Aspects of strong protomodularity, actions and quotients Giuseppe - PowerPoint PPT Presentation

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Aspects of strong protomodularity, actions and quotients Giuseppe Metere Universit` a degli Studi di Palermo June 15, 2015 Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Overview Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Overview Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro: actions on quotients Suppose we are given group action ξ and a surjective group homomorphism g ξ g � � Z � Y A × Y Y Question 1 : under what conditions does the action ξ induce an action of A on the quotient Z? ξ A × Y Y 1 × g g � Z A × Z ¯ ξ Answer: ξ is well defined on the cosets of Ker( g ) in Y ⇔ it is well defined on the 0-coset Ker( g ). (QA) An action passes to the quotient ⇔ it restricts to the kernel. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro: actions on quotients Suppose we are given group action ξ and a surjective group homomorphism g ξ g � � Z � Y A × Y Y Question 1 : under what conditions does the action ξ induce an action of A on the quotient Z? ξ A × Y Y 1 × g g � Z A × Z ¯ ξ Answer: ξ is well defined on the cosets of Ker( g ) in Y ⇔ it is well defined on the 0-coset Ker( g ). (QA) An action passes to the quotient ⇔ it restricts to the kernel. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro: actions of quotients Suppose we are given group action ξ and a surjective group homomorphism q ξ q � � Q � Y A × Y A Question 2 : under what conditions does the action ξ induce an action on Y of the quotient Q? ξ A × Y Y q × 1 1 � Y Q × Y ¯ ξ In this case, the restriction to Ker( q ) always exists. . . Answer: (AQ) The action of the quotient is well def. ⇔ Ker( q ) acts trivially on Y . Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro: actions of quotients Suppose we are given group action ξ and a surjective group homomorphism q ξ q � � Q � Y A × Y A Question 2 : under what conditions does the action ξ induce an action on Y of the quotient Q? ξ A × Y Y q × 1 1 � Y Q × Y ¯ ξ In this case, the restriction to Ker( q ) always exists. . . Answer: (AQ) The action of the quotient is well def. ⇔ Ker( q ) acts trivially on Y . Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) Y ⋊ ξ A ξ � Y �→ s p A × Y A We can address our questions in terms of split epimorphisms, even in contexts where the machinery of internal actions is not available. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) Y ⋊ ξ A ξ � Y �→ s p A × Y A We can address our questions in terms of split epimorphisms, even in contexts where the machinery of internal actions is not available. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Intro The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) Y ⋊ ξ A ξ � Y �→ s p A × Y A We can address our questions in terms of split epimorphisms, even in contexts where the machinery of internal actions is not available. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Overview Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� � � � Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Points From now on, C is a category with finite limits and initial object 0. Pt ( C ) is the category of points : f � B D s d s b d b � A C g widely studied by D. Bourn, G. Janelidze, F. Borceux. . . The codomain assignment ( B , A , b , s b ) �→ A gives rise to a fibration, the fibration of points : F : Pt ( C ) → C Fibers are denoted Pt A ( C ). Any g : C → A defines a change of base g ∗ : Pt A ( C ) → Pt C ( C ). Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Points Definition (Bourn) A category C is called protomodular when every change of base of the fibration of points is conservative, i.e. it reflects isomorphisms. If C admits an initial object 0, for any object A of C , we call kernel functor K A : Pt A ( C ) → Pt A (0) the change of base along the initial arrow ! A : 0 → A . Fact In the presence of an initial object, the protomodularity condition can be simplified by just requiring that kernel functors are conservative. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Points Definition (Bourn) A category C is called protomodular when every change of base of the fibration of points is conservative, i.e. it reflects isomorphisms. If C admits an initial object 0, for any object A of C , we call kernel functor K A : Pt A ( C ) → Pt A (0) the change of base along the initial arrow ! A : 0 → A . Fact In the presence of an initial object, the protomodularity condition can be simplified by just requiring that kernel functors are conservative. Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Internal actions Definition (Bourn - Janelidze) A protomodular category C admits semidirect products when, for every g : C → A , g ∗ : Pt A ( C ) → Pt C ( C ) is monadic. The algebras for the corresponding monads are called internal actions . Fact In the presence of an initial object, this definition can be simplified by just requiring that kernel functors are monadic. Act ( C ) ≃ Pt ( C ) � A ξ � Y A ♭ Y �→ Y ⋊ ξ A Giuseppe Metere Aspects of strong protomodularity, actions and quotients

� Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules Internal actions Definition (Bourn - Janelidze) A protomodular category C admits semidirect products when, for every g : C → A , g ∗ : Pt A ( C ) → Pt C ( C ) is monadic. The algebras for the corresponding monads are called internal actions . Fact In the presence of an initial object, this definition can be simplified by just requiring that kernel functors are monadic. Act ( C ) ≃ Pt ( C ) � A ξ � Y A ♭ Y �→ Y ⋊ ξ A Giuseppe Metere Aspects of strong protomodularity, actions and quotients