modules ( ∂ ′ , ξ ′ ) → ( ∂, ξ ) is a pair ( f , f 0 ) of maps that makes the following diagram commute: f 0 ♭ f � H 0 ♭ H G 0 ♭ G ξ ′ ξ f � G H ∂ ′ ∂ � G 0 H 0 f 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � Internal crossed modules Butterflies Extensions A morphism of crossed modules ( ∂ ′ , ξ ′ ) → ( ∂, ξ ) is a pair ( f , f 0 ) of maps that makes the following diagram commute: f 0 ♭ f � H 0 ♭ H G 0 ♭ G ξ ′ ξ f � G H ∂ ′ ∂ � G 0 H 0 f 0 These data form a category XMod( C ) equivalent to Grpd( C ) [G. Janelidze ’03]. Alan Cigoli Extention theory and the calculus of butterflies

� � � � Internal crossed modules Butterflies Extensions A morphism of crossed modules ( ∂ ′ , ξ ′ ) → ( ∂, ξ ) is a pair ( f , f 0 ) of maps that makes the following diagram commute: f 0 ♭ f � H 0 ♭ H G 0 ♭ G ξ ′ ξ f � G H ∂ ′ ∂ � G 0 H 0 f 0 These data form a category XMod( C ) equivalent to Grpd( C ) [G. Janelidze ’03]. This equivalence extends to a biequivalence of bicategories [Abbad, Mantovani, Metere, Vitale ’13]. Alan Cigoli Extention theory and the calculus of butterflies

� � � � � Internal crossed modules Butterflies Extensions We can define homotopy invariants: π 1( f ) � π 1 ( ∂ ′ ) π 1 ( ∂ ) ❴ ❴ � ker( ∂ ) ker( ∂ ′ ) � f � G H ∂ ′ � ∂ � G 0 H 0 f 0 coker( ∂ ′ ) ❴ coker( ∂ ) ❴ π 0 ( ∂ ′ ) π 0( f ) � π 0 ( ∂ ) Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � � � Internal crossed modules Butterflies Extensions We can define homotopy invariants: π 1( f ) � π 1 ( ∂ ′ ) π 1 ( ∂ ) ❴ ❴ � ker( ∂ ) ker( ∂ ′ ) � f � G H ∂ ′ � ∂ � G 0 H 0 f 0 coker( ∂ ′ ) ❴ coker( ∂ ) ❴ π 0 ( ∂ ′ ) π 0( f ) � π 0 ( ∂ ) π 1 ( ∂ ) is central in C . Bourn’s global direction of a groupoid translates in terms of crossed modules as: π 1 ( ∂ ) π 1 ( ∂ ) ❴ 0 ✤ � π 0 ( ∂ ) G ⋊ ξ G 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � � �� � � �� �� � � � �� � �� �� � � � � � �� � �� � �� �� � � � � Internal crossed modules Butterflies Extensions Translation of some special morphisms: final disc. fib. π 0 -cart. fully faith. weak equiv. ✤ � π 1 ∂ ′ π 1 ∂ ′ � � π 1 ∂ ′ π 1 ∂ ′ π 1 ∂ ′ π 1 ∂ π 1 ∂ π 1 ∂ π 1 ∂ π 1 ∂ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � G H H H H H H G H G � � pf � G 0 � G 0 � G 0 � G 0 � G 0 H 0 H 0 H 0 H 0 H 0 � ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ π 0 ∂ ′ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ π 0 ∂ π 0 ∂ [C., Mantovani, [Everaert, Kieboom, Metere ’13] Van der Linden ’04] Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � � �� �� � � � � � �� �� �� �� � � �� � � � �� �� � � � � � �� Internal crossed modules Butterflies Extensions Translation of some special morphisms: final disc. fib. π 0 -cart. fully faith. weak equiv. ✤ � π 1 ∂ ′ π 1 ∂ ′ � � π 1 ∂ ′ π 1 ∂ ′ π 1 ∂ ′ π 1 ∂ π 1 ∂ π 1 ∂ π 1 ∂ π 1 ∂ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � G H H H H H H G H G � � pf � G 0 � G 0 � G 0 � G 0 � G 0 H 0 H 0 H 0 H 0 H 0 � ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ π 0 ∂ ′ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ � π 0 ∂ π 0 ∂ ′ π 0 ∂ π 0 ∂ [C., Mantovani, [Everaert, Kieboom, Metere ’13] Van der Linden ’04] We have (among others) two factorization systems: (final, disc. fib.) ∩ ∪ ( π 0 -inv., π 0 -cart.) Alan Cigoli Extention theory and the calculus of butterflies

Internal crossed modules Butterflies Extensions Internal butterflies Alan Cigoli Extention theory and the calculus of butterflies

Internal crossed modules Butterflies Extensions Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � Internal crossed modules Butterflies Extensions Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly � E : ( ∂ ′ , ξ ′ ) � ( ∂, ξ ) is a commutative diagram of the form H G κ ι ∂ ′ E ∂ γ δ H 0 G 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � Internal crossed modules Butterflies Extensions Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly � E : ( ∂ ′ , ξ ′ ) � ( ∂, ξ ) is a commutative diagram of the form H G κ ι ∂ ′ E ∂ γ δ H 0 G 0 such that i. ( κ, γ ) is a complex, i.e. γ · κ = 0, ii. ( ι, δ ) is short exact, iii. The action of E on H induced by that of H 0 on H via δ makes κ : H → E a crossed module, iv. The action of E on G induced by that of G 0 on G via γ makes ι : g → E a crossed module. Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � Internal crossed modules Butterflies Extensions Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly � E : ( ∂ ′ , ξ ′ ) � ( ∂, ξ ) is a commutative diagram of the form H G κ ι ∂ ′ E ∂ γ δ H 0 G 0 such that i. ( κ, γ ) is a complex, i.e. γ · κ = 0, ii. ( ι, δ ) is short exact, iii. The action of E on H induced by that of H 0 on H via δ makes κ : H → E a crossed module, iv. The action of E on G induced by that of G 0 on G via γ makes ι : g → E a crossed module. E ′ : ( ∂ ′ , ξ ′ ) � ( ∂, ξ ) is an arrow α : E → E ′ commuting A morphism of butterflies � E , � with the κ ’s, the ι ’s, the δ ’s and the γ ’s. Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � � � � � � Internal crossed modules Butterflies Extensions Horizontal composition: H G K ι ′ κ ι κ ′ E ′ E ∂ ∂ ∂ γ δ ′ γ ′ δ H 0 G 0 K 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � � � � � � � � Internal crossed modules Butterflies Extensions Horizontal composition: E × γ,δ ′ E ′ r s H G K ι ′ κ ι κ ′ E ′ E ∂ ∂ ∂ γ δ ′ γ ′ δ H 0 G 0 K 0 Alan Cigoli Extention theory and the calculus of butterflies

� � � � � � � � � � � � � � Internal crossed modules Butterflies Extensions Horizontal composition: E × γ,δ ′ E ′ � ι,κ ′� r s H G K ι ′ κ ι κ ′ E ′ E ∂ ∂ ∂ γ δ ′ γ ′ δ H 0 G 0 K 0 Alan Cigoli Extention theory and the calculus of butterflies