The centralizer of a topos Joint work with Pieter Hofstra - - PowerPoint PPT Presentation

The centralizer of a topos

Joint work with

Pieter Hofstra

Symmetry: X

t

• August 16, 2016

2 / 28

Symmetry: X

t

• Conjugation:

A

f −1tf

• f

X

t

• August 16, 2016

2 / 28

Symmetry: X

t

• Conjugation:

A

f −1tf

• f

X

t

• However, the morphism f may not be invertible, and typically we cannot

expect it to be.

August 16, 2016 2 / 28

We do not wish to admit any restrictions on f , but rather provide t with more structure. B

g

• m
• tg

B

m

• g
• A

f

• tf

A

f

• X

t=tX X

∀m (fm = g) : mtg = tf m

August 16, 2016 3 / 28

We do not wish to admit any restrictions on f , but rather provide t with more structure. B

g

• m
• tg

B

m

• g
• A

f

• tf

A

f

• X

t=tX X

∀m (fm = g) : mtg = tf m t is an automorphism (endomorphism) of the functor X/X

f →A

X

August 16, 2016 3 / 28

We do not wish to admit any restrictions on f , but rather provide t with more structure. B

g

• m
• tg

B

m

• g
• A

f

• tf

A

f

• X

t=tX X

∀m (fm = g) : mtg = tf m t is an automorphism (endomorphism) of the functor X/X

f →A

X

Problem solved: move t along A

f X by whiskering.

X/A

m→fm

X/X

t

X

August 16, 2016 3 / 28

Addendum: X/X

t

• t

X/X

• X

t is a central automorphism of X iff tX = 1X .

August 16, 2016 4 / 28

Addendum: X/X

t

• t

X/X

• X

t is a central automorphism of X iff tX = 1X . Category: centralizer of X . Y

f

• s

Y

f

• X

t

X

August 16, 2016 4 / 28

Isotropy theory of toposes [2]

Symmetry

A symmetry of an object X of a topos E is an automorphism of the geometric morphism ΣX ⊣ X ∗ ⊣ ΠX : E /X

E .

August 16, 2016 5 / 28

Its an automorphism of the functor X ∗ : E

E /X

X ∗(E) = E × X

X ; t : X ∗ X ∗

for every object E , we have an ‘action’ tE : E × X

E ,

whose pairing E × X

E × X

with the projection to X is an isomorphism (as if X were a group).

August 16, 2016 6 / 28

Its an automorphism of the functor X ∗ : E

E /X

X ∗(E) = E × X

X ; t : X ∗ X ∗

for every object E , we have an ‘action’ tE : E × X

E ,

whose pairing E × X

E × X

with the projection to X is an isomorphism (as if X were a group).

Equivalently, its an automorphism of the left adjoint ΣX

ΣX(Y

f X) = Y ; t : ΣX

ΣX

This amounts to a compatible family Y

tf Y of automorphisms of E .

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The isotropy group Z of a topos E

Z is a group internal to E that classifies its symmetries in the sense that morphisms X

Z of E correspond naturally to automorphisms of

E /X

E .

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Z has a universal action θX in the every object X of E :

Y

(f ,t)

• tf

Y

f

• X × Z

θX

X

θX(f , t) = ftf

August 16, 2016 8 / 28

Z has a universal action θX in the every object X of E :

Y

(f ,t)

• tf

Y

f

• X × Z

θX

X

θX(f , t) = ftf On the other hand, for any X

t Z , Y f X or E , we have

E × X

tE

• E×t
• E × Z

θE

E

Y

(1Y ,tf )

• tf
• Y × Z

θY

Y

tE(e, x) = θE(e, t(x)) = et(x) ; tf (y) = θY (y, t(f (y))) = yt(f (y))

August 16, 2016 8 / 28

Z has a universal action θX in the every object X of E :

Y

(f ,t)

• tf

Y

f

• X × Z

θX

X

θX(f , t) = ftf On the other hand, for any X

t Z , Y f X or E , we have

E × X

tE

• E×t
• E × Z

θE

E

Y

(1Y ,tf )

• tf
• Y × Z

θY

Y

tE(e, x) = θE(e, t(x)) = et(x) ; tf (y) = θY (y, t(f (y))) = yt(f (y)) θZ : Z × Z

Z is conjugation: θZ(w, z) = z−1wz .

August 16, 2016 8 / 28

Z has a universal action θX in the every object X of E :

Y

(f ,t)

• tf

Y

f

• X × Z

θX

X

θX(f , t) = ftf On the other hand, for any X

t Z , Y f X or E , we have

E × X

tE

• E×t
• E × Z

θE

E

Y

(1Y ,tf )

• tf
• Y × Z

θY

Y

tE(e, x) = θE(e, t(x)) = et(x) ; tf (y) = θY (y, t(f (y))) = yt(f (y)) θZ : Z × Z

Z is conjugation: θZ(w, z) = z−1wz .

All maps of E are Z-equivariant.

August 16, 2016 8 / 28

Another perspective on symmetry: crossed sheaf

Definition

A crossed sheaf on E is a morphism X

t Z of E , but regarded as an

• bject of E /Z .

August 16, 2016 9 / 28

Another perspective on symmetry: crossed sheaf

Definition

A crossed sheaf on E is a morphism X

t Z of E , but regarded as an

• bject of E /Z .

E /Z is the topos of crossed sheaves on E .

August 16, 2016 9 / 28

Isotropy group of E /Z

What is it?

Z(Z)

• ζ
• Z × Z
• Z

Z(Z) = {(z, w) | zw = wz} , ζ(z, w) = z .

August 16, 2016 10 / 28

Isotropy group of E /Z

What is it?

Z(Z)

• ζ
• Z × Z
• Z

Z(Z) = {(z, w) | zw = wz} , ζ(z, w) = z .

Generic element: δ : 1Z

ζ

Z

z→(z,z)

• 1Z
• Z(Z)

ζ

• Z

(Not to be confused with the unit element u(z) = (z, u) of ζ .)

August 16, 2016 10 / 28

Twist automorphism α : central automorphism of E /Z corresponding to δ

What is α ?

Crossed sheaf X

t Z . Let αt(x) = θX(x, t(x)) = xt(x) .

X

αt

• t
• X

t

• Z

t(αt(x)) = t(xt(x)) = t(x)−1t(x)t(x) = t(x)

August 16, 2016 11 / 28

Quotient of a crossed sheaf

Definition

Let X

t Z be a crossed sheaf on E .

Coequifier E /X

• ⇓t ⇓1

E

ψ

Et

Et = { E | ∀(e, x) , tE(e, x) = et(x) = e , i.e., E × X

tE E is the

projection }

August 16, 2016 12 / 28

The geometric morphism ψ : E

Et

Et is a topos; ψ is connected atomic: ψ∗ is inclusion; ψ! is the orbit space of the action = coequalizer E × X

et(x)

• e

E

unit ψ!E ;

ψ∗E = {e | ∀x ∈ X , et(x) = e} = equalizer ψ∗E counit E

• E X (transposed maps)

August 16, 2016 13 / 28

Two special quotients

• 1. Quotient of the identity map Z

Z

E /Z

• ⇓θ ⇓1

E

ψ

Eθ

The topos Eθ consists of all isotropically trivial objects: those X for which the universal action θX is trivial. ψ∗(X) = X . ψ!(X) = orbit space of universal action of θX .

August 16, 2016 14 / 28

• 2. Quotient of the generic element δ : 1Z

ζ

E /Z

• ⇓α ⇓1

E /Z Z(E )

For any X

t Z , we have αt(x) = θX(x, t(x)) = xt(x) .

August 16, 2016 15 / 28

• 2. Quotient of the generic element δ : 1Z

ζ

E /Z

• ⇓α ⇓1

E /Z Z(E )

For any X

t Z , we have αt(x) = θX(x, t(x)) = xt(x) .

Definition

A crossed sheaf X

t Z is unital if ∀x ∈ X , xt(x) = x .

August 16, 2016 15 / 28

• 2. Quotient of the generic element δ : 1Z

ζ

E /Z

• ⇓α ⇓1

E /Z Z(E )

For any X

t Z , we have αt(x) = θX(x, t(x)) = xt(x) .

Definition

A crossed sheaf X

t Z is unital if ∀x ∈ X , xt(x) = x .

Proposition

The coequifier topos Z(E ) equals the full subcategory of E /Z on the unital crossed sheaves.

August 16, 2016 15 / 28

Pushout diagram

E /Z

• ⇓α ⇓1

E /Z

• conn. atomic

ϕ

Z(E )

atomic

• E /Z
• ⇓θ ⇓1

E

• conn. atomic

ψ

Eθ

The right-hand square is a pushout.

August 16, 2016 16 / 28

Equivalence of unital crossed sheaves with central automorphisms

Interpret the equivalence as a lifting property

If a crossed sheaf X

t Z is unital, then there is a central automorphism

α(t) of X such that the ‘whiskering’ diagram commutes. E /X

t

• α(t)

E /X

t

• E /Z

α

E /Z

August 16, 2016 17 / 28

Equivalence of unital crossed sheaves with central automorphisms

Interpret the equivalence as a lifting property

If a crossed sheaf X

t Z is unital, then there is a central automorphism

α(t) of X such that the ‘whiskering’ diagram commutes. E /X

t

• α(t)

E /X

t

• E /Z

α

E /Z

Proof

For any Y

s X , define α(t)s = αts . Verify sα(t)s = s :

s(α(t)s(y)) = s(α(ts(y))) = s(yts(y)) = s(y)ts(y) = s(y) . The second last equality holds because s is Z-equivariant. The last equality holds because t is unital.

August 16, 2016 17 / 28

X

t Z → (X, α(t)) is an equivalence

For every central automorphism β of X , ∃! unital X

t Z such that

β = α(t) . E /X

t

• β=α(t)

E /X

t

• E /Z

α

E /Z

August 16, 2016 18 / 28

X

t Z → (X, α(t)) is an equivalence

For every central automorphism β of X , ∃! unital X

t Z such that

β = α(t) . E /X

t

• β=α(t)

E /X

t

• E /Z

α

E /Z

Proof

Whiskering β with E /X

E corresponds to a morphism X

t Z . It

follows that t is unital.

August 16, 2016 18 / 28

Tensor product on crossed sheaves

X

s

• Y

t

• X × Y

s⊗t

• (x, y)

❴

• ⊗

= Z Z Z s(x)t(y) The identity object I is the unit 1

u Z .

August 16, 2016 19 / 28

Closed category

(E /Z, ⊗, u) is closed

The ‘hom’ object [s, t] of X

s Z and Y t Z is given by the pullback:

(f , z)

❴

• P
• [s,t]
• Y X

tX

• z

Z

Z X

The bottom horizontal morphism is the transpose of X × Z

s×Z

Z × Z mult Z ; (x, z) → s(x)z .

P = {(f , z) ∈ Y X × Z | ∀ x ∈ X : t(f (x)) = s(x)z } .

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For any U

r Z , there is a natural bijection between morphisms of

crossed sheaves: r

[s, t]

s ⊗ r

t .

The corresponding diagrams in E are: U

• r
• P

[s,t]

• Z

X × U

• s⊗r
• Y

t

• Z

August 16, 2016 21 / 28

Braiding of crossed sheaves

Definition (crossed G-sets: Freyd and Yetter [1])

X

s Z and Y t Z

bs,t : s ⊗ t

t ⊗ s ; bs,t(x, y) = (y, xt(y))

August 16, 2016 22 / 28

Braiding of crossed sheaves

Definition (crossed G-sets: Freyd and Yetter [1])

X

s Z and Y t Z

bs,t : s ⊗ t

t ⊗ s ; bs,t(x, y) = (y, xt(y))

bs,t is a morphism of crossed sheaves

X × Y

bs,t

• s⊗t
• Y × X

t⊗s

• Z

t ⊗ s(bs,t(x, y)) = t(y)s(xt(y)) = t(y)t(y)−1s(x)t(y) = s(x)t(y) .

August 16, 2016 22 / 28

Balanced (Joyal and Street [3])

αu = 1 and s ⊗ t

bs,t αs⊗t

• t ⊗ s

αt⊗αs

• s ⊗ t

t ⊗ s

bt,s

• (x, y) ✤
• ❴
• (y, xt(y))

❴

• (xs(x)t(y), ys(x)t(y))

(yt(y), xt(y)s(xt(y)))

✤

• NOTE: s and t unital
• αs⊗t = bt,sbs,t (full twist).

August 16, 2016 23 / 28

Unital tensor: s ⊗u t = ϕ!(s ⊗ t)

Lemma

The unital tensor s ⊗u t of unital X

s Z and Y t Z is the coequalizer

t ⊗ s

bt,s

• bs,t −1

s ⊗ t s ⊗u t .

August 16, 2016 24 / 28

Unital tensor: s ⊗u t = ϕ!(s ⊗ t)

Lemma

The unital tensor s ⊗u t of unital X

s Z and Y t Z is the coequalizer

t ⊗ s

bt,s

• bs,t −1

s ⊗ t s ⊗u t .

Proof

s ⊗ t

bs,t

• αs⊗t

1

s ⊗ t

1

• t ⊗ s

bt,s

• bs,t −1

s ⊗ t

August 16, 2016 24 / 28

Involution

A crossed sheaf X

s Z has a ‘dual’ X s∗ Z

s∗(x) = s(x)−1 . We have s∗∗ = s , (s ⊗ t)∗ ∼ = t∗ ⊗ s∗ s unital

• s∗ unital.

(INV) (s ⊗ t)∗

bs,t ∗

• (t ⊗ s)∗
• t∗ ⊗ s∗

s∗ ⊗ t∗

bs∗,t∗

• (x, y) ✤
• ❴
• (y, xt(y))

❴

• (y, xt(y)t(y)−1 = x)

(xt(y), y)

✤

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25 / 28

Mutually inverse

s∗

αs∗ s∗ αs ∗

• s

αs

s

αs∗ ∗

• ( )∗ = Σz−1 .

E /Z

z−1

• α−1

E /Z

z−1

• E /Z

α

E /Z

E /Z

( )∗

• α−1

E /Z

( )∗

• E /Z

α

E /Z

August 16, 2016 26 / 28

Theorem

The centralizer Z(E ) of a topos E is closed symmetric, with a covariant anti-involution.

August 16, 2016 27 / 28

Theorem

The centralizer Z(E ) of a topos E is closed symmetric, with a covariant anti-involution.

Proof

The braiding on crossed sheaves becomes (via ψ!) a symmetry of the unital tensor. We also have [s, t]u = ψ∗[s, t] . The involution satisfies (s ⊗u t)∗ ∼ = t∗ ⊗u s∗ by (INV) and the braid coequalizer formulation of s ⊗u t . ✷

August 16, 2016 27 / 28

Bibliography

[1] P. Freyd and D. Yetter. Braided compact closed categories with applications to low dimensional topology. Advances in Mathematics, 77:156–182, 1989. [2] J. Funk, P. Hofstra, and B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories, 26(24):660–709, 2012. [3] A. Joyal and R. Street. Braided tensor categories. Advances in Mathematics, 102:20–78, 1993.

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