Intrinsic Schreier split extensions Andrea Montoli Diana Rodelo - - PowerPoint PPT Presentation

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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 2 / 14

¨ [P] 2nd cohomology monoids Ø Schreier exts in Mon

Schreier (split) exts of monoids

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 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

• f split exts in Gp

( Split Short Five Lemma )

Schreier (split) exts of monoids

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 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

• f split exts in Gp

( Split Short Five Lemma ) ù Study Schreier (split) extensions categorically

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f suitable split epis

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f suitable split epis
• S -protomodular cat

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f suitable split epis
• S -protomodular cat

( S : pb-stable; strong; closed under finite lims )

S -protomodularity

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 3 / 14

¨ Gp : protomodular [B] Mon : non-protomodular Schreier ¨ [BM-FMS] - Study protomodularity wrt class S

• f 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 J´

• nsson–Tarski algebras

px ` 0 “ x “ 0 ` xq

Any J´

• nsson–Tarski variety V is S -protomodular

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS]

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

¨ Schreier split epi

pS1q

ñ pk, sq jointly extremal-epimorphic pair ô pf, sq strong point ñ Schreier split extension

Towards intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

¨ Schreier split epi

pS1q

ñ 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

¨ Schreier split epi

pS1q

ñ 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

¨ Schreier split epi

pS1q

ñ 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 4 / 14

¨ This categorical approach for S -protomodularity is not so categorical for S = Schreier split epis in Mon or J´

• nsson–Tarski variety V

¨ Definition of Schreier split epi depends on elements [BM-FMS] K ✤

k

pX, `, 0q

f

Y

s

• Schreier split epi

non-commutative

D q

❴ ❴ ❴

(S1) x “ kqpxq ` sfpxq, @xP X (S2) qpkpaq ` spyqq “ a, @aP K, y P Y

Schreier retraction ( qk “ 1K )

¨ Schreier split epi

pS1q

ñ 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX X

regular epi

projective

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX X

regular epi

projective

• @f : X Ñ Y , fεX “ εY P pfq

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX 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 Y , is a real morphism P pXq Ñ Y

Imaginary morphisms - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 5 / 14

¨ [BJ] Imaginary morphisms q : X function ❴ ❴ ❴ K ù P pXq

morphism K

rxs ÞÝ Ñ qpxq ¨ C regular cat w/ enough projectives

• P pXq

εX 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 Y , is a real morphism P pXq Ñ Y K ✤

k

X

f Y s

• in C

P pXq

q

P P P P imaginary (Schreier) retraction

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Y

1Y

Y real ù Y

1Y

• ❴

❴ ❴ Y imaginary ( P pY q

εY Y

)

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Y

1Y

Y real ù Y

1Y

• ❴

❴ ❴ Y imaginary ( P pY q

εY Y

) ¨ X

f

• ❴

❴ ❴

g˝f

• P ❯ ❩ ❴ ❞ ✐ ♥

Y

g

Z ù P pXq

f

Y

g

Z

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Y

1Y

Y real ù Y

1Y

• ❴

❴ ❴ Y imaginary ( P pY q

εY Y

) ¨ X

f

• ❴

❴ ❴

g˝f

• P ❯ ❩ ❴ ❞ ✐ ♥

Y

g

Z ù P pXq

f

Y

g

Z W

h

• f˝h
• ◗ ❱ ❩ ❴ ❞ ❤ ♠

X

f

• ❴

❴ ❴ Y ù P pW q

P phq P pXq f

Y

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Y

1Y

Y real ù Y

1Y

• ❴

❴ ❴ Y imaginary ( P pY q

εY Y

) ¨ X

f

• ❴

❴ ❴

g˝f

• P ❯ ❩ ❴ ❞ ✐ ♥

Y

g

Z ù P pXq

f

Y

g

Z W

h

• f˝h
• ◗ ❱ ❩ ❴ ❞ ❤ ♠

X

f

• ❴

❴ ❴ Y ù P pW q

P phq P pXq f

Y ¨ X

f Y

regular epi ô D imaginary splitting Y

s

• ❴

❴ ❴ X

Imaginary morphisms - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 6 / 14

¨ X

f

Y real ù X

f

• ❴

❴ ❴ Y imaginary ( P pXq

εX X f Y )

Y

1Y

Y real ù Y

1Y

• ❴

❴ ❴ Y imaginary ( P pY q

εY Y

) ¨ X

f

• ❴

❴ ❴

g˝f

• P ❯ ❩ ❴ ❞ ✐ ♥

Y

g

Z ù P pXq

f

Y

g

Z W

h

• f˝h
• ◗ ❱ ❩ ❴ ❞ ❤ ♠

X

f

• ❴

❴ ❴ Y ù P pW q

P phq P pXq f

Y ¨ X

f Y

regular epi ô D imaginary splitting Y

s

• ❴

❴ ❴ X

Y

s

❴ ❴

f˝s“1Y

• ❘

❴ ❧

X

f Y P pY q

s fs“εY

X

f Y

( )

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi iff D imaginary splitting ( + projs ) P pA ˆ Bq

D tA,B rA,BtA,B“εAˆB p˚q

A ` B

rA,B A ˆ B

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi iff D imaginary splitting ( + projs ) P pA ˆ Bq

D tA,B rA,BtA,B“εAˆB p˚q

A ` B

rA,B A ˆ B

¨ V J´

• nsson–Tarski variety ù D

tA,B : P pA ˆ Bq Ñ A ` B rpa, bqs ÞÑ a ` b

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi iff D imaginary splitting ( + projs ) P pA ˆ Bq

D tA,B rA,BtA,B“εAˆB p˚q

A ` B

rA,B A ˆ B

¨ V J´

• nsson–Tarski variety ù D

tA,B : P pA ˆ Bq Ñ A ` B rpa, bqs ÞÑ a ` b

Unital categories

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi iff D imaginary splitting ( + projs ) P pA ˆ Bq

D tA,B rA,BtA,B“εAˆB p˚q

A ` B

rA,B A ˆ B

¨ V J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 7 / 14

¨ Mon unital category ù J´

• nsson–Tarski variety

( x ` 0 “ x “ 0 ` x )

¨ C pointed + regular + binary coproducts is unital iff @rA,B “ v 1 0

0 1

w : A ` B ։ A ˆ B regular epi iff D imaginary splitting ( + projs ) P pA ˆ Bq

D tA,B rA,BtA,B“εAˆB p˚q

A ` B

rA,B A ˆ B

¨ V J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 8 / 14

¨ t ù µX : X ˆ X X natural imaginary addition

P pX ˆ Xq

tX,X X ` X p1 1q X

Imaginary addition - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 8 / 14

¨ t ù µX : X ˆ X X natural imaginary addition

P pX ˆ Xq

tX,X X ` X p1 1q X

¨ X

x1,0y

❙ ❙

• ❙

❙

µX˝x1,0y“1X

• ❵ ❴ ❴

❫ ❪ ❬ ❨ ❲ ❘ ❍ X ˆ X

µX

• ❴

❴ ❴ ❴ X X

x0,1y

❦ ❦ ❦

• ❦

❦ ❦

µX˝x0,1y“1X

• ❫ ❴ ❴

❵ ❛ ❝ ❡ ❤ ❧ ✈

Imaginary addition - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 8 / 14

¨ t ù µX : X ˆ X X natural imaginary addition

P pX ˆ Xq

tX,X X ` X p1 1q X

¨ X

x1,0y

❙ ❙

• ❙

❙

µX˝x1,0y“1X

• ❵ ❴ ❴

❫ ❪ ❬ ❨ ❲ ❘ ❍ X ˆ X

µX

• ❴

❴ ❴ ❴ X X

x0,1y

❦ ❦ ❦

• ❦

❦ ❦

µX˝x0,1y“1X

• ❫ ❴ ❴

❵ ❛ ❝ ❡ ❤ ❧ ✈ ( pps of t )

Imaginary addition - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 8 / 14

¨ t ù µX : X ˆ X X natural imaginary addition

P pX ˆ Xq

tX,X X ` X p1 1q X

¨ X

x1,0y

❙ ❙

• ❙

❙

µX˝x1,0y“1X

• ❵ ❴ ❴

❫ ❪ ❬ ❨ ❲ ❘ ❍ X ˆ X

µX

• ❴

❴ ❴ ❴ X X

x0,1y

❦ ❦ ❦

• ❦

❦ ❦

µX˝x0,1y“1X

• ❫ ❴ ❴

❵ ❛ ❝ ❡ ❤ ❧ ✈ ( pps of t ) ¨ @f : X Ñ Y, X ˆ X

µX

❴ ❴ ❴

fˆf

• X

f

• Y ˆ Y

µY

• ❴

❴ ❴ Y f ˝ µX “ µY ˝ pf ˆ fq

Imaginary addition - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 8 / 14

¨ t ù µX : X ˆ X X natural imaginary addition

P pX ˆ Xq

tX,X X ` X p1 1q X

¨ X

x1,0y

❙ ❙

• ❙

❙

µX˝x1,0y“1X

• ❵ ❴ ❴

❫ ❪ ❬ ❨ ❲ ❘ ❍ X ˆ X

µX

• ❴

❴ ❴ ❴ X X

x0,1y

❦ ❦ ❦

• ❦

❦ ❦

µX˝x0,1y“1X

• ❫ ❴ ❴

❵ ❛ ❝ ❡ ❤ ❧ ✈ ( pps of t ) ¨ @f : X Ñ Y, X ˆ X

µX

❴ ❴ ❴

fˆf

• X

f

• Y ˆ Y

µY

• ❴

❴ ❴ Y f ˝ µX “ µY ˝ pf ˆ fq ( naturality of t )

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

P pgˆhq

ssssssssssss

P pX ˆ Xq

tX,X

X ` X

p1 1q

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

P pX ˆ Xq

tX,X

X ` X

p1 1q

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X ¨ A

g

X

B

j

• gpaq ` jpbq

A ˆ B

gˆj X ˆ X µX

❴ ❴ ❴

X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X ¨ A

g

X

B

j

• gpaq ` jpbq

A ˆ B

gˆj X ˆ X µX

❴ ❴ ❴

X P pA ˆ Bq

P pgˆjq

• P pX ˆ Xq

tX,X

X ` X

p1 1q X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X ¨ A

g

X

B

j

• gpaq ` jpbq

A ˆ B

gˆj 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 ` X

p1 1q X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X ¨ A

g

X

B

j

• gpaq ` jpbq

A ˆ B

gˆj X ˆ X µX

❴ ❴ ❴

X P pA ˆ Bq

P pgˆjq

• tA,B

nt

A ` B

g`j

• pg jq
• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

P pX ˆ Xq

tX,X

X ` X

p1 1q X

Imaginary addition - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 9 / 14

¨ A

g

X

A

h

• gpaq ` hpaq

A

xg,hy X ˆ X µX

❴ ❴ ❴

X P pAq

P xg,hy

• P x1,1y P pA ˆ Aq

tA,A nt P pgˆhq

ssssssssssss

A ` A

g`h

✉✉✉✉✉✉✉✉✉✉✉✉

pg hq

• P pX ˆ Xq

tX,X

X ` X

p1 1q

X ¨ A

g

X

B

j

• gpaq ` jpbq

A ˆ B

gˆj X ˆ X µX

❴ ❴ ❴

X P pA ˆ Bq

P pgˆjq

• tA,B

nt

A ` B

g`j

• pg jq
• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

P pX ˆ Xq

tX,X

X ` X

p1 1q X

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• X

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX

X

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX δX

• X

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX δX

• X

(iS2)

a

pS2q

“ qpkpaq ` spyqq

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX δX

• X

(iS2)

a

pS2q

“ qpkpaq ` spyqq

P 2pK ˆ Y q

P ptK,Y q P pK ` Y q P pk sq P pXq q

• K

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX δX

• X

(iS2)

a

pS2q

“ qpkpaq ` spyqq

P 2pK ˆ Y q

P ptK,Y q P pK ` Y q P pk sq P pXq q

• P pK ˆ Y q

εKˆY

K ˆ Y

πK

K

Intrinsic Schreier split epis

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 10 / 14

¨ C regular unital category w/ binary coproducts, functorial projective covers and natural imaginary splitting t ¨ K ✤

k

X

f

Y

s

• intrinsic Schreier split epi

D q

❴ ❴ ❴ ❴

(iS1)

x

pS1q

“ kqpxq ` sfpxq

P 2pXq

P x1,1y P pP pXq ˆ P pXqq tP pXq,P pXq

P pXq ` P pXq

pkq sfεXq

• P pXq

εX δX

• X

(iS2)

a

pS2q

“ qpkpaq ` spyqq

P 2pK ˆ Y q

P ptK,Y q P pK ` Y q P pk sq P pXq q

• P pK ˆ Y q

εKˆY δKˆY

• K ˆ Y

πK

K

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X K imaginary Schreier retraction qk “ 1K in Mon ù q ˝ k “ 1K ô qP pkq “ εK in C

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù in C

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù in C ¨ q is unique

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù in C ¨ q is unique ¨ (iS1) ñ pk sq: K ` Y ։ X regular epi

Properties - I

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 11 / 14

¨ q : X 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

• bvious

in C kqpspyq ` kpaqq ` spyq “ spyq ` kpaq in Mon ù 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 ✤

x1X,0y

X ˆ Y

πY

Y

x0,1Y y

• intrinsic Schreier split extension

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ K

ρ

• ✤

k

X

f g

• Y

s

• h
• K1 ✤

k1

X1

f 1 Y 1 s1

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

f

Y

s

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

q˝πX

❴ ❴ ❴ ❴

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

q

❴ ❴ ❴ ❴ ❴

f

Y

s

• (iS1)

ñ

(iS1)

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

q˝πX

❴ ❴ ❴ ❴

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

q

❴ ❴ ❴ ❴ ❴

f

Y

s

• (iS1)

ñ

(iS1)

pf, sq strong pπZ, x1, sgyq strong

Properties - II

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

q˝πX

❴ ❴ ❴ ❴

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

q

❴ ❴ ❴ ❴ ❴

f

Y

s

• (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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

q˝πX

❴ ❴ ❴ ❴

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

q

❴ ❴ ❴ ❴ ❴

f

Y

s

• (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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 12 / 14

compatibility ¨ P pXq

q

K

ρ

• ✤

k

X

f g

• Y

s

• h
• P pX1q

q1

K1 ✤

k1

X1

f 1 Y 1 s1

• P pgq
• ρq “ q1P pgq

¨ K ✤

x0,ky

Z ˆY X

q˝πX

❴ ❴ ❴ ❴

πZ πX

• Z

x1,sgy

• g
• K ✤

k

X

q

❴ ❴ ❴ ❴ ❴

f

Y

s

• (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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 13 / 14

¨ Thm. In Mon (or any J´

• nsson–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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward )

Cohomological flavour

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f monoids

( push forward )

Cohomological flavour

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f monoids

( push forward ) protomodular

S-protomodular

Cohomological flavour

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f monoids

( push forward ) protomodular

S-protomodular

¨ [BM + MRVdL] S -protomodular: ´ Baer sums of special exts

Cohomological flavour

Schreier (split) exts of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f 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 of monoids S -protomodularity Towards intrinsic Schreier split epis Imaginary morphisms - I Imaginary morphisms - II Unital categories Imaginary addition - I Imaginary addition - II Intrinsic Schreier split epis Properties - I Properties - II Main results Cohomological flavour

CT2019 Intrinsic Schreier split extnesions – 14 / 14

¨ [EM] Gp : 2nd cohomology group ´ Baer sums of special exts of groups ( push forward ) ¨ [M-FMS] Mon : 2nd cohomology group ´

• f 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