f A E X e A E X f Theorem [Eckmann 1945-46; Eilenberg & Mac Lane, 1947] For any abelian group A we have H X A CentrExt X A , the group of equivalence classes of central extensions from A to X . H A is the first derived functor Gp op of Hom A Ab . By the Short Five Lemma, equivalence class = isomorphism class : The theorem remains true [Gran & VdL, 2008] in any semi-abelian category [Janelidze, Márki & Tholen, 2002] with enough projectives; centrality may be defined via commutator theory or via categorical Galois theory. Low-dimensional cohomology of groups, I An extension from A to X is a short exact sequence f ✤ � X � A ✤ � � E � 0 . 0 It is central if and only if [ A , E ] = 0 : all eae ´ 1 a ´ 1 vanish, a P A , e P E . Then, in particular, A is an abelian group.
f A E X e A E X f H A is the first derived functor Gp op of Hom A Ab . By the Short Five Lemma, equivalence class = isomorphism class : The theorem remains true [Gran & VdL, 2008] in any semi-abelian category [Janelidze, Márki & Tholen, 2002] with enough projectives; centrality may be defined via commutator theory or via categorical Galois theory. Low-dimensional cohomology of groups, I An extension from A to X is a short exact sequence f ✤ � X � A ✤ � � E � 0 . 0 It is central if and only if [ A , E ] = 0 : all eae ´ 1 a ´ 1 vanish, a P A , e P E . Then, in particular, A is an abelian group. Theorem [Eckmann 1945-46; Eilenberg & Mac Lane, 1947] For any abelian group A we have H 2 ( X , A ) – CentrExt 1 ( X , A ) , the group of equivalence classes of central extensions from A to X .
f A E X e A E X f By the Short Five Lemma, equivalence class = isomorphism class : The theorem remains true [Gran & VdL, 2008] in any semi-abelian category [Janelidze, Márki & Tholen, 2002] with enough projectives; centrality may be defined via commutator theory or via categorical Galois theory. Low-dimensional cohomology of groups, I An extension from A to X is a short exact sequence f ✤ � X � A ✤ � � E � 0 . 0 It is central if and only if [ A , E ] = 0 : all eae ´ 1 a ´ 1 vanish, a P A , e P E . Then, in particular, A is an abelian group. Theorem [Eckmann 1945-46; Eilenberg & Mac Lane, 1947] For any abelian group A we have H 2 ( X , A ) – CentrExt 1 ( X , A ) , the group of equivalence classes of central extensions from A to X . § H 2 ( ´ , A ) is the first derived functor of Hom ( ´ , A ): Gp op Ñ Ab .
The theorem remains true [Gran & VdL, 2008] in any semi-abelian category [Janelidze, Márki & Tholen, 2002] with enough projectives; centrality may be defined via commutator theory or via categorical Galois theory. � Low-dimensional cohomology of groups, I An extension from A to X is a short exact sequence f ✤ � X � A ✤ � � E � 0 . 0 It is central if and only if [ A , E ] = 0 : all eae ´ 1 a ´ 1 vanish, a P A , e P E . Then, in particular, A is an abelian group. Theorem [Eckmann 1945-46; Eilenberg & Mac Lane, 1947] For any abelian group A we have H 2 ( X , A ) – CentrExt 1 ( X , A ) , the group of equivalence classes of central extensions from A to X . § H 2 ( ´ , A ) is the first derived functor � A ✤ � � E f ✤ � X � 0 0 of Hom ( ´ , A ): Gp op Ñ Ab . e § By the Short Five Lemma, � A ✤ � � E 1 ✤ � X � 0 0 equivalence class = isomorphism class : f 1
� Low-dimensional cohomology of groups, I An extension from A to X is a short exact sequence f ✤ � X � A ✤ � � E � 0 . 0 It is central if and only if [ A , E ] = 0 : all eae ´ 1 a ´ 1 vanish, a P A , e P E . Then, in particular, A is an abelian group. Theorem [Eckmann 1945-46; Eilenberg & Mac Lane, 1947] For any abelian group A we have H 2 ( X , A ) – CentrExt 1 ( X , A ) , the group of equivalence classes of central extensions from A to X . § H 2 ( ´ , A ) is the first derived functor � A ✤ � � E f ✤ � X � 0 0 of Hom ( ´ , A ): Gp op Ñ Ab . e § By the Short Five Lemma, � A ✤ � � E 1 ✤ � X � 0 0 equivalence class = isomorphism class : f 1 The theorem remains true [Gran & VdL, 2008] in any semi-abelian category [Janelidze, Márki & Tholen, 2002] with enough projectives; centrality may be defined via commutator theory or via categorical Galois theory.
Eq f F f F X f Eq f F F Eq f ab Eq f ab F ab Basic analysis H is a derived functor of the reflector X ab Gp Ab X X X . The commutator R F occurs in/is determined by the reflector F ab Ext Gp CExt Gp f F X ab f X . R F Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. In fact, f is central iff the bottom right square is a pullback. All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] .
Eq f F f F X f Eq f F F Eq f ab Eq f ab F ab The commutator R F occurs in/is determined by the reflector F ab Ext Gp CExt Gp f F X ab f X . R F Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. In fact, f is central iff the bottom right square is a pullback. All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] .
Eq f F f F X f Eq f F F Eq f ab Eq f ab F ab Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. In fact, f is central iff the bottom right square is a pullback. All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . § The commutator [ R , F ] occurs in/is determined by the reflector F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) .
Eq f F f F X f Eq f F F Eq f ab Eq f ab F ab In fact, f is central iff the bottom right square is a pullback. All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . § The commutator [ R , F ] occurs in/is determined by the reflector F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) . § Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first.
All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … � � � � � � � Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis π 2 Eq ( f ) F § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . f π 1 � X § The commutator [ R , F ] occurs in/is determined by the reflector F f F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) . π 2 Eq ( f ) F § Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. η Eq ( f ) η F § In fact, f is central iff the bottom right square is a pullback. � ab ( F ) ab ( Eq ( f )) ab ( π 2 )
The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: X X is commutator ( Gp vs. Ab ), Lie bracket ( Lie vs. Vect ), product XX ( Alg R vs. Mod R ), or … � � � � � � � Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis π 2 Eq ( f ) F § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . f π 1 � X § The commutator [ R , F ] occurs in/is determined by the reflector F f F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) . π 2 Eq ( f ) F § Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. η Eq ( f ) η F § In fact, f is central iff the bottom right square is a pullback. � ab ( F ) ab ( Eq ( f )) ab ( π 2 ) All ingredients of the formula may be obtained from the reflector ab .
� � � � � � � Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis π 2 Eq ( f ) F § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . f π 1 � X § The commutator [ R , F ] occurs in/is determined by the reflector F f F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) . π 2 Eq ( f ) F § Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. η Eq ( f ) η F § In fact, f is central iff the bottom right square is a pullback. � ab ( F ) ab ( Eq ( f )) ab ( π 2 ) All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: [ X , X ] is commutator ( Gp vs. Ab ), Lie bracket ( Lie K vs. Vect K ), product XX ( Alg R vs. Mod R ), or …
� � � � � � � Low-dimensional homology of groups Theorem (Hopf formula for H 2 ( X ) , [Hopf, 1942] ) Consider a projective presentation X – F / R of X : an extension 0 Ñ R Ñ F Ñ X Ñ 0 where F is projective. Then the second integral homology group H 2 ( X ) is R ^ [ F , F ] [ R , F ] . Basic analysis π 2 Eq ( f ) F § H 2 is a derived functor of the reflector X ab : Gp Ñ Ab : X ÞÑ [ X , X ] . f π 1 � X § The commutator [ R , F ] occurs in/is determined by the reflector F f F ab 1 : Ext ( Gp ) Ñ CExt ( Gp ): ( f : F Ñ X ) ÞÑ ( ab 1 ( f ): [ R , F ] Ñ X ) . π 2 Eq ( f ) F § Through categorical Galois theory [Janelidze & Kelly, 1994] , the second adjunction may be obtained from the first. η Eq ( f ) η F § In fact, f is central iff the bottom right square is a pullback. � ab ( F ) ab ( Eq ( f )) ab ( π 2 ) All ingredients of the formula may be obtained from the reflector ab . The theorem remains true [Everaert & VdL, 2004] for reflectors of semi-abelian varieties of algebras to their subvarieties: [ X , X ] is commutator ( Gp vs. Ab ), Lie bracket ( Lie K vs. Vect K ), product XX ( Alg R vs. Mod R ), or …
All varieties of algebras and all elementary toposes are such. An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair.
An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such.
Examples: Mod R , sheaves of abelian groups. A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964]
A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups.
This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. § A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] .
coc , C - Alg , Set op , varieties of Examples: Gp , Lie , Alg , XMod , Loop , HopfAlg -groups. What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. § A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity.
What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. § A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity. Examples: Gp , Lie K , Alg K , XMod , Loop , HopfAlg K , coc , C ˚ - Alg , Set op ˚ , varieties of Ω -groups.
What is a semi-abelian category? A category is Barr-exact [Barr, 1971] when 1 finite limits and coequalisers of kernel pairs exist; 2 regular epimorphisms are pullback-stable; 3 every internal equivalence relation is a kernel pair. All varieties of algebras and all elementary toposes are such. § An abelian category is a Barr-exact category which is also additive : it has finitary biproducts and is enriched over Ab . [Buchsbaum, 1955; Grothendieck, 1957; Yoneda, 1960; Freyd, 1964] Examples: Mod R , sheaves of abelian groups. § A Barr-exact category is semi-abelian when it is pointed, has binary coproducts and is protomodular : the Split Short Five Lemma holds [Bourn, 1991] . This definition [Janelidze, Márki & Tholen, 2002] unifies “old” approaches towards an axiomatisation of categories “close to Gp ” such as [Higgins, 1956] and [Huq, 1968] with “new” categorical algebra—the concepts of Barr-exactness and Bourn-protomodularity . Examples: Gp , Lie K , Alg K , XMod , Loop , HopfAlg K , coc , C ˚ - Alg , Set op ˚ , varieties of Ω -groups.
B Z A point f s over X is a split epimorphism f Y X with a chosen splitting s X Y . s X Y A Y Pt X X is the X f X category of points over X in . X X The Split Short Five Lemma is precisely the condition that the pullback functor Pt X Pt reflects isomorphisms. X Points are actions. If is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories.
B Z s X Y A Y Pt X X is the X f X category of points over X in . X X The Split Short Five Lemma is precisely the condition that the pullback functor Pt X Pt reflects isomorphisms. X Points are actions. If is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y .
B Z A Y X The Split Short Five Lemma is precisely the condition that the pullback functor Pt X Pt reflects isomorphisms. X Points are actions. If is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . X
Points are actions. If is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f � � � � � � � � � � � � � � � � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. B ✤ � Z A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X A ✤ � Y Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . ❴ X � X 0 The Split Short Five Lemma is precisely the condition that the ❴ pullback functor Pt X ( X ) Ñ Pt 0 ( X ) – X reflects isomorphisms. � X 0
If is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f � � � � � � � � � � � � � � � � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. B ✤ � Z A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X A ✤ � Y Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . ❴ X � X 0 The Split Short Five Lemma is precisely the condition that the ❴ pullback functor Pt X ( X ) Ñ Pt 0 ( X ) – X reflects isomorphisms. � X 0 Points are actions.
; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via , we obtain s A A X X f � � � � � � � � � � � � � � � � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. B ✤ � Z A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X A ✤ � Y Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . ❴ X � X 0 The Split Short Five Lemma is precisely the condition that the ❴ pullback functor Pt X ( X ) Ñ Pt 0 ( X ) – X reflects isomorphisms. � X 0 Points are actions. If X is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998]
if X acts on A via , we obtain s A A X X f � � � � � � � � � � � � � � � � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. B ✤ � Z A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X A ✤ � Y Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . ❴ X � X 0 The Split Short Five Lemma is precisely the condition that the ❴ pullback functor Pt X ( X ) Ñ Pt 0 ( X ) – X reflects isomorphisms. � X 0 Points are actions. If X is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions:
� � � � � � � � � � � � � � � � � � More on protomodularity Protomodular categories [Bourn, 1991] arose out of the idea that in algebra, categories of points may be more fundamental than slice categories. B ✤ � Z A point ( f , s ) over X is a split epimorphism f : Y Ñ X with a chosen splitting s : X Ñ Y . s � Y X A ✤ � Y Pt X ( X ) = (1 X Ó ( X Ó X )) is the f 1 X � category of points over X in X . ❴ X � X 0 The Split Short Five Lemma is precisely the condition that the ❴ pullback functor Pt X ( X ) Ñ Pt 0 ( X ) – X reflects isomorphisms. � X 0 Points are actions. If X is semi-abelian, then this change-of-base functor is monadic [Bourn & Janelidze, 1998] ; the algebras for the monad are called internal actions , and correspond to split extensions: if X acts on A via ξ , we obtain s ξ � A ✤ � � A ¸ ξ X � 0 . ✤ � X 0 f ξ
R F X is a projective presentation. A priori, H is a derived functor of ab Ab . The Hopf formula is valid for any reflector I from a semi-abelian category to a Birkhoff subcategory ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial. Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ]
R F X is a projective presentation. A priori, H is a derived functor of ab Ab . The Hopf formula is valid for any reflector I from a semi-abelian category to a Birkhoff subcategory ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial. Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ]
A priori, H is a derived functor of ab Ab . The Hopf formula is valid for any reflector I from a semi-abelian category to a Birkhoff subcategory ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial. Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ] § 0 Ñ R Ñ F Ñ X Ñ 0 is a projective presentation.
The Hopf formula is valid for any reflector I from a semi-abelian category to a Birkhoff subcategory ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial. Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ] § 0 Ñ R Ñ F Ñ X Ñ 0 is a projective presentation. § A priori, H 2 is a derived functor of ab : X Ñ Ab ( X ) .
Also in the abelian case, this gives something non-trivial. Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ] § 0 Ñ R Ñ F Ñ X Ñ 0 is a projective presentation. § A priori, H 2 is a derived functor of ab : X Ñ Ab ( X ) . § The Hopf formula is valid for any reflector I : X Ñ Y from a semi-abelian category X to a Birkhoff subcategory Y ; then the commutators are relative with respect to I .
Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) see Julia’s talk! categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ] § 0 Ñ R Ñ F Ñ X Ñ 0 is a projective presentation. § A priori, H 2 is a derived functor of ab : X Ñ Ab ( X ) . § The Hopf formula is valid for any reflector I : X Ñ Y from a semi-abelian category X to a Birkhoff subcategory Y ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial.
Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ] § 0 Ñ R Ñ F Ñ X Ñ 0 is a projective presentation. § A priori, H 2 is a derived functor of ab : X Ñ Ab ( X ) . § The Hopf formula is valid for any reflector I : X Ñ Y from a semi-abelian category X to a Birkhoff subcategory Y ; then the commutators are relative with respect to I . Also in the abelian case, this gives something non-trivial.
Any action X Aut A of X on A pulls back along f to an action f E Aut A e f e of E on A . If A is abelian, then there is a unique action of X on A such that f is the conjugation action of E on A : put x a eae for e E with f e x . This action is called the direction of the given extension. It determines a left X -module structure on A . Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 .
If A is abelian, then there is a unique action of X on A such that f is the conjugation action of E on A : put x a eae for e E with f e x . This action is called the direction of the given extension. It determines a left X -module structure on A . Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A .
put x a eae for e E with f e x . This action is called the direction of the given extension. It determines a left X -module structure on A . Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A :
This action is called the direction of the given extension. It determines a left X -module structure on A . Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x .
It determines a left X -module structure on A . Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x . § This action ξ is called the direction of the given extension.
Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x . § This action ξ is called the direction of the given extension. It determines a left Z ( X ) -module structure on A .
f is central a eae This agrees with the above: an a A e E extension with abelian kernel is a f e a a A e E central iff its direction is trivial. x x X A How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x . § This action ξ is called the direction of the given extension. It determines a left Z ( X ) -module structure on A . Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) .
How to extend this to semi-abelian categories? Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x . § This action ξ is called the direction of the given extension. It determines a left Z ( X ) -module structure on A . Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) . a = eae ´ 1 f is central ô @ a P A @ e P E This agrees with the above: an extension with abelian kernel is a = ξ ( f ( e ))( a ) ô @ a P A @ e P E central iff its direction is trivial. 1 A = ξ ( x ) ô @ x P X
Low-dimensional cohomology of groups, II f ✤ � X � A ✤ � � E Consider an extension 0 � 0 . § Any action ξ : X Ñ Aut ( A ) of X on A pulls back along f to an action f ˚ ( ξ ): E Ñ Aut ( A ): e ÞÑ ξ ( f ( e )) of E on A . § If A is abelian, then there is a unique action ξ of X on A such that f ˚ ( ξ ) is the conjugation action of E on A : put ξ ( x )( a ) = eae ´ 1 for e P E with f ( e ) = x . § This action ξ is called the direction of the given extension. It determines a left Z ( X ) -module structure on A . Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) . a = eae ´ 1 f is central ô @ a P A @ e P E This agrees with the above: an extension with abelian kernel is a = ξ ( f ( e ))( a ) ô @ a P A @ e P E central iff its direction is trivial. 1 A = ξ ( x ) ô @ x P X How to extend this to semi-abelian categories?
� � � ✤ � � � � � � � � � ✤ � � � � � � � � � � Three commutators Huq & Higgins Smith-Pedicchio For K , L ◁ X , the Huq commutator [ K , L ] Q is the kernel of q : For equivalence relations R , S on X K r 1 s 2 ⑧ � x 1 K , 0 y � S , k R � X ∆ R ∆ S r 2 s 1 � Q K ˆ L X q the Smith-Pedicchio commutator [ R , S ] S is the kernel pair of t : l x 0 , 1 L y ❄ � L R The Higgins commutator [ K , L ] ď X is x 1 R , ∆ S ˝ r 1 y r 2 the image of p k l q ˝ ι K , L : � T R ˆ X S X t K ˛ L ✤ � ι K , L � ✤ � K ˆ L K + L s 2 x ∆ R ˝ s 1 , 1 S y p k l q S ❴ � X [ K , L ] �
� � � � � � � � � ✤ � � � � � � � � � � � Pregroupoids d c A span D X � C is a pregroupoid iff [ Eq ( d ) , Eq ( c )] S = ∆ X . [Kock, 1989] Smith-Pedicchio For equivalence relations R , S on X ( β, γ ) r 1 s 2 � S , Eq ( d ) R � X ∆ R ∆ S x 1 Eq ( d ) , x π 1 ,π 1 yy r 2 s 1 π 2 the Smith-Pedicchio commutator [ R , S ] S is � X Eq ( d ) ˆ X Eq ( c ) p the kernel pair of t : π 2 xx π 1 ,π 1 y , 1 Eq ( c ) y Eq ( c ) R x 1 R , ∆ S ˝ r 1 y r 2 ( β, α ) � T R ˆ X S X t ¨ β γ # p ( α, β, β ) = α s 2 x ∆ R ˝ s 1 , 1 S y ¨ ¨ S p ( β, β, γ ) = γ α p ( α,β,γ ) ¨
One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K L X do. r s r ker r s ker s K X L normalisations of R X S r s One implication is automatic [Bourn & Gran, 2002] . All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X C is a pregroupoid iff Ker d Ker c , d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”.
One implication is automatic [Bourn & Gran, 2002] . All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X C is a pregroupoid iff Ker d Ker c , d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. � � � � ✤ � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1
All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X C is a pregroupoid iff Ker d Ker c , d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. � � ✤ � � � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] .
By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X C is a pregroupoid iff Ker d Ker c , d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. � ✤ � � � � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] . § All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not.
This is, essentially, because d c then, a span D X C is a pregroupoid iff Ker d Ker c , d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. � � � ✤ � � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] . § All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. § By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies.
d so a reflexive graph G G is an internal groupoid iff Ker d Ker c . e c This is important when defining abelian extensions. � � � � � ✤ � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] . § All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. § By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X � C is a pregroupoid iff [ Ker ( d ) , Ker ( c )] = 0 ,
This is important when defining abelian extensions. � � � � � � � � ✤ The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] . § All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. § By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X � C is a pregroupoid iff [ Ker ( d ) , Ker ( c )] = 0 , d so a reflexive graph G 1 � G 0 is an internal groupoid iff [ Ker ( d ) , Ker ( c )] = 0 . e c
� � � � � � � ✤ � The Smith is Huq condition Several categorical-algebraic conditions have been considered which “make a semi-abelian category behave more like Gp does”. One (weak and well-studied) such is the Smith is Huq condition (SH) , which holds when two equivalence relations R and S on an object X commute iff their normalisations K , L ◁ X do. r 1 s 2 K ✤ � r 2 ˝ ker ( r 1 ) � X s 2 ˝ ker ( s 1 ) � X L normalisations of R S r 2 s 1 § One implication is automatic [Bourn & Gran, 2002] . § All Orzech categories of interest [Orzech, 1972] satisfy (SH). Loop does not. § By [Martins-Ferreira & VdL, 2012] and [Hartl & VdL, 2013] , under (SH) the description of internal crossed modules of [Janelidze, 2003] simplifies. This is, essentially, because d c then, a span D X � C is a pregroupoid iff [ Ker ( d ) , Ker ( c )] = 0 , d so a reflexive graph G 1 � G 0 is an internal groupoid iff [ Ker ( d ) , Ker ( c )] = 0 . e c This is important when defining abelian extensions.
this means that, equivalently, 1 the span f f is a pregroupoid; S is trivial; 2 the commutator Eq f Eq f E Eq f is a normal monomorphism f f in X ; 3 E E a a A Eq f is a normal monomorphism in . 4 Example: a split extension (a point f s with a ker f ) is abelian iff it is a Beck module [Beck, 1967] : an abelian group object in X . f A a a A A Eq f X Given an abelian extension, we may take cokernels as in the diagram on the left to find s f E E its direction : the X -module A . A E X a f The pullback f of along f is the conjugation action of E on A . The semi-abelian case: abelian extensions, I Let X be a semi-abelian category. An abelian extension in X is a short exact sequence � A ✤ � a � E f ✤ � X � 0 0 where f is an abelian object in ( X Ó X ) :
Example: a split extension (a point f s with a ker f ) is abelian iff it is a Beck module [Beck, 1967] : an abelian group object in X . f A a a A A Eq f X Given an abelian extension, we may take cokernels as in the diagram on the left to find s f E E its direction : the X -module A . A E X a f The pullback f of along f is the conjugation action of E on A . The semi-abelian case: abelian extensions, I Let X be a semi-abelian category. An abelian extension in X is a short exact sequence � A ✤ � a � E f ✤ � X � 0 0 where f is an abelian object in ( X Ó X ) : this means that, equivalently, 1 the span ( f , f ) is a pregroupoid; 2 the commutator [ Eq ( f ) , Eq ( f )] S is trivial; 3 x 1 E , 1 E y : E Ñ Eq ( f ) is a normal monomorphism f Ñ f π 1 in ( X Ó X ) ; 4 x a , a y : A Ñ Eq ( f ) is a normal monomorphism in X .
f A a a A A Eq f X Given an abelian extension, we may take cokernels as in the diagram on the left to find s f E E its direction : the X -module A . A E X a f The pullback f of along f is the conjugation action of E on A . The semi-abelian case: abelian extensions, I Let X be a semi-abelian category. An abelian extension in X is a short exact sequence � A ✤ � a � E f ✤ � X � 0 0 where f is an abelian object in ( X Ó X ) : this means that, equivalently, 1 the span ( f , f ) is a pregroupoid; 2 the commutator [ Eq ( f ) , Eq ( f )] S is trivial; 3 x 1 E , 1 E y : E Ñ Eq ( f ) is a normal monomorphism f Ñ f π 1 in ( X Ó X ) ; 4 x a , a y : A Ñ Eq ( f ) is a normal monomorphism in X . Example: a split extension (a point ( f , s ) with a = ker ( f ) ) is abelian iff it is a Beck module [Beck, 1967] : an abelian group object in ( X Ó X ) .
The pullback f of along f is the conjugation action of E on A . � � � � The semi-abelian case: abelian extensions, I Let X be a semi-abelian category. An abelian extension in X is a short exact sequence � A ✤ � a � E f ✤ � X � 0 0 where f is an abelian object in ( X Ó X ) : this means that, equivalently, 1 the span ( f , f ) is a pregroupoid; 2 the commutator [ Eq ( f ) , Eq ( f )] S is trivial; 3 x 1 E , 1 E y : E Ñ Eq ( f ) is a normal monomorphism f Ñ f π 1 in ( X Ó X ) ; 4 x a , a y : A Ñ Eq ( f ) is a normal monomorphism in X . Example: a split extension (a point ( f , s ) with a = ker ( f ) ) is abelian iff it is a Beck module [Beck, 1967] : an abelian group object in ( X Ó X ) . x a , a y � Eq ( f ) 1 A ¸ f � A ✤ � ✤ � � 0 A ¸ ξ X 0 Given an abelian extension, we may take cokernels as in the diagram on the left to find � s ξ f ξ ❴ π 1 ❴ � x 1 E , 1 E y its direction : the X -module ( A , ξ ) . � A ✤ � � E ✤ � X � 0 0 a f
� � � � The semi-abelian case: abelian extensions, I Let X be a semi-abelian category. An abelian extension in X is a short exact sequence � A ✤ � a � E f ✤ � X � 0 0 where f is an abelian object in ( X Ó X ) : this means that, equivalently, 1 the span ( f , f ) is a pregroupoid; 2 the commutator [ Eq ( f ) , Eq ( f )] S is trivial; 3 x 1 E , 1 E y : E Ñ Eq ( f ) is a normal monomorphism f Ñ f π 1 in ( X Ó X ) ; 4 x a , a y : A Ñ Eq ( f ) is a normal monomorphism in X . Example: a split extension (a point ( f , s ) with a = ker ( f ) ) is abelian iff it is a Beck module [Beck, 1967] : an abelian group object in ( X Ó X ) . x a , a y � Eq ( f ) 1 A ¸ f � A ✤ � ✤ � � 0 A ¸ ξ X 0 Given an abelian extension, we may take cokernels as in the diagram on the left to find � s ξ f ξ ❴ π 1 ❴ � x 1 E , 1 E y its direction : the X -module ( A , ξ ) . � A ✤ � � E ✤ � X � 0 0 a f The pullback f ˚ ( ξ ) of ξ along f is the conjugation action of E on A .
The condition (SH) implies that all extensions with abelian kernel are abelian, S is trivial. because A A Q implies that Eq f Eq f In particular then, any internal action on an abelian group object is a Beck module. (Actions are non-abelian modules.) Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . Under (SH), cohomology classifies all extensions with abelian kernel. By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] . X op H A is a derived functor of Hom A X X Ab . We assume that carries a comonad whose projectives are the regular projectives. The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not.
(Actions are non-abelian modules.) Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . Under (SH), cohomology classifies all extensions with abelian kernel. By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] . X op H A is a derived functor of Hom A X X Ab . We assume that carries a comonad whose projectives are the regular projectives. The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not. § The condition (SH) implies that all extensions with abelian kernel are abelian, because [ A , A ] Q = 0 implies that [ Eq ( f ) , Eq ( f )] S is trivial. In particular then, any internal action on an abelian group object is a Beck module.
Theorem (Cohomology with non-trivial coefficients) H X A OpExt X A , the group of equivalence classes of extensions from A to X with direction A . Under (SH), cohomology classifies all extensions with abelian kernel. By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] . X op H A is a derived functor of Hom A X X Ab . We assume that carries a comonad whose projectives are the regular projectives. The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not. § The condition (SH) implies that all extensions with abelian kernel are abelian, because [ A , A ] Q = 0 implies that [ Eq ( f ) , Eq ( f )] S is trivial. In particular then, any internal action on an abelian group object is a Beck module. (Actions are non-abelian modules.)
By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] . X op H A is a derived functor of Hom A X X Ab . We assume that carries a comonad whose projectives are the regular projectives. The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not. § The condition (SH) implies that all extensions with abelian kernel are abelian, because [ A , A ] Q = 0 implies that [ Eq ( f ) , Eq ( f )] S is trivial. In particular then, any internal action on an abelian group object is a Beck module. (Actions are non-abelian modules.) Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) . Under (SH), cohomology classifies all extensions with abelian kernel.
X op H A is a derived functor of Hom A X X Ab . We assume that carries a comonad whose projectives are the regular projectives. The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not. § The condition (SH) implies that all extensions with abelian kernel are abelian, because [ A , A ] Q = 0 implies that [ Eq ( f ) , Eq ( f )] S is trivial. In particular then, any internal action on an abelian group object is a Beck module. (Actions are non-abelian modules.) Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) . Under (SH), cohomology classifies all extensions with abelian kernel. § By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] .
The semi-abelian case: abelian extensions, II § There are examples (e.g. in Loop ) where A is abelian but f is not. § The condition (SH) implies that all extensions with abelian kernel are abelian, because [ A , A ] Q = 0 implies that [ Eq ( f ) , Eq ( f )] S is trivial. In particular then, any internal action on an abelian group object is a Beck module. (Actions are non-abelian modules.) Theorem (Cohomology with non-trivial coefficients) H 2 ( X , ( A , ξ )) – OpExt 1 ( X , A , ξ ) , the group of equivalence classes of extensions from A to X with direction ( A , ξ ) . Under (SH), cohomology classifies all extensions with abelian kernel. § By [Bourn & Janelidze, 2004] , abelian extensions are torsors , which by [Duskin, 1975] [Glenn, 1982] are classified by means of comonadic cohomology [Barr & Beck, 1969] . § H 2 ( ´ , ( A , ξ )) is a derived functor of Hom ( ´ , A ¸ ξ X Ñ X ): ( X Ó X ) op Ñ Ab . We assume that X carries a comonad G whose projectives are the regular projectives.
Overview, n = 1 Cohomology H 2 ( X , ( A , ξ )) Homology H 2 ( X ) trivial action ξ arbitrary action ξ R ^ [ F , F ] CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) Gp [ R , F ] abelian Ext 1 ( X , A ) 0 categories Barr-exact Tors 1 [ X , ( A , ξ )] categories R ^ [ F , F ] semi-abelian CentrExt 1 ( X , A ) OpExt 1 ( X , A , ξ ) categories [ R , F ]
Overview, arbitrary degrees ( n ě 1 ) Cohomology H n +1 ( X , ( A , ξ )) Homology H n +1 ( X ) trivial action ξ arbitrary action ξ i P n K i ^ [ F n , F n ] Ź CentrExt n ( X , A ) OpExt n ( X , A , ξ ) Gp i P I K i , Ź i P n z I K i ] Ž I Ď n [ Ź abelian Ext n ( X , A ) 0 categories Barr-exact Tors n [ X , ( A , ξ )] categories i P n K i ^ [ F n , F n ] Ź semi-abelian CentrExt n ( X , A ) OpExt n ( X , A , ξ ) categories L n [ F ]
Overview, arbitrary degrees ( n ě 1 ) Cohomology H n +1 ( X , ( A , ξ )) Homology H n +1 ( X ) trivial action ξ arbitrary action ξ i P n K i ^ [ F n , F n ] Ź CentrExt n ( X , A ) OpExt n ( X , A , ξ ) Gp i P I K i , Ź i P n z I K i ] Ž I Ď n [ Ź abelian Ext n ( X , A ) 0 categories Barr-exact Tors n [ X , ( A , ξ )] categories i P n K i ^ [ F n , F n ] Ź semi-abelian CentrExt n ( X , A ) OpExt n ( X , A , ξ ) categories L n [ F ]
Consider n . A Yoneda n -extension from A to X is an exact sequence f n f E n E n A X Taking commutative ladders between those as morphisms gives a category EXT n X A . Its set/abelian group of connected components is denoted Ext n X A . Theorem [Yoneda, 1960] Ext n X A . we have H n If has enough projectives, then for n X A op The cohomology on the left is a derived functor of Hom A Ab . How to extend this to semi-abelian categories? Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0
Taking commutative ladders between those as morphisms gives a category EXT n X A . Its set/abelian group of connected components is denoted Ext n X A . Theorem [Yoneda, 1960] Ext n X A . we have H n If has enough projectives, then for n X A op The cohomology on the left is a derived functor of Hom A Ab . How to extend this to semi-abelian categories? Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0 Consider n ě 2 . A Yoneda n -extension from A to X is an exact sequence f n f 1 ✤ � X � A ✤ � � E n � E n ´ 1 � ¨ ¨ ¨ � 0 . 0
Theorem [Yoneda, 1960] Ext n X A . we have H n If has enough projectives, then for n X A op The cohomology on the left is a derived functor of Hom A Ab . How to extend this to semi-abelian categories? Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0 Consider n ě 2 . A Yoneda n -extension from A to X is an exact sequence f n f 1 ✤ � X � A ✤ � � E n � E n ´ 1 � ¨ ¨ ¨ � 0 . 0 Taking commutative ladders between those as morphisms gives a category EXT n ( X , A ) . Its set/abelian group of connected components is denoted Ext n ( X , A ) .
op The cohomology on the left is a derived functor of Hom A Ab . How to extend this to semi-abelian categories? Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0 Consider n ě 2 . A Yoneda n -extension from A to X is an exact sequence f n f 1 ✤ � X � A ✤ � � E n � E n ´ 1 � ¨ ¨ ¨ � 0 . 0 Taking commutative ladders between those as morphisms gives a category EXT n ( X , A ) . Its set/abelian group of connected components is denoted Ext n ( X , A ) . Theorem [Yoneda, 1960] If A has enough projectives, then for n ě 1 we have H n +1 ( X , A ) – Ext n ( X , A ) .
How to extend this to semi-abelian categories? Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0 Consider n ě 2 . A Yoneda n -extension from A to X is an exact sequence f n f 1 ✤ � X � A ✤ � � E n � E n ´ 1 � ¨ ¨ ¨ � 0 . 0 Taking commutative ladders between those as morphisms gives a category EXT n ( X , A ) . Its set/abelian group of connected components is denoted Ext n ( X , A ) . Theorem [Yoneda, 1960] If A has enough projectives, then for n ě 1 we have H n +1 ( X , A ) – Ext n ( X , A ) . § The cohomology on the left is a derived functor of Hom ( ´ , A ): A op Ñ Ab .
Yoneda’s extensions Let X and A be objects in an abelian category A . A Yoneda 1 -extension from A to X is a short exact sequence f 1 ✤ � X � A ✤ � � E 1 � 0 . 0 Consider n ě 2 . A Yoneda n -extension from A to X is an exact sequence f n f 1 ✤ � X � A ✤ � � E n � E n ´ 1 � ¨ ¨ ¨ � 0 . 0 Taking commutative ladders between those as morphisms gives a category EXT n ( X , A ) . Its set/abelian group of connected components is denoted Ext n ( X , A ) . Theorem [Yoneda, 1960] If A has enough projectives, then for n ě 1 we have H n +1 ( X , A ) – Ext n ( X , A ) . § The cohomology on the left is a derived functor of Hom ( ´ , A ): A op Ñ Ab . How to extend this to semi-abelian categories?
The red square F Arr which determines it is a regular pushout : its arrows and the comparison F F F are regular epimorphisms. F op F is usually considered as a functor . n -diagram. An n -fold extension is a Arr n It is determined by an n -fold arrow F , n op an n -cube viewed as a functor . Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an n -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 f 0 ❴ ❴ ❴ � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 0 0 0
op F is usually considered as a functor . n -diagram. An n -fold extension is a Arr n It is determined by an n -fold arrow F , n op an n -cube viewed as a functor . Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an n -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 f 0 ❴ ❴ ❴ � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 0 0 0
n -diagram. An n -fold extension is a Arr n It is determined by an n -fold arrow F , n op an n -cube viewed as a functor . Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an n -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 F is usually considered as a functor P (2) op Ñ X . f 0 ❴ ❴ ❴ � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 0 0 0
Arr n It is determined by an n -fold arrow F , n op an n -cube viewed as a functor . Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an n -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 F is usually considered as a functor P (2) op Ñ X . f 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 0 0 0
Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an n -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 F is usually considered as a functor P (2) op Ñ X . f 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 It is determined by an n -fold arrow F P Arr n ( X ) , an n -cube viewed as a functor P ( n ) op Ñ X . 0 0 0
In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. B 1 ✤ � P 0 � K 1 ✤ � � 0 � P 1 0 F is usually considered as a functor P (2) op Ñ X . B 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ ✤ � X � 0 ✤ � � P 0 0 It is determined by an n -fold arrow F P Arr n ( X ) , an n -cube viewed as a functor P ( n ) op Ñ X . 0 0 0 Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an ( n + 1) -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] .
In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. B 1 ✤ � P 0 � K 1 ✤ � � 0 � P 1 0 F is usually considered as a functor P (2) op Ñ X . B 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ ✤ � X � 0 ✤ � � P 0 0 It is determined by an n -fold arrow F P Arr n ( X ) , an n -cube viewed as a functor P ( n ) op Ñ X . 0 0 0 Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution ) determines an ( n + 1) -fold extension ( presentation ). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] .
In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan). � � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 F is usually considered as a functor P (2) op Ñ X . f 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 It is determined by an n -fold arrow F P Arr n ( X ) , an n -cube viewed as a functor P ( n ) op Ñ X . 0 0 0 Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an ( n + 1) -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] .
� � � � � � � � � � � � � � � Non-abelian higher extensions: 3 n -diagrams A double extension is a 3 ˆ 3 diagram. 0 0 0 Its rows and columns are short exact sequences. The red square F P Arr 2 ( X ) which determines it � K 0 ^ K 1 � K 0 ✤ � ✤ � ¨ � 0 0 ❴ ❴ ❴ is a regular pushout : its arrows and the comparison F 2 Ñ F t 0 u ˆ F ∅ F t 1 u are regular epimorphisms. f 1 ✤ � F t 1 u � K 1 ✤ � � F 2 � 0 0 F is usually considered as a functor P (2) op Ñ X . f 0 ❴ ❴ ❴ An n -fold extension is a 3 n -diagram. � ¨ � F t 0 u ✤ � F ∅ � 0 ✤ � 0 It is determined by an n -fold arrow F P Arr n ( X ) , an n -cube viewed as a functor P ( n ) op Ñ X . 0 0 0 Example: the n -truncation of any aspherical augmented simplicial object (in particular, any simplicial resolution) determines an ( n + 1) -fold extension (presentation). In fact, the extension property characterises being aspherical [Everaert, Goedecke & VdL, 2012] . In the abelian case, Yoneda n -extensions are equivalent to n -fold extensions (by Dold-Kan).
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Abelian case: 3 -fold extension vs. Yoneda 3 -extension ✤ � ✤ � ¨ A ¨ ❄ ❄ ❄ ❴ ❴ ❴ ✤ � ✤ � ¨ ¨ ¨ ❴ ❴ ❴ ❄ ❄ ❄ ¨ ✤ � ✤ � ¨ ¨ ❴ ❴ ❴ ✤ � ✤ � ¨ E 3 ¨ ❄ ❄ ❄ ✤ � ✤ � ¨ ¨ F 3 f 0 f 1 ❄ ❄ ❄ ¨ ✤ � ✤ � ¨ ¨ f 2 ❴ ❴ ❴ ✤ � � ¨ ✤ � ¨ ¨ ❄ ❄ ❄ ❴ ❴ ❴ ✤ � � ¨ ✤ � ¨ E 2 ❴ ❄ ❴ ❄ ❴ ❄ � E 1 ✤ � X ¨ ✤ �
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