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Introduction A push forward construction A particular case A push forward construction and the comprehensive factorization for internal crossed modules I Alan Cigoli Universit` a degli Studi di Milano (joint work with S. Mantovani and G.

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Introduction A push forward construction A particular case

A push forward construction and the comprehensive factorization for internal crossed modules I

Alan Cigoli

Universit` a degli Studi di Milano

(joint work with S. Mantovani and G. Metere)

Workshop on Category Theory in honour of George Janelidze, on the occasion of his 60th birthday Coimbra, July 13, 2012

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Introduction

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A. Short exact sequences with kernel A and cokernel C form a groupoid EXT(C, A).

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A. Short exact sequences with kernel A and cokernel C form a groupoid EXT(C, A). Equivalence classes form an abelian group: Ext(C, A) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A. Short exact sequences with kernel A and cokernel C form a groupoid EXT(C, A). Equivalence classes form an abelian group: Ext(C, A) . Any morphism c : C ′ → C determines a functor c∗ : EXT(C, A) → EXT(C ′, A)

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A. Short exact sequences with kernel A and cokernel C form a groupoid EXT(C, A). Equivalence classes form an abelian group: Ext(C, A) . Any morphism c : C ′ → C determines a functor c∗ : EXT(C, A) → EXT(C ′, A) by means of the pullback along c: A B′

  • C ′
  • c
  • A

B C

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let A be an abelian category, A and C objects of A. Short exact sequences with kernel A and cokernel C form a groupoid EXT(C, A). Equivalence classes form an abelian group: Ext(C, A) . Any morphism c : C ′ → C determines a functor c∗ : EXT(C, A) → EXT(C ′, A) by means of the pullback along c: A B′

  • C ′
  • c
  • A

B C And this gives a group homomorphism Ext(C, A) → Ext(C ′, A) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Dually, any morphism a: A → A′ determines a functor: a∗ : EXT(C, A) → EXT(C, A′)

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Dually, any morphism a: A → A′ determines a functor: a∗ : EXT(C, A) → EXT(C, A′) by means of the pushout along a: A

  • a
  • B
  • C

A′ B′ C

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Dually, any morphism a: A → A′ determines a functor: a∗ : EXT(C, A) → EXT(C, A′) by means of the pushout along a: A

  • a
  • B
  • C

A′ B′ C And this gives a group homomorphism Ext(C, A) → Ext(C, A′) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated. Example: groups.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A: φ: G × A A

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A: φ: G × A A Short exact sequences inducing the same action of G on A form a groupoid OPEXT(G, A, φ).

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A: φ: G × A A Short exact sequences inducing the same action of G on A form a groupoid OPEXT(G, A, φ). Equivalence classes form an abelian group: Opext(G, A, φ) ∼ = H2

φ(G, A) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The non-abelian setting is more complicated. Example: groups. Any short exact sequence of abelian kernel A and cokernel G determines an action of G on A: φ: G × A A Short exact sequences inducing the same action of G on A form a groupoid OPEXT(G, A, φ). Equivalence classes form an abelian group: Opext(G, A, φ) ∼ = H2

φ(G, A) .

Again, for any group homomorphism g : G ′ → G, the pullback construction determines a functor: g ∗ : OPEXT(G, A, φ) → OPEXT(G ′, A, g ∗(φ)) ,

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

where g ∗(φ) is given by the composite: G ′ × A

g ∗(φ)

  • g×1
  • A

G × A

φ

  • Alan Cigoli

A push forward construction

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Introduction A push forward construction A particular case

where g ∗(φ) is given by the composite: G ′ × A

g ∗(φ)

  • g×1
  • A

G × A

φ

  • And again a group homomorphism:

H2

φ(G, A) → H2 g ∗(φ)(G ′, A) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works. Problems:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general;

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a: A → A′ does not determine an action of G on A′ in a canonical way.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a: A → A′ does not determine an action of G on A′ in a canonical way. So we need an action of G on A′: φ′ : G × A′ A′

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

The pushout contruction no longer works. Problems: the pushout of a normal mono is not a normal mono in general; a morphism a: A → A′ does not determine an action of G on A′ in a canonical way. So we need an action of G on A′: φ′ : G × A′ A′ and we require that a is equivariant, i.e.: G × A

φ

  • 1×a
  • A

a

  • G × A′

φ′

  • A′

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

These data allow to construct the so called push forward along a: A

k

  • a
  • p.f .

E

f

  • e
  • G

A′

k′

E ′

f ′

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

These data allow to construct the so called push forward along a: A

k

  • a
  • p.f .

E

f

  • e
  • G

A′

k′

E ′

f ′

G which determines a functor: a∗ : OPEXT(G, A, φ) → OPEXT(G, A′, φ′)

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

These data allow to construct the so called push forward along a: A

k

  • a
  • p.f .

E

f

  • e
  • G

A′

k′

E ′

f ′

G which determines a functor: a∗ : OPEXT(G, A, φ) → OPEXT(G, A′, φ′) and a group homomorphism: H2

φ(G, A) → H2 φ′(G, A′) .

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Construction of the push forward (for groups): E

iE

  • A

k

  • a
  • E ⋊f ∗(φ′) A′

q E ′

A′

iA′

  • where q = coeq(iEk, iA′a).

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Construction of the push forward (for groups): E

iE

  • A

k

  • a
  • E ⋊f ∗(φ′) A′

q E ′

A′

iA′

  • where q = coeq(iEk, iA′a).

Universal property: A

k

  • a
  • p.f .

E

f

  • G

A′

k′

E ′

f ′

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Construction of the push forward (for groups): E

iE

  • A

k

  • a
  • E ⋊f ∗(φ′) A′

q E ′

A′

iA′

  • where q = coeq(iEk, iA′a).

Universal property: A

k

  • a
  • p.f .

E

f

  • G

A′

k′

E ′

f ′

G A′

k′′ E ′′ f ′′

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Construction of the push forward (for groups): E

iE

  • A

k

  • a
  • E ⋊f ∗(φ′) A′

q E ′

A′

iA′

  • where q = coeq(iEk, iA′a).

Universal property: A

k

  • a
  • p.f .

E

f

  • G

A′

k′

E ′

f ′

  • ∃!
  • G

A′

k′′ E ′′ f ′′

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Questions:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Questions: Is there a push forward construction for extensions with non-abelian kernel?

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Questions: Is there a push forward construction for extensions with non-abelian kernel? Is there an internal version of this construction?

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Questions: Is there a push forward construction for extensions with non-abelian kernel? Is there an internal version of this construction? Can it be extended to crossed modules?

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

A push forward construction in semi-abelian categories

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category. A precrossed module in C is a morphism ∂ : H → H0 , together with an internal action ξ : H0♭H → H , such that the following diagram commutes: H0♭H

ξ

  • 1♭∂
  • H

∂

  • H0♭H0

χ

H0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category. A precrossed module in C is a morphism ∂ : H → H0 , together with an internal action ξ : H0♭H → H , such that the following diagram commutes: H0♭H

ξ

  • 1♭∂
  • H

∂

  • H0♭H0

χ

H0 If we want ∂ to be a crossed module, we need a further condition, which is not in general the straightforward generalization of the Peiffer condition for crossed modules of groups.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

  • G. Janelidze in ’03 gave a definition of internal crossed module, showing

the equivalence with internal groupoids.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

  • G. Janelidze in ’03 gave a definition of internal crossed module, showing

the equivalence with internal groupoids. However, if C satisfies the “Smith is Huq” property, the Peiffer condition: H♭H

χ

  • ∂♭1
  • H

H0♭H

ξ

H turns out to be sufficient to characterize internal crossed modules among precrossed modules (Martins-Ferreira and Van der Linden, ’10).

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Hartl (unpublished preprint ’10): push forward of a normal monomorphism in semi-abelian setting with conditions expressed in terms of cross effects;

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Hartl (unpublished preprint ’10): push forward of a normal monomorphism in semi-abelian setting with conditions expressed in terms of cross effects; He should have presented here a generalization of his result to crossed modules;

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Hartl (unpublished preprint ’10): push forward of a normal monomorphism in semi-abelian setting with conditions expressed in terms of cross effects; He should have presented here a generalization of his result to crossed modules; Meanwhile we reinterpreted conditions in terms of internal actions and semi-direct products, obtaining, for push forward of (pre)crossed modules, an equivalent result.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category, ∂ and p two morphisms in C: H

∂ p

  • H0

G satisfying the following conditions:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category, ∂ and p two morphisms in C: H

∂ p

  • H0

G satisfying the following conditions: 1) there is an action ξ : H0♭H → H such that (∂, ξ) is a precrossed module;

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be a semi-abelian category, ∂ and p two morphisms in C: H

∂ p

  • H0

G satisfying the following conditions: 1) there is an action ξ : H0♭H → H such that (∂, ξ) is a precrossed module; 2) there is an action α : H0♭G → G, and p is equivariant: H0♭H

ξ

  • 1♭p
  • H

p

  • H0♭G

α

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

3) the following diagram commutes: (H ⋊ξ H0)♭G

ϕ♭1 (p⋊1)♭1

  • H0♭G

α

  • (G ⋊α H0)♭G

χ

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

3) the following diagram commutes: (H ⋊ξ H0)♭G

ϕ♭1 (p⋊1)♭1

  • H0♭G

α

  • (G ⋊α H0)♭G

χ

G where ϕ is defined by the universal property of semi-direct product: H

iH ∂

  • H ⋊ξ H0

ϕ

  • H0

iH0

  • 1
  • H0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

3) the following diagram commutes: (H ⋊ξ H0)♭G

ϕ♭1 (p⋊1)♭1

  • H0♭G

α

  • (G ⋊α H0)♭G

χ

G where ϕ is defined by the universal property of semi-direct product: H

iH ∂

  • H ⋊ξ H0

ϕ

  • H0

iH0

  • 1
  • H0

These conditions are sufficient to obtain a push forward construction.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Theorem There exists an object G ⋊H H0, together with a crossed module

  • ∂ : G → G ⋊H H0, with coker(

∂) ∼ = coker(∂), and a morphism

  • p0 : H0 → G ⋊H H0, such that the following diagram is a morphism of

precrossed modules: H

∂

  • p
  • H0
  • p0
  • G
  • ∂

G ⋊H H0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Theorem There exists an object G ⋊H H0, together with a crossed module

  • ∂ : G → G ⋊H H0, with coker(

∂) ∼ = coker(∂), and a morphism

  • p0 : H0 → G ⋊H H0, such that the following diagram is a morphism of

precrossed modules: H

∂

  • p
  • H0
  • p0
  • p0
  • G
  • ∂
  • ∂′
  • G ⋊H H0

G0 which is universal in the following sense: for any other morphism (p, p0)

  • f precrossed modules, where (∂′, ξ′) is a crossed module and p∗

0(ξ′) = α,

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Theorem There exists an object G ⋊H H0, together with a crossed module

  • ∂ : G → G ⋊H H0, with coker(

∂) ∼ = coker(∂), and a morphism

  • p0 : H0 → G ⋊H H0, such that the following diagram is a morphism of

precrossed modules: H

∂

  • p
  • H0
  • p0
  • p0
  • G
  • ∂
  • ∂′
  • G ⋊H H0

t

  • G0

which is universal in the following sense: for any other morphism (p, p0)

  • f precrossed modules, where (∂′, ξ′) is a crossed module and p∗

0(ξ′) = α,

there exists a unique factorization t, with t p0 = p0 and (1G, t) a morphism of crossed modules.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

A remark about the notation.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

A remark about the notation. If H = 0, conditions 1)–3) reduce to the request of existence of the action α, and the above construction is nothing but semi-direct product:

  • H0

iH0

  • G

iG

G ⋊α H0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

A remark about the notation. If H = 0, conditions 1)–3) reduce to the request of existence of the action α, and the above construction is nothing but semi-direct product:

  • H0

iH0

  • p0
  • G

iG

  • ∂′
  • G ⋊α H0

t

  • G0

The universal property reduces to the universal property of semi-direct product.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

If the category C is moreover action accessible (e.g. groups, Lie algebras, rings, any category of interest), we can replace condition 3) with the following condition:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

If the category C is moreover action accessible (e.g. groups, Lie algebras, rings, any category of interest), we can replace condition 3) with the following condition: 3′) H♭G

∂♭1 p♭1

  • H0♭G

α

  • G♭G

χ

G

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

If the category C is moreover action accessible (e.g. groups, Lie algebras, rings, any category of interest), we can replace condition 3) with the following condition: 3′) H♭G

∂♭1 p♭1

  • H0♭G

α

  • G♭G

χ

G which is in fact weaker: H♭G

iH♭1 p♭1

  • (H ⋊ξ H0)♭G

ϕ♭1 (p⋊1)♭1

  • H0♭G

α

  • G♭G

iG ♭1

(G ⋊α H0)♭G

χ

G and we obtain the same result.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Sketch of the proof (action accessible case).

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Sketch of the proof (action accessible case). As a consequence of condition 3’) the semi-direct product G ⋊p∗(χ) H is isomorphic to the product G × H:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Sketch of the proof (action accessible case). As a consequence of condition 3’) the semi-direct product G ⋊p∗(χ) H is isomorphic to the product G × H: G × H

πG

  • πH
  • τ
  • G

iG

  • 1,0
  • H

j

  • 0,1
  • G ⋊p∗(χ) H

ρ

  • πH
  • Alan Cigoli

A push forward construction

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Introduction A push forward construction A particular case

Sketch of the proof (action accessible case). As a consequence of condition 3’) the semi-direct product G ⋊p∗(χ) H is isomorphic to the product G × H: G × H

πG

  • πH
  • τ
  • G

iG

  • 1,0
  • iG
  • H

j

  • 0,1
  • n
  • G ⋊p∗(χ) H

ρ

  • πH
  • 1⋊∂
  • G ⋊α H0

Hence the morphisms n = (1 ⋊ ∂)j and iG cooperate in G ⋊α H0.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Sketch of the proof (action accessible case). As a consequence of condition 3’) the semi-direct product G ⋊p∗(χ) H is isomorphic to the product G × H: G × H

πG

  • πH
  • τ
  • G

iG

  • 1,0
  • iG
  • H

j

  • 0,1
  • n
  • G ⋊p∗(χ) H

ρ

  • πH
  • 1⋊∂
  • G ⋊α H0

Hence the morphisms n = (1 ⋊ ∂)j and iG cooperate in G ⋊α H0. And consequently [n(H), G] = 0.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible:

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible: [n(H), G] = 0 ⇒

  • n(H), G
  • = 0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible: [n(H), G] = 0 ⇒

  • n(H), G
  • = 0

and the “Smith is Huq” property holds.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible: [n(H), G] = 0 ⇒

  • n(H), G
  • = 0

and the “Smith is Huq” property holds. These conditions allow to construct a split butterfly: n(H)

  • n
  • G
  • ∂
  • iG
  • G ⋊α H0

πH0

  • q
  • H0

iH0

  • G ⋊H H0

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible: [n(H), G] = 0 ⇒

  • n(H), G
  • = 0

and the “Smith is Huq” property holds. These conditions allow to construct a split butterfly: n(H)

  • n
  • G
  • ∂
  • iG
  • G ⋊α H0

πH0

  • q
  • H0

iH0

  • G ⋊H H0

which produces a morphism of crossed modules.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Since the category is action accessible: [n(H), G] = 0 ⇒

  • n(H), G
  • = 0

and the “Smith is Huq” property holds. These conditions allow to construct a split butterfly: H

  • ∂
  • n(H)
  • n
  • G
  • ∂
  • iG
  • G ⋊α H0

πH0

  • q
  • H0

H0

iH0

  • G ⋊H H0

which produces a morphism of crossed modules. By composition we get the required morphism of precrossed modules.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

A particular case

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be semi-abelian.

Alan Cigoli A push forward construction

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Introduction A push forward construction A particular case

Let C be semi-abelian. In the case p = 1: H

∂ H0

H

Alan Cigoli A push forward construction

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SLIDE 75

Introduction A push forward construction A particular case

Let C be semi-abelian. In the case p = 1: H

∂ H0

H 1) as before: (∂, ξ) is a precrossed module;

Alan Cigoli A push forward construction

slide-76
SLIDE 76

Introduction A push forward construction A particular case

Let C be semi-abelian. In the case p = 1: H

∂ H0

H 1) as before: (∂, ξ) is a precrossed module; 2) disappears (equivariance of 1);

Alan Cigoli A push forward construction

slide-77
SLIDE 77

Introduction A push forward construction A particular case

Let C be semi-abelian. In the case p = 1: H

∂ H0

H 1) as before: (∂, ξ) is a precrossed module; 2) disappears (equivariance of 1); 3) becomes: (H ⋊ξ H0)♭H

ϕ♭1 H0♭H ξ

  • (H ⋊ξ H0)♭H

χ

H

Alan Cigoli A push forward construction

slide-78
SLIDE 78

Introduction A push forward construction A particular case

Let C be semi-abelian. In the case p = 1: H

∂ H0

H 1) as before: (∂, ξ) is a precrossed module; 2) disappears (equivariance of 1); 3) becomes: (H ⋊ξ H0)♭H

ϕ♭1 H0♭H ξ

  • (H ⋊ξ H0)♭H

χ

H and gives a condition for a precrossed module to be a crossed module (“Super-Peiffer”).

Alan Cigoli A push forward construction

slide-79
SLIDE 79

Introduction A push forward construction A particular case

In the action accessible context, 3) is replaced by 3’) and the previous condition reduces to Peiffer condition.

Alan Cigoli A push forward construction

slide-80
SLIDE 80

Introduction A push forward construction A particular case

In the action accessible context, 3) is replaced by 3’) and the previous condition reduces to Peiffer condition. Conclusion: Action accessible ⇒ 3’) instead of 3) ⇓ ⇓ “Smith is Huq” ⇔ Peiffer ⇒ “Super-Peiffer”

Alan Cigoli A push forward construction

slide-81
SLIDE 81

Introduction A push forward construction A particular case

In the action accessible context, 3) is replaced by 3’) and the previous condition reduces to Peiffer condition. Conclusion: Action accessible ⇒ 3’) instead of 3) ⇓ ⇓ “Smith is Huq” ⇔ Peiffer ⇒ “Super-Peiffer” Observe that the implication on the top depends on the property: [H, K] = 0 ⇒

  • H, K
  • = 0

Alan Cigoli A push forward construction