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Introduction A push forward construction A particular case A push forward construction and the comprehensive factorization for internal crossed modules I Alan Cigoli Universit` a degli Studi di Milano (joint work with S. Mantovani and G. Metere) Workshop on Category Theory in honour of George Janelidze, on the occasion of his 60th birthday Coimbra, July 13, 2012 Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Introduction Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Short exact sequences with kernel A and cokernel C form a groupoid EXT ( C , A ). Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Short exact sequences with kernel A and cokernel C form a groupoid EXT ( C , A ). Equivalence classes form an abelian group: Ext ( C , A ) . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Short exact sequences with kernel A and cokernel C form a groupoid EXT ( C , A ). Equivalence classes form an abelian group: Ext ( C , A ) . Any morphism c : C ′ → C determines a functor c ∗ : EXT ( C , A ) → EXT ( C ′ , A ) Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Short exact sequences with kernel A and cokernel C form a groupoid EXT ( C , A ). Equivalence classes form an abelian group: Ext ( C , A ) . Any morphism c : C ′ → C determines a functor c ∗ : EXT ( C , A ) → EXT ( C ′ , A ) by means of the pullback along c : � A � B ′ C ′ 0 0 � c � A � B � C � 0 0 Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case Let A be an abelian category, A and C objects of A . Short exact sequences with kernel A and cokernel C form a groupoid EXT ( C , A ). Equivalence classes form an abelian group: Ext ( C , A ) . Any morphism c : C ′ → C determines a functor c ∗ : EXT ( C , A ) → EXT ( C ′ , A ) by means of the pullback along c : � A � B ′ C ′ 0 0 � c � A � B � C � 0 0 And this gives a group homomorphism Ext ( C , A ) → Ext ( C ′ , A ) . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case Dually, any morphism a : A → A ′ determines a functor: a ∗ : EXT ( C , A ) → EXT ( C , A ′ ) Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case Dually, any morphism a : A → A ′ determines a functor: a ∗ : EXT ( C , A ) → EXT ( C , A ′ ) by means of the pushout along a : � A � 0 0 B C a � � A ′ � B ′ � C � 0 0 Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case Dually, any morphism a : A → A ′ determines a functor: a ∗ : EXT ( C , A ) → EXT ( C , A ′ ) by means of the pushout along a : � A � 0 0 B C a � � A ′ � B ′ � C � 0 0 And this gives a group homomorphism Ext ( C , A ) → Ext ( C , A ′ ) . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Example: groups. Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A : φ : G × A ��� A Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A : φ : G × A ��� A Short exact sequences inducing the same action of G on A form a groupoid OPEXT ( G , A , φ ). Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A : φ : G × A ��� A Short exact sequences inducing the same action of G on A form a groupoid OPEXT ( G , A , φ ). Equivalence classes form an abelian group: Opext ( G , A , φ ) ∼ = H 2 φ ( G , A ) . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A : φ : G × A ��� A Short exact sequences inducing the same action of G on A form a groupoid OPEXT ( G , A , φ ). Equivalence classes form an abelian group: Opext ( G , A , φ ) ∼ = H 2 φ ( G , A ) . Again, for any group homomorphism g : G ′ → G , the pullback construction determines a functor: g ∗ : OPEXT ( G , A , φ ) → OPEXT ( G ′ , A , g ∗ ( φ )) , Alan Cigoli A push forward construction

� � � Introduction A push forward construction A particular case where g ∗ ( φ ) is given by the composite: g ∗ ( φ ) G ′ × A � � � � � � � � A � � � � � � � � � � g × 1 � φ � � G × A Alan Cigoli A push forward construction

� � � Introduction A push forward construction A particular case where g ∗ ( φ ) is given by the composite: g ∗ ( φ ) G ′ × A � � � � � � � � A � � � � � � � � � � g × 1 � φ � � G × A And again a group homomorphism: H 2 φ ( G , A ) → H 2 g ∗ ( φ ) ( G ′ , A ) . Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The pushout contruction no longer works. Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The pushout contruction no longer works. Problems: Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a : A → A ′ does not determine an action of G on A ′ in a canonical way. Alan Cigoli A push forward construction

Introduction A push forward construction A particular case The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a : A → A ′ does not determine an action of G on A ′ in a canonical way. So we need an action of G on A ′ : φ ′ : G × A ′ ��� A ′ Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a : A → A ′ does not determine an action of G on A ′ in a canonical way. So we need an action of G on A ′ : φ ′ : G × A ′ ��� A ′ and we require that a is equivariant, i.e.: φ � � � G × A A 1 × a a G × A ′ A ′ � � � φ ′ Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case These data allow to construct the so called push forward along a : k f � A � 0 0 E G a p . f . e � 0 � A ′ � E ′ � G 0 k ′ f ′ Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case These data allow to construct the so called push forward along a : k f � A � 0 0 E G a p . f . e � 0 � A ′ � E ′ � G 0 k ′ f ′ which determines a functor: a ∗ : OPEXT ( G , A , φ ) → OPEXT ( G , A ′ , φ ′ ) Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case These data allow to construct the so called push forward along a : k f � A � 0 0 E G a p . f . e � 0 � A ′ � E ′ � G 0 k ′ f ′ which determines a functor: a ∗ : OPEXT ( G , A , φ ) → OPEXT ( G , A ′ , φ ′ ) and a group homomorphism: H 2 φ ( G , A ) → H 2 φ ′ ( G , A ′ ) . Alan Cigoli A push forward construction

� � � � Introduction A push forward construction A particular case Construction of the push forward (for groups): i E k E q � � E ′ E ⋊ f ∗ ( φ ′ ) A ′ A A ′ a i A ′ where q = coeq ( i E k , i A ′ a ). Alan Cigoli A push forward construction