[PPT] - Extention theory and the calculus of butterflies Alan Cigoli PowerPoint Presentation

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Internal crossed modules Butterflies Extensions

Extention theory and the calculus of butterflies

Alan Cigoli

Universit` a degli Studi di Milano

(joint work with G. Metere) Categorical Methods in Algebra and Topology

workshop in honour of Manuela Sobral on the occasion of her 70th birthday

Coimbra, January 26, 2014

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Internal crossed modules

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute).

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute). An internal crossed module (∂, ξ) in C is a morphism ∂ together with an action ξ G0♭G

ξ

G

∂

G0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 5

Internal crossed modules Butterflies Extensions

Let C be a semi-abelian category satisfying (SH) (i.e. where two equivalence relation centralize each other as soon as their normalizations commute). An internal crossed module (∂, ξ) in C is a morphism ∂ together with an action ξ G0♭G

ξ

G

∂

G0

such that the following squares commute: G♭G

χG ∂♭1

G G0♭G

ξ

1♭∂

G

∂

G0♭G0

χG0

G0

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

A morphism of crossed modules (∂′, ξ′) → (∂, ξ) is a pair (f , f0) of maps that makes the following diagram commute: H0♭H

f0♭f ξ′

G0♭G

ξ

H

∂′

f

G

∂

H0

f0

G0

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

A morphism of crossed modules (∂′, ξ′) → (∂, ξ) is a pair (f , f0) of maps that makes the following diagram commute: H0♭H

f0♭f ξ′

G0♭G

ξ

H

∂′

f

G

∂

H0

f0

G0

These data form a category XMod(C) equivalent to Grpd(C) [G. Janelidze ’03].

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

A morphism of crossed modules (∂′, ξ′) → (∂, ξ) is a pair (f , f0) of maps that makes the following diagram commute: H0♭H

f0♭f ξ′

G0♭G

ξ

H

∂′

f

G

∂

H0

f0

G0

These data form a category XMod(C) equivalent to Grpd(C) [G. Janelidze ’03]. This equivalence extends to a biequivalence of bicategories [Abbad, Mantovani, Metere, Vitale ’13].

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

We can define homotopy invariants:

π1(∂′)

π1(f )

❴

ker(∂′)

π1(∂)

❴ ker(∂)

H

∂′ f

G

∂

H0

coker(∂′)❴

f0

G0

coker(∂)

❴

π0(∂′)

π0(f ) π0(∂) Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

We can define homotopy invariants:

π1(∂′)

π1(f )

❴

ker(∂′)

π1(∂)

❴ ker(∂)

H

∂′ f

G

∂

H0

coker(∂′)❴

f0

G0

coker(∂)

❴

π0(∂′)

π0(f ) π0(∂)

π1(∂) is central in C. Bourn’s global direction of a groupoid translates in terms of crossed modules as: π1(∂)

❴

π1(∂)
G ⋊ξ G0

✤ π0(∂)

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Translation of some special morphisms:

final

disc. fib.

π0-cart. fully faith. weak equiv. π1∂′

✤ ❴

π1∂

❴

H
pf

G

H0

❴

G0

❴

π0∂′

π0∂ π1∂′

❴

π1∂

❴

H
H
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
H
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
G
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
G
H0

❴

G0

❴

π0∂′

π0∂ [C., Mantovani, Metere ’13] [Everaert, Kieboom, Van der Linden ’04]

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Translation of some special morphisms:

final

disc. fib.

π0-cart. fully faith. weak equiv. π1∂′

✤ ❴

π1∂

❴

H
pf

G

H0

❴

G0

❴

π0∂′

π0∂ π1∂′

❴

π1∂

❴

H
H
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
H
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
G
H0

❴

G0

❴

π0∂′

π0∂

π1∂′

❴

π1∂

❴

H
G
H0

❴

G0

❴

π0∂′

π0∂ [C., Mantovani, Metere ’13] [Everaert, Kieboom, Van der Linden ’04]

We have (among others) two factorization systems: (final,

disc. fib.)

∩ ∪ (π0-inv., π0-cart.)

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Internal butterflies

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13].

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly E : (∂′, ξ′) (∂, ξ) is a commutative diagram of the form

H

κ

∂′
G

∂

ι
E

δ

γ
H0

G0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 16

Internal crossed modules Butterflies Extensions

Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly E : (∂′, ξ′) (∂, ξ) is a commutative diagram of the form

H

κ

∂′
G

∂

ι
E

δ

γ
H0

G0

such that

i. (κ, γ) is a complex, i.e. γ · κ = 0,
ii. (ι, δ) is short exact,
iii. The action of E on H induced by that of H0 on H via δ makes κ: H → E a

crossed module,

iv. The action of E on G induced by that of G0 on G via γ makes ι: g → E a

crossed module.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 17

Internal crossed modules Butterflies Extensions

Introduced by Noohi in the category of groups, further developed in the semi-abelian context [Abbad, Mantovani, Metere, Vitale ’13]. A butterfly E : (∂′, ξ′) (∂, ξ) is a commutative diagram of the form

H

κ

∂′
G

∂

ι
E

δ

γ
H0

G0

such that

i. (κ, γ) is a complex, i.e. γ · κ = 0,
ii. (ι, δ) is short exact,
iii. The action of E on H induced by that of H0 on H via δ makes κ: H → E a

crossed module,

iv. The action of E on G induced by that of G0 on G via γ makes ι: g → E a

crossed module. A morphism of butterflies E, E ′ : (∂′, ξ′) (∂, ξ) is an arrow α: E → E ′ commuting with the κ’s, the ι’s, the δ’s and the γ’s.

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Horizontal composition:

H

∂

κ
G

ι

κ′
∂
K

ι′

∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Horizontal composition:

E ×γ,δ′ E ′

r

s
H

∂

κ
G

ι

κ′
∂
K

ι′

∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 20

Internal crossed modules Butterflies Extensions

Horizontal composition:

E ×γ,δ′ E ′

r

s
H

∂

κ
G

ι,κ′

ι
κ′
∂
K

ι′

∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 21

Internal crossed modules Butterflies Extensions

Horizontal composition:

Q E ×γ,δ′ E ′

q

❴

r
s
H

∂

κ
G

ι,κ′

ι
κ′
∂
K

ι′

∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Horizontal composition:

Q

γ′s

δr
E ×γ,δ′ E ′

q

❴

r
s
H

κ,0

∂
κ
G

ι,κ′

ι
κ′
∂
K

0,ι′

ι′
∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 23

Internal crossed modules Butterflies Extensions

Horizontal composition:

Q

γ′s

δr
E ×γ,δ′ E ′

q

❴

r
s
H

κ,0

∂
κ
G

ι,κ′

ι
κ′
∂
K

0,ι′

ι′
∂
E

δ

γ
E ′

δ′

γ′
H0

G0 K0

Identity butterfly:

G

ker c

∂
G

∂

ker d
G ⋊ξ G0

d

c
G0

G0

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Butterfly composition extends to 2-cells, and these data form a bicategory Bfly(C) whose hom-categories are groupoids.

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Butterfly composition extends to 2-cells, and these data form a bicategory Bfly(C) whose hom-categories are groupoids. The 2-category of crossed modules embeds in the bicategory of butterflies: XMod(C) → Bfly(C)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 26

Internal crossed modules Butterflies Extensions

Butterfly composition extends to 2-cells, and these data form a bicategory Bfly(C) whose hom-categories are groupoids. The 2-category of crossed modules embeds in the bicategory of butterflies: XMod(C) → Bfly(C) Butterflies coming from morphisms:

H

κ

∂′
G

∂

ι
E

δ

γ
H0

s

G0

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 27

Internal crossed modules Butterflies Extensions

Butterfly composition extends to 2-cells, and these data form a bicategory Bfly(C) whose hom-categories are groupoids. The 2-category of crossed modules embeds in the bicategory of butterflies: XMod(C) → Bfly(C) Butterflies coming from morphisms:

H

κ

∂′
G

∂

ι
E

δ

γ
H0

s

G0

In fact, Bfly(C) is the bicategory of fractions of XMod(C) with respect to weak equivalences.

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Every butterfly is associated with a span of crossed module morphisms:

H × G

p1

p2
κ♯ι
H

κ

∂′
G

∂

ι
E

δ

γ
H0

G0

where the morphism (p1, δ) is a weak equivalence.

Alan Cigoli Extention theory and the calculus of butterflies

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Internal crossed modules Butterflies Extensions

Every butterfly is associated with a span of crossed module morphisms:

π1∂′

❴

π1(κ♯ι)
❴
π1∂

❴

H × G

p1

p2
κ♯ι
H

κ

∂′
G

∂

ι
E

δ

γ
❴
H0

❴

G0

❴

π0∂′

π0(κ♯ι)

π0∂

where the morphism (p1, δ) is a weak equivalence. This allows us to extend the definition of π0 and π1 to butterflies.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 30

Internal crossed modules Butterflies Extensions

Extensions

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 31

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 32

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Every extension K ✤

k

X

f

✤ Y

is associated with a butterfly:

∆Y
K

IK

k
X

f

α
Y

Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 33

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Every extension K ✤

k

X

f

✤ Y

is associated with a butterfly:

∆Y
K

IK

k
X

f

α
Y

❴

Aut(K)

❴

π0∆Y = Y

φ

π0IK = Out(K)

whose image under π0 is the so called “abstract kernel” of the extension.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 34

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Every extension K ✤

k

X

f

✤ Y

is associated with a butterfly:

∆Y
K

IK

k
X

f

α
Y

❴

Aut(K)

❴

π0∆Y = Y

φ

π0IK = Out(K)

whose image under π0 is the so called “abstract kernel” of the extension. We can denote by Ext(Y , K, φ) the set of isomorphism classes of butterflies in Bfly(∆Y , IK ) inducing φ on π0.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 35

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Every extension K ✤

k

X

f

✤ Y

is associated with a butterfly:

∆Y
K

IK

k
X

f

α
Y

❴

Aut(K)

❴

π0∆Y = Y

φ

π0IK = Out(K)

whose image under π0 is the so called “abstract kernel” of the extension. We can denote by Ext(Y , K, φ) the set of isomorphism classes of butterflies in Bfly(∆Y , IK ) inducing φ on π0. Global direction of IK : (ZK → Out(K), ξ)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 36

Internal crossed modules Butterflies Extensions

If C is action representative, for any object K there is a canonical crossed module: K

IK

Aut(K)

Every extension K ✤

k

X

f

✤ Y

is associated with a butterfly:

∆Y
K

IK

k
X

f

α
Y

❴

Aut(K)

❴

π0∆Y = Y

φ

π0IK = Out(K)

whose image under π0 is the so called “abstract kernel” of the extension. We can denote by Ext(Y , K, φ) the set of isomorphism classes of butterflies in Bfly(∆Y , IK ) inducing φ on π0. Global direction of IK : (ZK → Out(K), ξ) induces (ZK → Y , φ∗ξ)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 37

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 38

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 39

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1. In particular, we consider butterflies of this kind where domain and codomain are (ZK → Y , φ∗ξ). We can denote by H2(Y , ZK, φ∗ξ) the abelian group of isomorphism classes of such butterflies.

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 40

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1. In particular, we consider butterflies of this kind where domain and codomain are (ZK → Y , φ∗ξ). We can denote by H2(Y , ZK, φ∗ξ) the abelian group of isomorphism classes of such butterflies. Going back to general extensions

∆Y

K

IK

k
X

f

α
Y

Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 41

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1. In particular, we consider butterflies of this kind where domain and codomain are (ZK → Y , φ∗ξ). We can denote by H2(Y , ZK, φ∗ξ) the abelian group of isomorphism classes of such butterflies. By factorizing

∆Y
ZK

−h

K

IK

k
X

f

α
Y

Y Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 42

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1. In particular, we consider butterflies of this kind where domain and codomain are (ZK → Y , φ∗ξ). We can denote by H2(Y , ZK, φ∗ξ) the abelian group of isomorphism classes of such butterflies. By factorizing and composing...

∆Y
ZK

−a

ZK

a

−h
K

IK

k
X ′

p

p
X

f

α
Y

Y Y Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 43

Internal crossed modules Butterflies Extensions

Alternative embedding for abelian extensions:

A

−k

A
k
X

f

f
Y

Y

With this choice we have identity on π0 and π1. In particular, we consider butterflies of this kind where domain and codomain are (ZK → Y , φ∗ξ). We can denote by H2(Y , ZK, φ∗ξ) the abelian group of isomorphism classes of such butterflies. By factorizing and composing...

∆Y

K

IK

k′
X ′

f ′

α
Y

Aut(K)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 44

Internal crossed modules Butterflies Extensions

We get a simply transitive action: H2(Y , ZK, φ∗ξ) × Ext(Y , K, φ) → Ext(Y , K, φ)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 45

Internal crossed modules Butterflies Extensions

We get a simply transitive action: H2(Y , ZK, φ∗ξ) × Ext(Y , K, φ) → Ext(Y , K, φ) This is the intrinsic Schreier-Mac Lane theorem [Bourn ’08].

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 46

Internal crossed modules Butterflies Extensions

Given a crossed module K0♭K

ξ

K

∂

K0

and a morphism Y

φ

π0(∂)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 47

Internal crossed modules Butterflies Extensions

Given a crossed module K0♭K

ξ

K

∂

K0

and a morphism Y

φ

π0(∂)

π1(∂) is endowed with a Y -module structure (let φ∗ξ be the corresponding action)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 48

Internal crossed modules Butterflies Extensions

Given a crossed module K0♭K

ξ

K

∂

K0

and a morphism Y

φ

π0(∂)

π1(∂) is endowed with a Y -module structure (let φ∗ξ be the corresponding action) We denote by Ext(Y , ∂, φ) the set of isomorphism classes of extensions of Y by (∂, ξ) inducing φ on π0. That is, the triples (k, f , α) that make the following diagram a butterfly with its π0 image:

K

k

∂
X

f

α
Y

K0

❴

Y

φ

π0(∂)

Alan Cigoli Extention theory and the calculus of butterflies

SLIDE 49

Internal crossed modules Butterflies Extensions

Given a crossed module K0♭K

ξ

K

∂

K0

and a morphism Y

φ

π0(∂)

π1(∂) is endowed with a Y -module structure (let φ∗ξ be the corresponding action) We denote by Ext(Y , ∂, φ) the set of isomorphism classes of extensions of Y by (∂, ξ) inducing φ on π0. That is, the triples (k, f , α) that make the following diagram a butterfly with its π0 image:

K

k

∂
X

f

α
Y

K0

❴

Y

φ

π0(∂)

Theorem Either Ext(Y , ∂, φ) is empty, or it is a simply transitive H2(Y , π1(∂), φ∗ξ)-set.

Alan Cigoli Extention theory and the calculus of butterflies