Post on 22-Aug-2021
CT2019 Intrinsic Schreier split extnesions – 1 / 14
Intrinsic Schreier split extensions
Andrea Montoli
Diana Rodelo
Tim van der Linden
Centre for Mathematics of the University of Coimbra
University of Algarve, Portugal
Schreier (split) exts of monoids
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 2 / 14
¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon
Schreier (split) exts of monoids
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 2 / 14
¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon
¨ [P, M-FMS] Schreier split extensions in Mon Ø monoid actions
Schreier (split) exts of monoids
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 2 / 14
¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon
¨ [P, M-FMS] Schreier split extensions in Mon Ø monoid actions
B Ñ EndpXq
Schreier (split) exts of monoids
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 2 / 14
¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon
¨ [P, M-FMS] Schreier split extensions in Mon Ø monoid actions
B Ñ EndpXq
¨ [BM-FMS] Schreier split exts in Mon have classical (co)homological pps
of split exts in Gp ( Split Short Five Lemma )
Schreier (split) exts of monoids
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 2 / 14
¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon
¨ [P, M-FMS] Schreier split extensions in Mon Ø monoid actions
B Ñ EndpXq
¨ [BM-FMS] Schreier split exts in Mon have classical (co)homological pps
of split exts in Gp ( Split Short Five Lemma )
ù Study Schreier (split) extensions categorically
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodular
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
- S -protomodular cat
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
- S -protomodular cat ( S : pb-stable; strong; closed under finite lims )
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
- S -protomodular cat ( S : pb-stable; strong; closed under finite lims )
¨ Protomodular cats Ø pps wrt split epis
S -protomodular cats Ø pps wrt split epis in S ( SS5L )
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
- S -protomodular cat ( S : pb-stable; strong; closed under finite lims )
¨ Protomodular cats Ø pps wrt split epis
S -protomodular cats Ø pps wrt split epis in S ( SS5L )
¨ Ex: S = class of Schreier split epis of monoids
Mon is S -protomodular
S -protomodularity
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 3 / 14
¨ Gp : protomodular [B] Mon : non-protomodularooSchreier
¨ [BM-FMS] - Study protomodularity wrt class S of suitable split epis
- S -protomodular cat ( S : pb-stable; strong; closed under finite lims )
¨ Protomodular cats Ø pps wrt split epis
S -protomodular cats Ø pps wrt split epis in S ( SS5L )
¨ Ex: S = class of Schreier split epis of monoids
Mon is S -protomodular
¨ Ex: S = Schreier split epis of Jonsson–Tarski algebras px ` 0 “ x “ 0 ` xq
Any Jonsson–Tarski variety V is S -protomodular
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epi
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutative
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P Y
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
¨ Schreier split epipS1qñ pk, sq jointly extremal-epimorphic pair
ô pf, sq strong point ñ Schreier split extension
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
¨ Schreier split epipS1qñ pk, sq jointly extremal-epimorphic pair
ô pf, sq strong point ñ Schreier split extension
¨ Categorically: - How to define q?
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
¨ Schreier split epipS1qñ pk, sq jointly extremal-epimorphic pair
ô pf, sq strong point ñ Schreier split extension
¨ Categorically: - How to define q? ( not a morphism )
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
¨ Schreier split epipS1qñ pk, sq jointly extremal-epimorphic pair
ô pf, sq strong point ñ Schreier split extension
¨ Categorically: - How to define q? ( not a morphism )
- What diagrams give (S1) and (S2)?
Towards intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 4 / 14
¨ This categorical approach for S -protomodularity is not so categorical for
S = Schreier split epis in Mon or Jonsson–Tarski variety V
¨ Definition of Schreier split epi depends on elements [BM-FMS]
K ✤ ,2k
// pX, `, 0qf
// // Ysoo
Schreier split epinon-commutativeD qoo❴ ❴ ❴
(S1) x “ kqpxq ` sfpxq, @xP X
(S2) qpkpaq ` spyqq “ a, @aP K, y P YSchreier retraction
( qk “ 1K )
¨ Schreier split epipS1qñ pk, sq jointly extremal-epimorphic pair
ô pf, sq strong point ñ Schreier split extension
¨ Categorically: - How to define q? ( not a morphism )
- What diagrams give (S1) and (S2)?( recover Mon{V )
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
- @f : X Ñ Y , fεX “ εY P pfq
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
- @f : X Ñ Y , fεX “ εY P pfq
- pP : C Ñ C, δ : P ñ P 2, ε : P ñ 1Cq comonad
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
- @f : X Ñ Y , fεX “ εY P pfq
- pP : C Ñ C, δ : P ñ P 2, ε : P ñ 1Cq comonad
C has functorial
(comonadic)
projective covers
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
- @f : X Ñ Y , fεX “ εY P pfq
- pP : C Ñ C, δ : P ñ P 2, ε : P ñ 1Cq comonad
C has functorial
(comonadic)
projective covers
¨ Def. An imaginary morphism from X to Y , denoted X 99K Y , is a
real morphism P pXq Ñ Y
Imaginary morphisms - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 5 / 14
¨ [BJ] Imaginary morphisms q : Xfunction//❴❴❴ K ù P pXq
morphism // K
rxs ÞÝÑ qpxq
¨ C regular cat w/ enough projectives
- P pXqεX // // X regular epi
projective
- @f : X Ñ Y , fεX “ εY P pfq
- pP : C Ñ C, δ : P ñ P 2, ε : P ñ 1Cq comonad
C has functorial
(comonadic)
projective covers
¨ Def. An imaginary morphism from X to Y , denoted X 99K Y , is a
real morphism P pXq Ñ Y
K✤ ,2
k// X
f// // Y
soo in C
P pXqq
hhPPPPimaginary (Schreier) retraction
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Y1Y // Y real ù Y
1Y //❴❴❴ Y imaginary ( P pY qεY // // Y )
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Y1Y // Y real ù Y
1Y //❴❴❴ Y imaginary ( P pY qεY // // Y )
¨ Xf //❴❴❴
g˝f
77P❯ ❩ ❴ ❞ ✐
♥Y
g // Z ù P pXqf // Y
g // Z
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Y1Y // Y real ù Y
1Y //❴❴❴ Y imaginary ( P pY qεY // // Y )
¨ Xf //❴❴❴
g˝f
77P❯ ❩ ❴ ❞ ✐
♥Y
g // Z ù P pXqf // Y
g // Z
Wh //
f˝h
77◗❱ ❩ ❴ ❞ ❤
♠X
f //❴❴❴ Y ù P pW qP phq // P pXq
f // Y
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Y1Y // Y real ù Y
1Y //❴❴❴ Y imaginary ( P pY qεY // // Y )
¨ Xf //❴❴❴
g˝f
77P❯ ❩ ❴ ❞ ✐
♥Y
g // Z ù P pXqf // Y
g // Z
Wh //
f˝h
77◗❱ ❩ ❴ ❞ ❤
♠X
f //❴❴❴ Y ù P pW qP phq // P pXq
f // Y
¨ Xf // // Y regular epi ô D imaginary splitting Y
s //❴❴❴ X
Imaginary morphisms - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 6 / 14
¨ Xf // Y real ù X
f //❴❴❴ Y imaginary ( P pXqεX// // X
f // Y )
Y1Y // Y real ù Y
1Y //❴❴❴ Y imaginary ( P pY qεY // // Y )
¨ Xf //❴❴❴
g˝f
77P❯ ❩ ❴ ❞ ✐
♥Y
g // Z ù P pXqf // Y
g // Z
Wh //
f˝h
77◗❱ ❩ ❴ ❞ ❤
♠X
f //❴❴❴ Y ù P pW qP phq // P pXq
f // Y
¨ Xf // // Y regular epi ô D imaginary splitting Y
s //❴❴❴ X
Ys //❴❴
f˝s“1Y
99❘❴ ❧
Xf // Y P pY q
s //
fs“εY
99 99Xf // Y( )
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
iff D imaginary splitting
( + projs )
P pA ˆ BqD tA,B //
rA,BtA,B“εAˆB
p˚q66 66A ` B
rA,B // // A ˆ B
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
iff D imaginary splitting
( + projs )
P pA ˆ BqD tA,B //
rA,BtA,B“εAˆB
p˚q66 66A ` B
rA,B // // A ˆ B
¨ V Jonsson–Tarski variety ù D tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
iff D imaginary splitting
( + projs )
P pA ˆ BqD tA,B //
rA,BtA,B“εAˆB
p˚q66 66A ` B
rA,B // // A ˆ B
¨ V Jonsson–Tarski variety ù D tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
iff D imaginary splitting
( + projs )
P pA ˆ BqD tA,B //
rA,BtA,B“εAˆB
p˚q66 66A ` B
rA,B // // A ˆ B
¨ V Jonsson–Tarski variety ù D tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
natural transformation
Unital categories
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 7 / 14
¨ Mon unital category ù Jonsson–Tarski variety ( x ` 0 “ x “ 0 ` x )
¨ C pointed + regular + binary coproducts is unital
iff @rA,B “v
1 00 1
w
: A ` B ։ A ˆ B regular epi
iff D imaginary splitting
( + projs )
P pA ˆ BqD tA,B //
rA,BtA,B“εAˆB
p˚q66 66A ` B
rA,B // // A ˆ B
¨ V Jonsson–Tarski variety ù D tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
natural transformation
¨ natural imaginary splitting: t : P pp¨qˆp¨qq ñ p¨q`p¨q sth (*) in C
Imaginary addition - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 8 / 14
¨ t ù µX: X ˆ X 99K X natural imaginary addition
P pX ˆ XqtX,X // X ` X
p1 1q // X
Imaginary addition - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 8 / 14
¨ t ù µX: X ˆ X 99K X natural imaginary addition
P pX ˆ XqtX,X // X ` X
p1 1q // X
¨ Xx1,0y❙❙
))❙❙
µX˝x1,0y“1X
��
❵ ❴ ❴ ❫ ❪ ❬ ❨ ❲ ❘❍
X ˆ XµX
//❴❴❴❴ X
Xx0,1y❦❦❦
55❦❦❦
µX˝x0,1y“1X
FF
❫ ❴ ❴ ❵ ❛ ❝ ❡ ❤ ❧✈
Imaginary addition - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 8 / 14
¨ t ù µX: X ˆ X 99K X natural imaginary addition
P pX ˆ XqtX,X // X ` X
p1 1q // X
¨ Xx1,0y❙❙
))❙❙
µX˝x1,0y“1X
��
❵ ❴ ❴ ❫ ❪ ❬ ❨ ❲ ❘❍
X ˆ XµX
//❴❴❴❴ X
Xx0,1y❦❦❦
55❦❦❦
µX˝x0,1y“1X
FF
❫ ❴ ❴ ❵ ❛ ❝ ❡ ❤ ❧✈
( pps of t )
Imaginary addition - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 8 / 14
¨ t ù µX: X ˆ X 99K X natural imaginary addition
P pX ˆ XqtX,X // X ` X
p1 1q // X
¨ Xx1,0y❙❙
))❙❙
µX˝x1,0y“1X
��
❵ ❴ ❴ ❫ ❪ ❬ ❨ ❲ ❘❍
X ˆ XµX
//❴❴❴❴ X
Xx0,1y❦❦❦
55❦❦❦
µX˝x0,1y“1X
FF
❫ ❴ ❴ ❵ ❛ ❝ ❡ ❤ ❧✈
( pps of t )
¨ @f : X Ñ Y, X ˆ XµX
//❴❴❴
fˆf
��
X
f
��Y ˆ Y
µY
//❴❴❴ Y
f ˝ µX “ µY ˝ pf ˆ fq
Imaginary addition - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 8 / 14
¨ t ù µX: X ˆ X 99K X natural imaginary addition
P pX ˆ XqtX,X // X ` X
p1 1q // X
¨ Xx1,0y❙❙
))❙❙
µX˝x1,0y“1X
��
❵ ❴ ❴ ❫ ❪ ❬ ❨ ❲ ❘❍
X ˆ XµX
//❴❴❴❴ X
Xx0,1y❦❦❦
55❦❦❦
µX˝x0,1y“1X
FF
❫ ❴ ❴ ❵ ❛ ❝ ❡ ❤ ❧✈
( pps of t )
¨ @f : X Ñ Y, X ˆ XµX
//❴❴❴
fˆf
��
X
f
��Y ˆ Y
µY
//❴❴❴ Y
f ˝ µX “ µY ˝ pf ˆ fq
( naturality of t )
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ Aq
P pgˆhq
yyssssssssssss
P pX ˆ XqtX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
P pX ˆ XqtX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
¨ Ag // X B
joo
gpaq ` jpbq
A ˆ Bgˆj // X ˆ X
µX
//❴❴❴ X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
¨ Ag // X B
joo
gpaq ` jpbq
A ˆ Bgˆj // X ˆ X
µX
//❴❴❴ X
P pA ˆ Bq
P pgˆjq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
¨ Ag // X B
joo
gpaq ` jpbq
A ˆ Bgˆj // X ˆ X
µX
//❴❴❴ X
P pA ˆ Bq
P pgˆjq
��
tA,B //
nt
A ` B
g`j
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
¨ Ag // X B
joo
gpaq ` jpbq
A ˆ Bgˆj // X ˆ X
µX
//❴❴❴ X
P pA ˆ Bq
P pgˆjq
��
tA,B //
nt
A ` B
g`j
��
pg jq
""❉❉❉❉❉❉❉❉❉❉
P pX ˆ XqtX,X
// X ` Xp1 1q
// X
Imaginary addition - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 9 / 14
¨ Ag // X A
hoo
gpaq ` hpaq
Axg,hy // X ˆ X
µX
//❴❴❴ X
P pAq
P xg,hy
��
P x1,1y // P pA ˆ AqtA,A //
ntP pgˆhq
yyssssssssssss
A ` A
g`h
zz✉✉✉✉✉✉✉✉✉✉✉✉
pg hq
��P pX ˆ Xq
tX,X
// X ` Xp1 1q
// X
¨ Ag // X B
joo
gpaq ` jpbq
A ˆ Bgˆj // X ˆ X
µX
//❴❴❴ X
P pA ˆ Bq
P pgˆjq
��
tA,B //
nt
A ` B
g`j
��
pg jq
""❉❉❉❉❉❉❉❉❉❉
P pX ˆ XqtX,X
// X ` Xp1 1q
// X
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��X
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// // X
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// //
δX
OO
X
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// //
δX
OO
X
(iS2)
apS2q“ qpkpaq ` spyqq
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// //
δX
OO
X
(iS2)
apS2q“ qpkpaq ` spyqq
P 2pK ˆ Y qP ptK,Y q// P pK ` Y q
P pk sq // P pXq
q
��K
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// //
δX
OO
X
(iS2)
apS2q“ qpkpaq ` spyqq
P 2pK ˆ Y qP ptK,Y q// P pK ` Y q
P pk sq // P pXq
q
��P pK ˆ Y q
εKˆY
// // K ˆ YπK
// // K
Intrinsic Schreier split epis
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 10 / 14
¨ C regular unital category w/ binary coproducts, functorial projective
covers and natural imaginary splitting t
¨ K ✤ ,2k
// Xf
// // Ysoo intrinsic Schreier split epi
D qoo❴ ❴ ❴ ❴
(iS1)
xpS1q“ kqpxq ` sfpxq
P 2pXqP x1,1y // P pP pXq ˆ P pXqq
tP pXq,P pXq// P pXq ` P pXq
pkq sfεX q
��P pXq
εX
// //
δX
OO
X
(iS2)
apS2q“ qpkpaq ` spyqq
P 2pK ˆ Y qP ptK,Y q// P pK ` Y q
P pk sq // P pXq
q
��P pK ˆ Y q
εKˆY
// //
δKˆY
OO
K ˆ YπK
// // K
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
¨ q is unique
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
¨ q is unique
¨ (iS1) ñ pk sq : K ` Y ։ X regular epi
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
¨ q is unique
¨ (iS1) ñ pk sq : K ` Y ։ X regular epi
ñ pk, sq jointly extremal-epimorphic pair { pf, sq strong
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
¨ q is unique
¨ (iS1) ñ pk sq : K ` Y ։ X regular epi
ñ pk, sq jointly extremal-epimorphic pair { pf, sq strong
ñ Schreier split epi ñ Schreier split extension
Properties - I
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 11 / 14
¨ q : X 99K K imaginary Schreier retraction
qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C
¨ qs “ 0 in Mon ù qP psq “ 0 in C
qp0q “ 0 in Mon ù obvious in C
kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù X in C
¨ q is unique
¨ (iS1) ñ pk sq : K ` Y ։ X regular epi
ñ pk, sq jointly extremal-epimorphic pair { pf, sq strong
ñ Schreier split epi ñ Schreier split extension
¨ X✤ ,2x1X ,0y
// X ˆ YπY
// // Yx0,1Y yoo intrinsic Schreier split extension
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY XπZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
f// // Y
soo
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY Xq˝πXoo❴ ❴ ❴ ❴
πZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
qoo❴ ❴ ❴ ❴ ❴
f// // Y
soo(iS1)
ñ
(iS1)
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY Xq˝πXoo❴ ❴ ❴ ❴
πZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
qoo❴ ❴ ❴ ❴ ❴
f// // Y
soo(iS1)
ñ
(iS1)
pf, sq strong
pπZ , x1, sgyq strong
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY Xq˝πXoo❴ ❴ ❴ ❴
πZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
qoo❴ ❴ ❴ ❴ ❴
f// // Y
soo(iS1)
ñ
(iS1)
pf, sq strong
pπZ , x1, sgyq strong
ù pf, sq satisfies (iS1) ñ pf, sq stably strong
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY Xq˝πXoo❴ ❴ ❴ ❴
πZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
qoo❴ ❴ ❴ ❴ ❴
f// // Y
soo(iS1)
ñ
(iS1)
pf, sq strong
pπZ , x1, sgyq strong
ù pf, sq satisfies (iS1) ñ pf, sq stably strong
¨ [MRVdL] Y protomodular object: all points X Ô Y are stably strong
Properties - II
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 12 / 14
compatibility¨ P pXqq // K
ρ
��
✤ ,2 k // Xf
// //
g
��
Ysoo
h
��P pX 1q
q1// K 1 ✤ ,2
k1// X 1
f 1// // Y 1
s1oo
P pgq
��ρq “ q1P pgq
¨ K ✤ ,2x0,ky
// Z ˆY Xq˝πXoo❴ ❴ ❴ ❴
πZ
// //
πX
��
Zx1,sgyoo
g
��K ✤ ,2
k// X
qoo❴ ❴ ❴ ❴ ❴
f// // Y
soo(iS1)
ñ
(iS1)
pf, sq strong
pπZ , x1, sgyq strong
ù pf, sq satisfies (iS1) ñ pf, sq 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
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ b ` a
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ b ` a
= left homogeneous split epi [BM-FMS]
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ b ` a
= left homogeneous split epi [BM-FMS]
( (S1)’ x “ sfpxq ` kqpxq; (S2)’ qpspyq ` kpaqq “ a )
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ b ` a
= left homogeneous split epi [BM-FMS]
( (S1)’ x “ sfpxq ` kqpxq; (S2)’ qpspyq ` kpaqq “ 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
Main results
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 13 / 14
¨ Thm. In Mon (or any Jonsson–Tarski variety V):
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ a ` b
= Schreier split epi / right homogeneous split epi [BM-FMS]
( (S1) x “ kqpxq ` sfpxq; (S2) qpkpaq ` spyqq “ a )
- intrinsic Schreier split epi wrt tA,B : P pA ˆ Bq Ñ A ` B
rpa, bqs ÞÑ b ` a
= left homogeneous split epi [BM-FMS]
( (S1)’ x “ sfpxq ` kqpxq; (S2)’ qpspyq ` kpaqq “ 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
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts
- S = class of intrinsic Schreier split epis
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts
- S = class of intrinsic Schreier split epis
- Baer sums through direction functor (+ Barr-exact)
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts
- S = class of intrinsic Schreier split epis
- Baer sums through direction functor (+ Barr-exact)
¨ Good categorical context towards cohomology
Cohomological flavour
Schreier (split) exts ofmonoids
S -protomodularity
Towards intrinsicSchreier split epis
Imaginary morphisms - I
Imaginary morphisms - II
Unital categories
Imaginary addition - I
Imaginary addition - II
Intrinsic Schreier splitepis
Properties - I
Properties - II
Main results
Cohomological flavour
CT2019 Intrinsic Schreier split extnesions – 14 / 14
¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups
( push forward )
¨ [M-FMS] Mon : 2nd cohomology group ´ q q of monoids
( push forward )
protomodular
S -protomodular
¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts
- S = class of intrinsic Schreier split epis
- Baer sums through direction functor (+ Barr-exact)
¨ Good categorical context towards cohomology
¨ Future: higher-order cohomology groups