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Intrinsic Schreier split extensions Andrea Montoli Diana Rodelo Tim van der Linden Centre for Mathematics of the University of Coimbra University of Algarve, Portugal CT2019 Intrinsic Schreier split extnesions 1 / 14 Schreier (split)

  1. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I s Imaginary addition - II K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 4 / 14

  2. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I non-commutative s Imaginary addition - II K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 4 / 14

  3. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 4 / 14

  4. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 4 / 14

  5. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 4 / 14

  6. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour p S1 q ¨ Schreier split epi ñ p k, s q jointly extremal-epimorphic pair ô p f, s q strong point ñ Schreier split extension CT2019 Intrinsic Schreier split extnesions – 4 / 14

  7. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour p S1 q ¨ Schreier split epi ñ p k, s q jointly extremal-epimorphic pair ô p f, s q strong point ñ Schreier split extension ¨ Categorically: - How to define q ? CT2019 Intrinsic Schreier split extnesions – 4 / 14

  8. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour p S1 q ¨ Schreier split epi ñ p k, s q jointly extremal-epimorphic pair ô p f, s q strong point ñ Schreier split extension ¨ Categorically: - How to define q ? ( not a morphism ) CT2019 Intrinsic Schreier split extnesions – 4 / 14

  9. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour p S1 q ¨ Schreier split epi ñ p k, s q jointly extremal-epimorphic pair ô p f, s q strong point ñ Schreier split extension ¨ Categorically: - How to define q ? ( not a morphism ) - What diagrams give (S1) and (S2) ? CT2019 Intrinsic Schreier split extnesions – 4 / 14

  10. � Towards intrinsic Schreier split epis ¨ This categorical approach for S -protomodularity is not so categorical for Schreier (split) exts of monoids S = Schreier split epis in Mon or J´ onsson–Tarski variety V S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ Definition of Schreier split epi depends on elements [BM-FMS] Unital categories Imaginary addition - I D q non-commutative s Imaginary addition - II � ❴ ❴ ❴ K ✤ � � p X, ` , 0 q � � Y Schreier split epi Intrinsic Schreier split k f epis Properties - I (S1) x “ kq p x q ` sf p x q , @ x P X Properties - II Schreier retraction Main results (S2) q p k p a q ` s p y qq “ a , @ a P K, y P Y ( qk “ 1 K ) Cohomological flavour p S1 q ¨ Schreier split epi ñ p k, s q jointly extremal-epimorphic pair ô p f, s q strong point ñ Schreier split extension ¨ Categorically: - How to define q ? ( not a morphism ) ( recover Mon { V ) - What diagrams give (S1) and (S2) ? CT2019 Intrinsic Schreier split extnesions – 4 / 14

  11. Imaginary morphisms - I ¨ [BJ] Imaginary morphisms Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  12. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  13. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  14. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  15. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II Intrinsic Schreier split epis - @ f : X Ñ Y , fε X “ ε Y P p f q Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  16. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II Intrinsic Schreier split epis - @ f : X Ñ Y , fε X “ ε Y P p f q Properties - I Properties - II - p P : C Ñ C , δ : P ñ P 2 , ε : P ñ 1 C q comonad Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  17. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective C has functorial ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II (comonadic) Intrinsic Schreier split epis - @ f : X Ñ Y , fε X “ ε Y P p f q projective covers Properties - I Properties - II - p P : C Ñ C , δ : P ñ P 2 , ε : P ñ 1 C q comonad Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 5 / 14

  18. Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective C has functorial ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II (comonadic) Intrinsic Schreier split epis - @ f : X Ñ Y , fε X “ ε Y P p f q projective covers Properties - I Properties - II - p P : C Ñ C , δ : P ñ P 2 , ε : P ñ 1 C q comonad Main results Cohomological flavour ¨ Def. An imaginary morphism from X to Y , denoted X ��� Y , is a real morphism P p X q Ñ Y CT2019 Intrinsic Schreier split extnesions – 5 / 14

  19. � Imaginary morphisms - I morphism � K q : X function � ¨ [BJ] Imaginary morphisms ù P p X q K ❴ ❴ ❴ Schreier (split) exts of monoids r x s ÞÝ Ñ q p x q S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I ¨ C regular cat w/ enough projectives Imaginary morphisms - II Unital categories projective C has functorial ε X � � X Imaginary addition - I - P p X q regular epi Imaginary addition - II (comonadic) Intrinsic Schreier split epis - @ f : X Ñ Y , fε X “ ε Y P p f q projective covers Properties - I Properties - II - p P : C Ñ C , δ : P ñ P 2 , ε : P ñ 1 C q comonad Main results Cohomological flavour ¨ Def. An imaginary morphism from X to Y , denoted X ��� Y , is a real morphism P p X q Ñ Y s K ✤ � � X f � � Y in C k � P P P P q P p X q imaginary (Schreier) retraction CT2019 Intrinsic Schreier split extnesions – 5 / 14

  20. � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 6 / 14

  21. � � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity 1 Y 1 Y ε Y � � Y Towards intrinsic � Y real ù imaginary ( P p Y q ) Y Y Y ❴ ❴ ❴ Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 6 / 14

  22. � � � � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity 1 Y 1 Y ε Y � � Y Towards intrinsic � Y real ù imaginary ( P p Y q ) Y Y Y ❴ ❴ ❴ Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I f g f g � Z � Y � Z ¨ X ù P p X q Imaginary addition - II Y ❴ ❴ ❴ P ❯ ❩ ❴ ❞ ✐ ♥ Intrinsic Schreier split epis g ˝ f Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 6 / 14

  23. � � � � � � � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity 1 Y 1 Y ε Y � � Y Towards intrinsic � Y real ù imaginary ( P p Y q ) Y Y Y ❴ ❴ ❴ Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I f g f g � Z � Y � Z ¨ X ù P p X q Imaginary addition - II Y ❴ ❴ ❴ P ❯ ❩ ❴ ❞ ✐ ♥ Intrinsic Schreier split epis g ˝ f Properties - I Properties - II P p h q � P p X q Main results f f h � Y P p W q W X Y ù ❴ ❴ ❴ Cohomological flavour ◗ ❱ ❩ ❴ ❞ ❤ ♠ f ˝ h CT2019 Intrinsic Schreier split extnesions – 6 / 14

  24. � � � � � � � � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity 1 Y 1 Y ε Y � � Y Towards intrinsic � Y real ù imaginary ( P p Y q ) Y Y Y ❴ ❴ ❴ Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I f g f g � Z � Y � Z ¨ X ù P p X q Imaginary addition - II Y ❴ ❴ ❴ P ❯ ❩ ❴ ❞ ✐ ♥ Intrinsic Schreier split epis g ˝ f Properties - I Properties - II P p h q � P p X q Main results f f h � Y P p W q W X Y ù ❴ ❴ ❴ Cohomological flavour ◗ ❱ ❩ ❴ ❞ ❤ ♠ f ˝ h f � � Y s ¨ X regular epi ô D imaginary splitting Y X ❴ ❴ ❴ CT2019 Intrinsic Schreier split extnesions – 6 / 14

  25. � � � � � � � � � Imaginary morphisms - II f f f � Y ) ε X � � X � Y ¨ X real ù imaginary ( P p X q X Y ❴ ❴ ❴ Schreier (split) exts of monoids S -protomodularity 1 Y 1 Y ε Y � � Y Towards intrinsic � Y real ù imaginary ( P p Y q ) Y Y Y ❴ ❴ ❴ Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I f g f g � Z � Y � Z ¨ X ù P p X q Imaginary addition - II Y ❴ ❴ ❴ P ❯ ❩ ❴ ❞ ✐ ♥ Intrinsic Schreier split epis g ˝ f Properties - I Properties - II P p h q � P p X q Main results f f h � Y P p W q W X Y ù ❴ ❴ ❴ Cohomological flavour ◗ ❱ ❩ ❴ ❞ ❤ ♠ f ˝ h f � � Y s ¨ X regular epi ô D imaginary splitting Y X ❴ ❴ ❴ f � Y f � Y s � s � ( ) Y X ❴ ❴ P p Y q X � � ❘ ❧ ❴ fs “ εY f ˝ s “ 1 Y CT2019 Intrinsic Schreier split extnesions – 6 / 14

  26. Unital categories ¨ Mon unital category Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 7 / 14

  27. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 7 / 14

  28. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 7 / 14

  29. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I D t A,B � r A,B � � A ˆ B Imaginary addition - II iff D imaginary splitting P p A ˆ B q A ` B Intrinsic Schreier split � � epis ( + projs ) p˚q Properties - I Properties - II r A,B t A,B “ ε A ˆ B Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 7 / 14

  30. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I D t A,B � r A,B � � A ˆ B Imaginary addition - II iff D imaginary splitting P p A ˆ B q A ` B Intrinsic Schreier split � � epis ( + projs ) p˚q Properties - I Properties - II r A,B t A,B “ ε A ˆ B Main results Cohomological flavour ¨ V J´ onsson–Tarski variety ù D t A,B : P p A ˆ B q Ñ A ` B rp a, b qs ÞÑ a ` b CT2019 Intrinsic Schreier split extnesions – 7 / 14

  31. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I D t A,B � r A,B � � A ˆ B Imaginary addition - II iff D imaginary splitting P p A ˆ B q A ` B Intrinsic Schreier split � � epis ( + projs ) p˚q Properties - I Properties - II r A,B t A,B “ ε A ˆ B Main results Cohomological flavour ¨ V J´ onsson–Tarski variety ù D t A,B : P p A ˆ B q Ñ A ` B rp a, b qs ÞÑ a ` b CT2019 Intrinsic Schreier split extnesions – 7 / 14

  32. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I D t A,B � r A,B � � A ˆ B Imaginary addition - II iff D imaginary splitting P p A ˆ B q A ` B Intrinsic Schreier split � � epis ( + projs ) p˚q Properties - I Properties - II r A,B t A,B “ ε A ˆ B Main results Cohomological flavour ¨ V J´ onsson–Tarski variety ù D t A,B : P p A ˆ B q Ñ A ` B rp a, b qs ÞÑ a ` b natural transformation CT2019 Intrinsic Schreier split extnesions – 7 / 14

  33. Unital categories ¨ Mon unital category ù J´ onsson–Tarski variety ( x ` 0 “ x “ 0 ` x ) Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis ¨ C pointed + regular + binary coproducts is unital Imaginary morphisms - I Imaginary morphisms - II v 1 0 w iff @ r A,B “ : A ` B ։ A ˆ B regular epi Unital categories 0 1 Imaginary addition - I D t A,B � r A,B � � A ˆ B Imaginary addition - II iff D imaginary splitting P p A ˆ B q A ` B Intrinsic Schreier split � � epis ( + projs ) p˚q Properties - I Properties - II r A,B t A,B “ ε A ˆ B Main results Cohomological flavour ¨ V J´ onsson–Tarski variety ù D t A,B : P p A ˆ B q Ñ A ` B rp a, b qs ÞÑ a ` b natural transformation ¨ natural imaginary splitting: t : P pp¨qˆp¨qq ñ p¨q`p¨q sth (*) in C CT2019 Intrinsic Schreier split extnesions – 7 / 14

  34. Imaginary addition - I ù µ X : X ˆ X ��� X ¨ t natural imaginary addition Schreier (split) exts of monoids S -protomodularity t X,X � X ` X p 1 1 q � X Towards intrinsic P p X ˆ X q Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 8 / 14

  35. � � � � � Imaginary addition - I ù µ X : X ˆ X ��� X ¨ t natural imaginary addition Schreier (split) exts of monoids S -protomodularity t X,X � X ` X p 1 1 q � X Towards intrinsic P p X ˆ X q Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I µ X ˝x 1 , 0 y“ 1 X Imaginary addition - II ❵ ❴ ❴ ¨ X ❫ ❪ ❬ ❨ ❲ ❘ ❍ Intrinsic Schreier split ❙ ❙ epis x 1 , 0 y ❙ Properties - I ❙ µ X Properties - II X ˆ X X ❴ ❴ ❴ ❴ Main results ❦ ❦ ❦ Cohomological flavour ❝ ❡ ❤ ❧ ✈ x 0 , 1 y ❦ ❦ ❦ X ❛ ❫ ❴ ❴ ❵ µ X ˝x 0 , 1 y“ 1 X CT2019 Intrinsic Schreier split extnesions – 8 / 14

  36. � � � � � Imaginary addition - I ù µ X : X ˆ X ��� X ¨ t natural imaginary addition Schreier (split) exts of monoids S -protomodularity t X,X � X ` X p 1 1 q � X Towards intrinsic P p X ˆ X q Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I µ X ˝x 1 , 0 y“ 1 X Imaginary addition - II ❵ ❴ ❴ ¨ X ❫ ❪ ❬ ❨ ❲ ❘ ❍ Intrinsic Schreier split ❙ ❙ epis x 1 , 0 y ❙ Properties - I ❙ µ X ( pps of t ) Properties - II X ˆ X X ❴ ❴ ❴ ❴ Main results ❦ ❦ ❦ Cohomological flavour ❝ ❡ ❤ ❧ ✈ x 0 , 1 y ❦ ❦ ❦ X ❛ ❫ ❴ ❴ ❵ µ X ˝x 0 , 1 y“ 1 X CT2019 Intrinsic Schreier split extnesions – 8 / 14

  37. � � � � � � � � Imaginary addition - I ù µ X : X ˆ X ��� X ¨ t natural imaginary addition Schreier (split) exts of monoids S -protomodularity t X,X � X ` X p 1 1 q � X Towards intrinsic P p X ˆ X q Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I µ X ˝x 1 , 0 y“ 1 X Imaginary addition - II ❵ ❴ ❴ ¨ X ❫ ❪ ❬ ❨ ❲ ❘ ❍ Intrinsic Schreier split ❙ ❙ epis x 1 , 0 y ❙ Properties - I ❙ µ X ( pps of t ) Properties - II X ˆ X X ❴ ❴ ❴ ❴ Main results ❦ ❦ ❦ Cohomological flavour ❝ ❡ ❤ ❧ ✈ x 0 , 1 y ❦ ❦ ❦ X ❛ ❫ ❴ ❴ ❵ µ X ˝x 0 , 1 y“ 1 X µ X � f ˝ µ X “ µ Y ˝ p f ˆ f q ¨ @ f : X Ñ Y, X ˆ X X ❴ ❴ ❴ f f ˆ f Y ˆ Y Y ❴ ❴ ❴ µ Y CT2019 Intrinsic Schreier split extnesions – 8 / 14

  38. � � � � � � � � Imaginary addition - I ù µ X : X ˆ X ��� X ¨ t natural imaginary addition Schreier (split) exts of monoids S -protomodularity t X,X � X ` X p 1 1 q � X Towards intrinsic P p X ˆ X q Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I µ X ˝x 1 , 0 y“ 1 X Imaginary addition - II ❵ ❴ ❴ ¨ X ❫ ❪ ❬ ❨ ❲ ❘ ❍ Intrinsic Schreier split ❙ ❙ epis x 1 , 0 y ❙ Properties - I ❙ µ X ( pps of t ) Properties - II X ˆ X X ❴ ❴ ❴ ❴ Main results ❦ ❦ ❦ Cohomological flavour ❝ ❡ ❤ ❧ ✈ x 0 , 1 y ❦ ❦ ❦ X ❛ ❫ ❴ ❴ ❵ µ X ˝x 0 , 1 y“ 1 X µ X � f ˝ µ X “ µ Y ˝ p f ˆ f q ¨ @ f : X Ñ Y, X ˆ X X ❴ ❴ ❴ f f ˆ f ( naturality of t ) Y ˆ Y Y ❴ ❴ ❴ µ Y CT2019 Intrinsic Schreier split extnesions – 8 / 14

  39. � Imaginary addition - II g h ¨ A � X A Schreier (split) exts of monoids S -protomodularity g p a q ` h p a q Towards intrinsic Schreier split epis Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II A X ❴ ❴ ❴ Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  40. � � Imaginary addition - II g h ¨ A � X P p A q A Schreier (split) exts of monoids S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y Schreier split epis Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  41. � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q g h ¨ A � X P p A q A Schreier (split) exts of monoids � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y Schreier split epis P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  42. � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  43. � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  44. � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 9 / 14

  45. � � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour g j ¨ A � X B g p a q ` j p b q µ X � g ˆ j � X ˆ X A ˆ B X ❴ ❴ ❴ CT2019 Intrinsic Schreier split extnesions – 9 / 14

  46. � � � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour g j ¨ A � X P p A ˆ B q B g p a q ` j p b q P p g ˆ j q µ X � g ˆ j � X ˆ X � X ` X p 1 1 q � X P p X ˆ X q A ˆ B X ❴ ❴ ❴ t X,X CT2019 Intrinsic Schreier split extnesions – 9 / 14

  47. � � � � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour t A,B � g j ¨ A � X P p A ˆ B q A ` B B g p a q ` j p b q P p g ˆ j q g ` j nt µ X � g ˆ j � X ˆ X � X ` X p 1 1 q � X P p X ˆ X q A ˆ B X ❴ ❴ ❴ t X,X CT2019 Intrinsic Schreier split extnesions – 9 / 14

  48. � � � � � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour t A,B � g j ¨ A � X P p A ˆ B q A ` B B ❉ ❉ ❉ ❉ p g j q g p a q ` j p b q ❉ P p g ˆ j q g ` j ❉ nt ❉ ❉ ❉ ❉ µ X � g ˆ j � X ˆ X � X ` X p 1 1 q � X P p X ˆ X q A ˆ B X ❴ ❴ ❴ t X,X CT2019 Intrinsic Schreier split extnesions – 9 / 14

  49. � � � � � � � Imaginary addition - II P x 1 , 1 y � P p A ˆ A q t A,A � g h ¨ A � X P p A q A ` A A Schreier (split) exts of monoids � ✉✉✉✉✉✉✉✉✉✉✉✉ � ssssssssssss S -protomodularity g p a q ` h p a q Towards intrinsic P x g,h y p g h q nt Schreier split epis g ` h P p g ˆ h q Imaginary morphisms - I µ X � x g,h y � X ˆ X Imaginary morphisms - II � X ` X � X P p X ˆ X q A X ❴ ❴ ❴ Unital categories t X,X p 1 1 q Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour t A,B � g j ¨ A � X P p A ˆ B q A ` B B ❉ ❉ ❉ ❉ p g j q g p a q ` j p b q ❉ P p g ˆ j q g ` j ❉ nt ❉ ❉ ❉ ❉ µ X � g ˆ j � X ˆ X � X ` X p 1 1 q � X P p X ˆ X q A ˆ B X ❴ ❴ ❴ t X,X CT2019 Intrinsic Schreier split extnesions – 9 / 14

  50. Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 10 / 14

  51. � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I s Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 10 / 14

  52. � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 10 / 14

  53. � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis (iS1) Properties - I Properties - II p S1 q Main results x “ kq p x q ` sf p x q Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 10 / 14

  54. � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q x “ kq p x q ` sf p x q Cohomological flavour X CT2019 Intrinsic Schreier split extnesions – 10 / 14

  55. � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q x “ kq p x q ` sf p x q Cohomological flavour � � X P p X q ε X CT2019 Intrinsic Schreier split extnesions – 10 / 14

  56. � � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q δ X x “ kq p x q ` sf p x q Cohomological flavour P p X q � � X ε X CT2019 Intrinsic Schreier split extnesions – 10 / 14

  57. � � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q δ X x “ kq p x q ` sf p x q Cohomological flavour P p X q � � X ε X (iS2) p S2 q “ q p k p a q ` s p y qq a CT2019 Intrinsic Schreier split extnesions – 10 / 14

  58. � � � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q δ X x “ kq p x q ` sf p x q Cohomological flavour P p X q � � X ε X P p t K,Y q � P p K ` Y q P p k s q � P p X q P 2 p K ˆ Y q (iS2) p S2 q q “ q p k p a q ` s p y qq a K CT2019 Intrinsic Schreier split extnesions – 10 / 14

  59. � � � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q δ X x “ kq p x q ` sf p x q Cohomological flavour P p X q � � X ε X P p t K,Y q � P p K ` Y q P p k s q � P p X q P 2 p K ˆ Y q (iS2) p S2 q q “ q p k p a q ` s p y qq a � � K ˆ Y � � K P p K ˆ Y q ε K ˆ Y π K CT2019 Intrinsic Schreier split extnesions – 10 / 14

  60. � � � � � Intrinsic Schreier split epis ¨ C regular unital category w/ binary coproducts, functorial projective Schreier (split) exts of monoids covers and natural imaginary splitting t S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I D q s � ❴ ❴ ❴ ❴ Imaginary morphisms - II K ✤ � ¨ � X � � Y intrinsic Schreier split epi Unital categories k f Imaginary addition - I Imaginary addition - II Intrinsic Schreier split t P p X q ,P p X q P x 1 , 1 y � P p P p X q ˆ P p X qq � P p X q ` P p X q epis P 2 p X q (iS1) Properties - I Properties - II p S1 q Main results p kq sfε X q δ X x “ kq p x q ` sf p x q Cohomological flavour P p X q � � X ε X P p t K,Y q � P p K ` Y q P p k s q � P p X q P 2 p K ˆ Y q (iS2) p S2 q δ K ˆ Y q “ q p k p a q ` s p y qq a � � K P p K ˆ Y q � � K ˆ Y ε K ˆ Y π K CT2019 Intrinsic Schreier split extnesions – 10 / 14

  61. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 11 / 14

  62. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 11 / 14

  63. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 11 / 14

  64. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 11 / 14

  65. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results ¨ q is unique Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 11 / 14

  66. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results ¨ q is unique Cohomological flavour ¨ (iS1) ñ p k s q : K ` Y ։ X regular epi CT2019 Intrinsic Schreier split extnesions – 11 / 14

  67. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results ¨ q is unique Cohomological flavour ¨ (iS1) ñ p k s q : K ` Y ։ X regular epi ñ p k, s q jointly extremal-epimorphic pair { p f, s q strong CT2019 Intrinsic Schreier split extnesions – 11 / 14

  68. Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results ¨ q is unique Cohomological flavour ¨ (iS1) ñ p k s q : K ` Y ։ X regular epi ñ p k, s q jointly extremal-epimorphic pair { p f, s q strong ñ Schreier split epi ñ Schreier split extension CT2019 Intrinsic Schreier split extnesions – 11 / 14

  69. � Properties - I ¨ q : X ��� K imaginary Schreier retraction Schreier (split) exts of monoids qk “ 1 K in Mon ù q ˝ k “ 1 K ô qP p k q “ ε K in C S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II ¨ qs “ 0 in Mon ù qP p s q “ 0 in C Unital categories Imaginary addition - I q p 0 q “ 0 in Mon ù obvious in C Imaginary addition - II Intrinsic Schreier split kq p s p y q ` k p a qq ` s p y q “ s p y q ` k p a q in Mon ù � in C epis Properties - I Properties - II Main results ¨ q is unique Cohomological flavour ¨ (iS1) ñ p k s q : K ` Y ։ X regular epi ñ p k, s q jointly extremal-epimorphic pair { p f, s q strong ñ Schreier split epi ñ Schreier split extension x 0 , 1 Y y ¨ X ✤ � � X ˆ Y � � Y intrinsic Schreier split extension x 1 X , 0 y π Y CT2019 Intrinsic Schreier split extnesions – 11 / 14

  70. � � � � � Properties - II s k ✤ � � X ¨ compatibility K Y � � Schreier (split) exts of monoids f S -protomodularity ρ g h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I K 1 ✤ � � X 1 f 1 � � Y 1 Imaginary morphisms - II k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 12 / 14

  71. � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 12 / 14

  72. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y epis K ✤ � ¨ � Z ˆ Y X Z Properties - I � � Properties - II π Z x 0 ,k y Main results g π X Cohomological flavour s K ✤ � � X � � Y k f CT2019 Intrinsic Schreier split extnesions – 12 / 14

  73. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y q ˝ π X epis K ✤ � � ❴ ❴ ❴ ❴ ¨ � Z ˆ Y X Z (iS1) Properties - I � � Properties - II π Z x 0 ,k y ñ Main results g π X Cohomological flavour q s K ✤ � � ❴ ❴ ❴ ❴ ❴ � X � � Y (iS1) k f CT2019 Intrinsic Schreier split extnesions – 12 / 14

  74. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y q ˝ π X epis K ✤ � � ❴ ❴ ❴ ❴ ¨ � Z ˆ Y X Z (iS1) Properties - I p π Z , x 1 , sg yq strong � � Properties - II π Z x 0 ,k y ñ Main results g π X Cohomological flavour q s K ✤ � � ❴ ❴ ❴ ❴ ❴ � X � � Y (iS1) p f, s q strong k f CT2019 Intrinsic Schreier split extnesions – 12 / 14

  75. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y q ˝ π X epis K ✤ � � ❴ ❴ ❴ ❴ ¨ � Z ˆ Y X Z (iS1) Properties - I p π Z , x 1 , sg yq strong � � Properties - II π Z x 0 ,k y ñ Main results g π X Cohomological flavour q s K ✤ � � ❴ ❴ ❴ ❴ ❴ � X � � Y (iS1) p f, s q strong k f ù p f, s q satisfies (iS1) ñ p f, s q stably strong CT2019 Intrinsic Schreier split extnesions – 12 / 14

  76. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y q ˝ π X epis K ✤ � � ❴ ❴ ❴ ❴ ¨ � Z ˆ Y X Z (iS1) Properties - I p π Z , x 1 , sg yq strong � � Properties - II π Z x 0 ,k y ñ Main results g π X Cohomological flavour q s K ✤ � � ❴ ❴ ❴ ❴ ❴ � X � � Y (iS1) p f, s q strong k f ù p f, s q satisfies (iS1) ñ p f, s q stably strong ¨ [MRVdL] Y protomodular object: all points X Ô Y are stably strong CT2019 Intrinsic Schreier split extnesions – 12 / 14

  77. � � � � � � � � � � Properties - II s q k ✤ � � K � X ¨ P p X q compatibility Y � � Schreier (split) exts of monoids f S -protomodularity ρq “ q 1 P p g q ρ g P p g q h Towards intrinsic Schreier split epis s 1 Imaginary morphisms - I � K 1 ✤ � � X 1 P p X 1 q f 1 � � Y 1 Imaginary morphisms - II q 1 k 1 Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split x 1 ,sg y q ˝ π X epis K ✤ � � ❴ ❴ ❴ ❴ ¨ � Z ˆ Y X Z (iS1) Properties - I p π Z , x 1 , sg yq strong � � Properties - II π Z x 0 ,k y ñ Main results g π X Cohomological flavour q s K ✤ � � ❴ ❴ ❴ ❴ ❴ � X � � Y (iS1) p f, s q strong k f ù p f, s q satisfies (iS1) ñ p f, s q stably strong ¨ [MRVdL] Y protomodular object: all points X Ô Y are stably strong ù If all X Ô Y satisfy (iS1) , then Y is a protomodular object CT2019 Intrinsic Schreier split extnesions – 12 / 14

  78. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 13 / 14

  79. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 13 / 14

  80. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi / right homogeneous split epi [BM-FMS] Unital categories Imaginary addition - I Imaginary addition - II ( (S1) x “ kq p x q ` sf p x q ; (S2) q p k p a q ` s p y qq “ a ) Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 13 / 14

  81. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi / right homogeneous split epi [BM-FMS] Unital categories Imaginary addition - I Imaginary addition - II ( (S1) x “ kq p x q ` sf p x q ; (S2) q p k p a q ` s p y qq “ a ) Intrinsic Schreier split epis Properties - I - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B Properties - II Main results rp a, b qs ÞÑ b ` a Cohomological flavour CT2019 Intrinsic Schreier split extnesions – 13 / 14

  82. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi / right homogeneous split epi [BM-FMS] Unital categories Imaginary addition - I Imaginary addition - II ( (S1) x “ kq p x q ` sf p x q ; (S2) q p k p a q ` s p y qq “ a ) Intrinsic Schreier split epis Properties - I - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B Properties - II Main results rp a, b qs ÞÑ b ` a Cohomological flavour = left homogeneous split epi [BM-FMS] CT2019 Intrinsic Schreier split extnesions – 13 / 14

  83. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi / right homogeneous split epi [BM-FMS] Unital categories Imaginary addition - I Imaginary addition - II ( (S1) x “ kq p x q ` sf p x q ; (S2) q p k p a q ` s p y qq “ a ) Intrinsic Schreier split epis Properties - I - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B Properties - II Main results rp a, b qs ÞÑ b ` a Cohomological flavour = left homogeneous split epi [BM-FMS] ( (S1) ’ x “ sf p x q ` kq p x q ; (S2) ’ q p s p y q ` k p a qq “ a ) CT2019 Intrinsic Schreier split extnesions – 13 / 14

  84. Main results ¨ Thm. In Mon (or any J´ onsson–Tarski variety V ): Schreier (split) exts of monoids - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B S -protomodularity Towards intrinsic Schreier split epis rp a, b qs ÞÑ a ` b Imaginary morphisms - I Imaginary morphisms - II = Schreier split epi / right homogeneous split epi [BM-FMS] Unital categories Imaginary addition - I Imaginary addition - II ( (S1) x “ kq p x q ` sf p x q ; (S2) q p k p a q ` s p y qq “ a ) Intrinsic Schreier split epis Properties - I - intrinsic Schreier split epi wrt t A,B : P p A ˆ B q Ñ A ` B Properties - II Main results rp a, b qs ÞÑ b ` a Cohomological flavour = left homogeneous split epi [BM-FMS] ( (S1) ’ x “ sf p x q ` kq p x q ; (S2) ’ q p s p y q ` k p a qq “ a ) ¨ Thm. C regular unital category w/ binary coproducts, functorial proj covers and a nat imaginary splitting. C is S -protomodular for S = the class of intrinsic Schreier split epis CT2019 Intrinsic Schreier split extnesions – 13 / 14

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