A new characterisation of higher central extensions in semi-abelian categories Cyrille Sandry Simeu Université Catholique de Louvain Joint work with T. Everaert and T. Van der Linden July 11, 2018 Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 1/ 28
Outline Introduction 1 Categorical Galois theory 2 Semi-abelian categories § Higher central extensions § The Smith is Huq condition § The higher-order Higgins commutator 3 Definitions and examples § Some properties of the n -fold Higgins commutator § A new characterisation of higher central extensions 4 The known results § The main results § Some perspectives 5 Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-a 2/ 28
� � Introduction The concept of higher centrality is useful and unavoidable in the recent approach to homology and cohomology of non-abelian structures based on categorical Galois theory. In our work, higher central extensions are the covering morphisms with respect to certain Galois structures induced by a refletion Ab Ab p X q X K Ą and can also be defined more generally, for any semi-abelian category X and any Birkhoff subcategory B of X . The descriptions of higher central extensions in terms of algebraic conditions using "generalised commutators" is in general a non-trivial problem. Today, I am going to: give a new characterisation of higher central extensions in terms of higher-order § Higgins commutators in semi-abelian categories which do not satisfy the Smith is Huq condition. give some perspectives for future work. § Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 3/ 28
Semi-abelian categories Throughout this presentation, X is a semi-abelian category. Definition [G. Janelidze, L. Márki, and W. Tholen] A category X is semi-abelian when it is pointed; 1 has binary coproducts; 2 is Barr-exact; 3 is Bourn-protomodular: the Split Short Five Lemma holds. 4 Examples: Grp, Lie K , Alg K , XMod, varieties of Ω -groups, Loops, Near-Rings. Definition [G. Janelidze, G.M. Kelly] A subcategory B of X is a Birkhoff subcategory when it is closed under subobjects and regular quotients. Examples: ‚ Any subvariety B of a variety of universal algebras V . ‚ The subcategory Ab p X q of abelian objects in X . Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 4/ 28
Semi-abelian categories Throughout this presentation, X is a semi-abelian category. Definition [G. Janelidze, L. Márki, and W. Tholen] A category X is semi-abelian when it is pointed; 1 has binary coproducts; 2 is Barr-exact; 3 is Bourn-protomodular: the Split Short Five Lemma holds. 4 Examples: Grp, Lie K , Alg K , XMod, varieties of Ω -groups, Loops, Near-Rings. Definition [G. Janelidze, G.M. Kelly] A subcategory B of X is a Birkhoff subcategory when it is closed under subobjects and regular quotients. Examples: ‚ Any subvariety B of a variety of universal algebras V . ‚ The subcategory Ab p X q of abelian objects in X . Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 4/ 28
� � � � Higher extensions An n -fold arrow in X is a functor F : p 2 n q op Ý Ñ X . § Arr n p X q “ Fun pp 2 n q op , X q An n -fold arrow F is an n -fold extension when for all H ‰ I Ď n the arrow § F I ։ lim J Ĺ I F J is a regular epimorphism. Ext n p X q is the category of n -fold extensions The adjunction § ab Ab p X q X K Ą induces a Galois structure Γ 0 “ p X , Ab p X q , ab , Ă , E , F q in the sense of G. Janelidze. A 1-fold extension f : B ։ A P E is central w.r.t Γ 0 if and only if the square § π 1 ✤ � Eq p f q B η 0 η 0 Eq p f q ❴ B ❴ ✤ � ab 0 p B q ab 0 p Eq p f qq ab p π 1 q is a pullback. Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 5/ 28
� � � � Higher extensions An n -fold arrow in X is a functor F : p 2 n q op Ý Ñ X . § Arr n p X q “ Fun pp 2 n q op , X q An n -fold arrow F is an n -fold extension when for all H ‰ I Ď n the arrow § F I ։ lim J Ĺ I F J is a regular epimorphism. Ext n p X q is the category of n -fold extensions The adjunction § ab Ab p X q X K Ą induces a Galois structure Γ 0 “ p X , Ab p X q , ab , Ă , E , F q in the sense of G. Janelidze. A 1-fold extension f : B ։ A P E is central w.r.t Γ 0 if and only if the square § π 1 ✤ � Eq p f q B η 0 η 0 Eq p f q ❴ B ❴ ✤ � ab 0 p B q ab 0 p Eq p f qq ab p π 1 q is a pullback. Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 5/ 28
� � � � Higher central extensions The category CExt p X q of 1-fold central extensions in X is a strongly E 1 -Birkhoff § subcategory of Ext p X q Inductively, for any n ě 1, this gives an adjunction § ab n Ext n p X q CExt n p X q K Ą which induce a Galois structure Γ n “ p Ext n p X q , CExt n p X q , ab n , Ă , E n , F n q The category CExt n p X q of n -fold central extensions in X w.r.t Γ n ´ 1 is a strongly § E n -Birkhoff subcategory of Ext n p X q An n -fold extension f : B Ñ A is central w.r.t Γ n ´ 1 if and only if the square § π 1 ✤ � Eq p f q B η n ´ 1 η n ´ 1 Eq p f q ❴ B ❴ ✤ � ab n ´ 1 p B q ab n ´ 1 p Eq p f qq ab n ´ 1 p π 1 q is a pullback. Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 6/ 28
� � � � � � � � � � � � � � � � The reflection The reflection ab n : Ext n p X q Ñ CExt n p X q is built as follows: [T. Everaert, M. Gran, and T. Van der Linden, 2008] J n r F s ker r π 1 s CExt n ´ 1 p X q µ n ´ 1 η n ´ 1 Eq p f q Eq p F q � r Eq p F qs CExt n ´ 1 p X q � 0 0 Eq p F q ab n ´ 1 p Eq p F qq r π 2 s CExt n ´ 1 p X q r π 1 s CExt n ´ 1 p X q π 2 π 1 ab n ´ 1 p π 2 q ab n ´ 1 p π 1 q µ n ´ 1 η n ´ 1 B B � r B s CExt n ´ 1 p X q 0 B ab n ´ 1 p B q 0 r F s CExt n ´ 1 p X q ab n ´ 1 p F q F � r A s CExt n ´ 1 p X q � A � ab n ´ 1 p A q � 0 0 µ n ´ 1 η n ´ 1 A A Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 7/ 28
� � � � The object L n r F s This yields a morphism of short exact sequences in Arr n ´ 1 p X q ρ n � J n r F s F 0 B ab n r F s 0 J n F � F � ab n F � A � 0 0 A § L n r F s is the inital object of the n -fold extension J n F denoted by L n r F s “ p J n F q n § J n F is zero everywhere, except on its initial object L n r F s . Remark [T. Everaert, M. Gran, and T. Van der Linden, 2008] An n -fold extension F is central w.r.t Γ n ´ 1 if and only if L n r F s “ 0 What is L n r F s ? Our goal is to give an explicite description of this object in terms of § "generalised commutators". Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 8/ 28
� � � � � � � �� � � � � � � � � � � � The Smith is Huq condition For subojects K , L of X , the Huq-Bourn For equivalence relations R , S on X commutator r K , L s Q is the kernel of the π R π S morphism q , 1 1 � S R � X ∆ R ∆ S π R π S K 2 2 ă 1 K , 0 ą h The Smith-Pedicchio commutator r R , S s S , q m � Q K ˆ L A is the kernel pair of ψ ă 0 , 1 L ą k L R x 1 R , ∆ S ˝ π R π R ‚ R and S Smith commute iff 1 y 2 ϕ ψ � T r R , S s S R ˆ X S X r R , S s S “ ∆ X x ∆ R ˝ π S π S 2 , 1 S y ‚ K and L Huq commute iff r K , L s Q “ 0 1 S The Smith is Huq condition A semi-abelian category X satisfies the Smith is Huq condition (SH) , when two equivalence relations on the same object Smith-commute if and only if their normalisations Huq commute. Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 9/ 28
� � � � � � � �� � � � � � � � � � � � The Smith is Huq condition For subojects K , L of X , the Huq-Bourn For equivalence relations R , S on X commutator r K , L s Q is the kernel of the π R π S morphism q , 1 1 � S R � X ∆ R ∆ S π R π S K 2 2 ă 1 K , 0 ą h The Smith-Pedicchio commutator r R , S s S , q m � Q K ˆ L A is the kernel pair of ψ ă 0 , 1 L ą k L R x 1 R , ∆ S ˝ π R π R ‚ R and S Smith commute iff 1 y 2 ϕ ψ � T r R , S s S R ˆ X S X r R , S s S “ ∆ X x ∆ R ˝ π S π S 2 , 1 S y ‚ K and L Huq commute iff r K , L s Q “ 0 1 S The Smith is Huq condition A semi-abelian category X satisfies the Smith is Huq condition (SH) , when two equivalence relations on the same object Smith-commute if and only if their normalisations Huq commute. Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 9/ 28
When the condition p SH q holds, the object L n r F s has a characterisation in terms of § binary Higgins or binary Huq commutators. Examples of categories with p SH q Grp; § Lie K ; § Action accessible categories; § Categories of interest in the sense of Orzech. § Cyrille Sandry Simeu A new characterisation of higher central extensions in semi-ab 10/ 28
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