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