Aspects of strong protomodularity, actions and quotients Giuseppe - - PowerPoint PPT Presentation

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Aspects of strong protomodularity, actions and quotients

Giuseppe Metere

Universit` a degli Studi di Palermo

June 15, 2015

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro: actions on quotients

Suppose we are given group action ξ and a surjective group homomorphism g A × Y

ξ

Y Y

g Z

Question 1: under what conditions does the action ξ induce an action of A on the quotient Z? A × Y

ξ

• 1×g
• Y

g

• A × Z

¯ ξ

Z Answer: ξ is well defined on the cosets of Ker(g) in Y ⇔ it is well defined on the 0-coset Ker(g). (QA) An action passes to the quotient ⇔ it restricts to the kernel.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro: actions on quotients

Suppose we are given group action ξ and a surjective group homomorphism g A × Y

ξ

Y Y

g Z

Question 1: under what conditions does the action ξ induce an action of A on the quotient Z? A × Y

ξ

• 1×g
• Y

g

• A × Z

¯ ξ

Z Answer: ξ is well defined on the cosets of Ker(g) in Y ⇔ it is well defined on the 0-coset Ker(g). (QA) An action passes to the quotient ⇔ it restricts to the kernel.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro: actions of quotients

Suppose we are given group action ξ and a surjective group homomorphism q A × Y

ξ

Y A

q Q

Question 2: under what conditions does the action ξ induce an action

• n Y of the quotient Q?

A × Y

ξ

• q×1
• Y

1

• Q × Y

¯ ξ

Y In this case, the restriction to Ker(q) always exists. . . Answer: (AQ) The action of the quotient is well def. ⇔ Ker(q) acts trivially on Y .

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro: actions of quotients

Suppose we are given group action ξ and a surjective group homomorphism q A × Y

ξ

Y A

q Q

Question 2: under what conditions does the action ξ induce an action

• n Y of the quotient Q?

A × Y

ξ

• q×1
• Y

1

• Q × Y

¯ ξ

Y In this case, the restriction to Ker(q) always exists. . . Answer: (AQ) The action of the quotient is well def. ⇔ Ker(q) acts trivially on Y .

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro

The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) A × Y

ξ

Y → Y ⋊ξ A

p

• A

s

• We can address our questions in terms of split epimorphisms, even in

contexts where the machinery of internal actions is not available.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro

The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) A × Y

ξ

Y → Y ⋊ξ A

p

• A

s

• We can address our questions in terms of split epimorphisms, even in

contexts where the machinery of internal actions is not available.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Intro

The aim of this talk is to discuss Question 1 and Question 2 internally, in categorical contexts that extend the case of groups. These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category Fact: actions correspond to split epimorphisms (with chosen sections) A × Y

ξ

Y → Y ⋊ξ A

p

• A

s

• We can address our questions in terms of split epimorphisms, even in

contexts where the machinery of internal actions is not available.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Points

From now on, C is a category with finite limits and initial object 0. Pt(C) is the category of points: D

d

• f

B

b

• C

sd

• g

A

sb

• widely studied by D. Bourn, G. Janelidze, F. Borceux. . .

The codomain assignment (B, A, b, sb) → A gives rise to a fibration, the fibration of points: F : Pt(C) → C Fibers are denoted PtA(C). Any g : C → A defines a change of base g ∗ : PtA(C) → PtC(C).

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Points

Definition (Bourn) A category C is called protomodular when every change of base of the fibration of points is conservative, i.e. it reflects isomorphisms. If C admits an initial object 0, for any object A of C, we call kernel functor KA : PtA(C) → PtA(0) the change of base along the initial arrow !A : 0 → A. Fact In the presence of an initial object, the protomodularity condition can be simplified by just requiring that kernel functors are conservative.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Points

Definition (Bourn) A category C is called protomodular when every change of base of the fibration of points is conservative, i.e. it reflects isomorphisms. If C admits an initial object 0, for any object A of C, we call kernel functor KA : PtA(C) → PtA(0) the change of base along the initial arrow !A : 0 → A. Fact In the presence of an initial object, the protomodularity condition can be simplified by just requiring that kernel functors are conservative.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Internal actions

Definition (Bourn - Janelidze) A protomodular category C admits semidirect products when, for every g : C → A, g ∗ : PtA(C) → PtC(C) is monadic. The algebras for the corresponding monads are called internal actions. Fact In the presence of an initial object, this definition can be simplified by just requiring that kernel functors are monadic. Act(C) ≃ Pt(C) A♭Y

ξ

Y → Y ⋊ξ A A

• Giuseppe Metere

Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Internal actions

Definition (Bourn - Janelidze) A protomodular category C admits semidirect products when, for every g : C → A, g ∗ : PtA(C) → PtC(C) is monadic. The algebras for the corresponding monads are called internal actions. Fact In the presence of an initial object, this definition can be simplified by just requiring that kernel functors are monadic. Act(C) ≃ Pt(C) A♭Y

ξ

Y → Y ⋊ξ A A

• Giuseppe Metere

Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Kernels

Definition C is quasi-pointed if 0 → 1 is a monomorphism C is pointed if 0 → 1 is an isomorphism Definition Let C be a quasi-pointed. X

• f
• (∗)
• Y

g

Z (i) f = ker(g) if (∗) is a pullback (ii) g = coker(f ) if (∗) is a pushout (iii) the pair (f , g) is a short exact sequence if both (i) and (ii) hold.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Kernels

Definition C is quasi-pointed if 0 → 1 is a monomorphism C is pointed if 0 → 1 is an isomorphism Definition Let C be a quasi-pointed. X

• f
• (∗)
• Y

g

Z (i) f = ker(g) if (∗) is a pullback (ii) g = coker(f ) if (∗) is a pushout (iii) the pair (f , g) is a short exact sequence if both (i) and (ii) hold.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Sequentiable, homological, semi-abelian cats

Fact (Bourn) If C is quasi-pointed and protomodular, every regular epimorphism is the cokernel of its kernel In this case (f , g) is short exact precisely when g is a regular epimorphism, and f = ker(g). Quasi-pointed protomodular regular categories are called sequentiable. Pointed sequentiable categories are called homological. Barr exact homological categories with finite coproducts are called semi-abelian.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

C quasi-pointed ⇒ C0 = Pt0(C) ֒ → C C0 := category of objects with null support The kernel functor lands in C: KA : PtA(C) → C C    protomodular regular Barr exact ⇒ PtA(C)    protomodular regular Barr exact The aim of this part of the talk is to explore some properties of the functor KA w.r.t. the short exact sequences.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

C quasi-pointed ⇒ C0 = Pt0(C) ֒ → C C0 := category of objects with null support The kernel functor lands in C: KA : PtA(C) → C C    protomodular regular Barr exact ⇒ PtA(C)    protomodular regular Barr exact The aim of this part of the talk is to explore some properties of the functor KA w.r.t. the short exact sequences.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

C quasi-pointed ⇒ C0 = Pt0(C) ֒ → C C0 := category of objects with null support The kernel functor lands in C: KA : PtA(C) → C C    protomodular regular Barr exact ⇒ PtA(C)    protomodular regular Barr exact The aim of this part of the talk is to explore some properties of the functor KA w.r.t. the short exact sequences.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

KA : PtA(C) → C clearly preserves limits and 0 ⇒ It preserves kernels. We can get more from some additional assumptions on C. Proposition Let C be a quasi-pointed protomodular category with pullback stable regular epimorphisms. Then KA preserves short exact sequences. Question: does it also reflect short exact sequences? In order to answer this question, we need more info about kernels and cokernels in PtA(C).

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

KA : PtA(C) → C clearly preserves limits and 0 ⇒ It preserves kernels. We can get more from some additional assumptions on C. Proposition Let C be a quasi-pointed protomodular category with pullback stable regular epimorphisms. Then KA preserves short exact sequences. Question: does it also reflect short exact sequences? In order to answer this question, we need more info about kernels and cokernels in PtA(C).

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

Lemma In a quasi-pointed protomodular category C, let us consider a morphism ϕ in PtA(C), together with its restriction to kernels: X

kb

• f
• B

b

• ϕ
• A

sb

• Y

kc

C

c

A

sc

• Then

1

ϕ is a kernel in PtA(C) ⇔ ϕ · kb is a kernel in C;

2

In this case, the cokernel of ϕ in PtA(C) (provided it exists) is given by the cokernel of ϕ · kb in C.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

Proposition Let C be a quasi-pointed regular category. TFAE: (1) C is protomodular, (2) KA reflects short exact sequences, for every A in C. The two Propositions together imply immediately: Corollary Let C be sequentiable. For any map e : E → A, e∗ : PtA(C) → PtE(C) preserves and reflects short exact sequences.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Exactness properties of kernel functors

Proposition Let C be a quasi-pointed regular category. TFAE: (1) C is protomodular, (2) KA reflects short exact sequences, for every A in C. The two Propositions together imply immediately: Corollary Let C be sequentiable. For any map e : E → A, e∗ : PtA(C) → PtE(C) preserves and reflects short exact sequences.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

So far we have examined the behaviour of kernel functors KA w.r.t. kernel-cokernel sequences. What about kernels alone? Fact In general KA does not reflect kernels KA(b)

kb

• KA(ϕ)
• B

b

• ϕ
• A

sb

• KA(c)

kc

C

c

A

sc

• Giuseppe Metere

Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

So far we have examined the behaviour of kernel functors KA w.r.t. kernel-cokernel sequences. What about kernels alone? Fact In general KA does not reflect kernels KA(b)

kb

• KA(ϕ)
• B

b

• ϕ
• A

sb

• KA(c)

kc

C

c

A

sc

• Giuseppe Metere

Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Recall: f : X → Y is normal to an equiv. rel. R on X when X × X

• R

r1,r2

• X × X

f ×f Y × Y

X × X

• p1
• R

r1

• X

f

Y ⋄ C protomodular, ⇒ R is uniq. det. by f . ⋄ All kernels are normal (to kernel pair relations). C is strongly protomodular when change-of-base functors reflect normal monos. C is strongly semi-abelian when it is semi-abelian and strongly protomodular

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Recall: f : X → Y is normal to an equiv. rel. R on X when X × X

• R

r1,r2

• X × X

f ×f Y × Y

X × X

• p1
• R

r1

• X

f

Y ⋄ C protomodular, ⇒ R is uniq. det. by f . ⋄ All kernels are normal (to kernel pair relations). C is strongly protomodular when change-of-base functors reflect normal monos. C is strongly semi-abelian when it is semi-abelian and strongly protomodular

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Recall: f : X → Y is normal to an equiv. rel. R on X when X × X

• R

r1,r2

• X × X

f ×f Y × Y

X × X

• p1
• R

r1

• X

f

Y ⋄ C protomodular, ⇒ R is uniq. det. by f . ⋄ All kernels are normal (to kernel pair relations). C is strongly protomodular when change-of-base functors reflect normal monos. C is strongly semi-abelian when it is semi-abelian and strongly protomodular

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Recall: f : X → Y is normal to an equiv. rel. R on X when X × X

• R

r1,r2

• X × X

f ×f Y × Y

X × X

• p1
• R

r1

• X

f

Y ⋄ C protomodular, ⇒ R is uniq. det. by f . ⋄ All kernels are normal (to kernel pair relations). C is strongly protomodular when change-of-base functors reflect normal monos. C is strongly semi-abelian when it is semi-abelian and strongly protomodular

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

In the semi-abelian case, normal monos coincide with kernels, so that strong protomodularity becomes a matter of reflection of kernels. Hence we can find categories that are semi-abelian, but not strongly semi-abelian, where the kernel functors fail to reflect kernels. On the other side, more generally, kernels may form a proper subclass of normal monos, so that reflection of kernels becomes an issue that deserves a specific investigation.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

In the semi-abelian case, normal monos coincide with kernels, so that strong protomodularity becomes a matter of reflection of kernels. Hence we can find categories that are semi-abelian, but not strongly semi-abelian, where the kernel functors fail to reflect kernels. On the other side, more generally, kernels may form a proper subclass of normal monos, so that reflection of kernels becomes an issue that deserves a specific investigation.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Proposition Let C be sequentiable, and KA : PtA(C) → C the kernel functor. TFAE: (1) KA reflects kernel maps, (2) KA lifts the cokernels of the maps ϕ such that KA(ϕ) is a kernel: KA(b)

• KA(ϕ)

B

b

• ϕ

A

• KA(c)
• KA(γ)

C

c

• γ

A

• KA(d)

D

d

A

• ∀ϕ in PtA(C) s.t. KA(ϕ) = f is a kernel in C,

∃!γ in PtA(C) s.t. γ = coker(ϕ), and KA(γ) = coker(f ).

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Do kernels reflect kernels?

Proposition Let C be sequentiable, and KA : PtA(C) → C the kernel functor. TFAE: (1) KA reflects kernel maps, (2) KA lifts the cokernels of the maps ϕ such that KA(ϕ) is a kernel: KA(b)

• KA(ϕ)

B

b

• ϕ

A

• KA(c)
• KA(γ)

C

c

• γ

A

• KA(d)

D

d

A

• ∀ϕ in PtA(C) s.t. KA(ϕ) = f is a kernel in C,

∃!γ in PtA(C) s.t. γ = coker(ϕ), and KA(γ) = coker(f ).

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Back to actions

If C admits semidirect products, we can translate condition (2) in terms

• f internal actions:

X

• f

B

b

• ϕ

A

• Y
• g

C

c

• γ

A

• Z

D

d

A

• ⇒

A♭X

ξ|

• 1♭f

X

f

• A♭Y

ξ

• 1♭g

Y

g

• A♭Z

¯ ξ

Z (3) If an A-action ξ restricts to the kernel f , then it induces an A-action ¯ ξ on the cokernel of f .

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Action on quotients

Finally, we have a setting to answer our first question, and a characterization of strongly semi-abelian categories. Proposition Let C be a semi-abelian category. TFAE: (1) C is strongly protomodular, (2) for every object A of C, the kernel functor KA lifts the cokernels of the maps ϕ such that KA(ϕ) is a kernel (3) For any pair (ξ, g) A♭Y

ξ

Y

g

Z with ξ action and g regular epimorphism, if ξ restricts to ker(g), then it induces an A-action ¯ ξ on Z

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

The other question

It turns out that strongly semi-abelian categories are also suitable for an answer to our second question. Suppose we are given a pair A♭Y

ξ

Y A

q

Q with ξ action and q regular epimorphism. Question 2: under what conditions does the action ξ induce an action

• n Y of the quotient Q?

As before, we will work out in terms of points the required properties, and this will unveil some more properties of the change of base along q.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

The other question

It turns out that strongly semi-abelian categories are also suitable for an answer to our second question. Suppose we are given a pair A♭Y

ξ

Y A

q

Q with ξ action and q regular epimorphism. Question 2: under what conditions does the action ξ induce an action

• n Y of the quotient Q?

As before, we will work out in terms of points the required properties, and this will unveil some more properties of the change of base along q.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Change of base: the other way

Let’s translate in terms of points: Question 2’: under what conditions, given a point (C, c, sc, A) and a regular epimorphism q : A → Q can we get a point (D, d, sd, Q) such that q∗(D, d, sD, Q) = (C, c, sc, A)? A♭Y

ξ

• q♭1
• Y

Q♭Y Y ⇒ Y C

c

• A

q

• sc
• Y

D

d

Q

sd

• Giuseppe Metere

Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Change of base: the other way

Proposition In a strongly semi-abelian category C, we consider a point (C, c, sc, A), and a regular epi q : A → Q. We denote by (K, k) and (Y , kc) the kernels of q and of c respectively. Then TFAE: (1) the pullback along k of (C, c, sc, A) is trivial (2) there exist a (D, d, sd, Q) and a cartesian morphism (γ, q): (C, c, sc, A) → (D, d, sd, Q) Y

kπ2 Y × K π2

• ϕ

K

0,1

• k
• Y

kc

C

c

• γ

A

sc

• q
• Y

kd

D

d

Q

sd

• ⇔

(3) K♭Y

k♭1 τ

Y A♭Y

ξ

• q♭1

Y Q♭Y Y

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Change of base: the other way

Proposition In a strongly semi-abelian category C, we consider a point (C, c, sc, A), and a regular epi q : A → Q. We denote by (K, k) and (Y , kc) the kernels of q and of c respectively. Then TFAE: (1) the pullback along k of (C, c, sc, A) is trivial (2) there exist a (D, d, sd, Q) and a cartesian morphism (γ, q): (C, c, sc, A) → (D, d, sd, Q) Y

kπ2 Y × K π2

• ϕ

K

0,1

• k
• Y

kc

C

c

• γ

A

sc

• q
• Y

kd

D

d

Q

sd

• ⇔

(3) K♭Y

k♭1 τ

Y A♭Y

ξ

• q♭1

Y Q♭Y Y

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Change of base: the other way

This gives a description of the change of base q∗ when q, is a regular epi. Proposition Given a regular epi q : A → Q in a sequentiable category C, the change of base q∗ : PtQ(C) → PtA(C) is fully faithful. If C is strongly semi-abelian and k = ker(q), we have a factorization q∗ : PtQ(C) → PtA(C)|k ֌ PtA(C) where PtA(C)|k is the full subcategory of PtA(C) with objects the points that are sent to trivial ones by k∗. Indeed, the point (2) of the previous proposition defines precisely a quasi-inverse for the equivalence PtQ(C) → PtA(C)|k

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Change of base: the other way

This gives a description of the change of base q∗ when q, is a regular epi. Proposition Given a regular epi q : A → Q in a sequentiable category C, the change of base q∗ : PtQ(C) → PtA(C) is fully faithful. If C is strongly semi-abelian and k = ker(q), we have a factorization q∗ : PtQ(C) → PtA(C)|k ֌ PtA(C) where PtA(C)|k is the full subcategory of PtA(C) with objects the points that are sent to trivial ones by k∗. Indeed, the point (2) of the previous proposition defines precisely a quasi-inverse for the equivalence PtQ(C) → PtA(C)|k

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Overview

Introduction Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

An application to crossed modules

Internal (pre)crossed modules in a semi-abelian category were introduced by G. Janelidze. They form a category equivalent to that of internal groupoids (reflexive graphs). In the strongly semi-abelian setting, they can be described this way. Definition A precrossed module G = (∂, ξ) in C is a map ∂ together with an action ξ s.t. diagram (pre) below commutes. G0♭G

ξG

• 1♭∂G
• (pre)

G

∂G

• G0♭G0

χG0

G0 G♭G

χG

• ∂G ♭1
• (pff)

G G0♭G

ξG

G G is a crossed module when also diagram (pff) above commutes.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

An application to crossed modules

Definition A morphism of (pre)crossed modules H → G is a pair (h, h0) of maps that makes the following diagram commute. H0♭H

h0♭h ξ

• (i)

G0♭G

ξ

• H

∂

• h
• (ii)

G

∂

• H0

h0

G0

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

An application to crossed modules

A (pre)crossed submodule N of a (pre)crossed module G is just subobject (n, n0): N → G in the category of (pre)crossed modules in C, N0♭N

n0♭n ξN (i)

G0♭G

ξG

• N

∂N n

• (ii)

G

∂G

• N0

n0

G0 with n, n0 mono. Definition The commutative diagram (ii) defines a normal (pre)crossed submodule N of G when both n and n0 are kernels in C,

1

q · ξG · (1♭n) = 0,

2

q · ξG · (n0♭1) = q · τG, where τG is the trivial action of N0 on G. where (Q, q) is the cokernel of n.

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

An application to crossed modules

The previous definition makes sense because of the following: Proposition Let (n, n0) be a normal monomorphism of (pre)crossed modules. Then it is the kernel of its cokernel (q, q0) which can be computed levelwise. N

(n,n0) G (q,q0) Q

N0♭N

n0♭n ξN (i)

G0♭G

ξG

• q0♭q

(iii)

Q0♭Q

ξQ

• N

∂N n

• (ii)

G

∂G

• q
• (iv)

Q

∂Q

• N0

n0

G0

q0

Q0

Giuseppe Metere Aspects of strong protomodularity, actions and quotients

Intro Some fine points Exactness properties of kernel functors Change of base: the other way An application to crossed modules

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Giuseppe Metere Aspects of strong protomodularity, actions and quotients