Fibrations and Polynomial Functors

Tamara von Glehn

University of Cambridge

BMC - 27 March 2013

Plan

1 Polynomials

2 Spans and Fibrations

3 Sums and Products

4 Polynomials

Polynomial Functors

Let B be a locally cartesian closed category.A polynomial in B is a diagram

F = B f //

szztttttttt

At$$JJJJJJJJ

I J .

F represents the polynomial functor

PF = B/B∏

f // B/A ∑t

$$JJJJJJJ

B/I

s∗ ::uuuuuuuB/J .

In the internal language of B,

(Xi )i∈I 7→ (∑a∈Aj

∏b∈Ba

Xs(b))j∈J .

Polynomial FunctorsExamples:

The identity functor B/A→ B/A is represented by

A

wwwwww

wwwwww AHHHHHH

HHHHHH

A A.

The functor A×− : B → B is represented by

A

{{xxxxxxx A##GGGGGG

1 1.

The list monad Σn∈N(−)n : Set→ Set is represented by

{(i , n) | i ≤ n ∈ N} π2 //

xxppppppppN

""FFFFFFF

1 1.

Moerdijk & Palmgren (2000); Abbott, Altenkirch & Ghani (2003);Gambino & Hyland (2004); Gambino & Kock (2009)

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

2-cells:I Boo // A // J

E

I ′ B′oo // A′ // J ′.

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

2-cells:I

��

Boo // A // J

��

E

I ′ B′oo // A′ // J ′.

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

2-cells:I

��

Boo // A

��

// J

��

EOO

77ooooooo

��

_�

I ′ B′oo // A′ // J ′.

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N g∗M M

E

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

F G

PG◦F ∼= PGPF

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrrrrrrrrrrrrrrrrr _� g∗M //

εyyrrrrrr

��

_� M∏g h

��

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

F G

PG◦F ∼= PGPF

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrrrrrrrrrrrrrrrrr _� g∗M //

εyyrrrrrr

��

_� M

��

��111111111111111

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

G◦F

PG◦F ∼= PGPF

Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrrrrrrrrrrrrrrrrr _� g∗M //

εyyrrrrrr

��

_� M

��

��111111111111111

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

G◦F

PG◦F ∼= PGPF

The Bicategory of Spans

Categories and spans of functors form a bicategory Span(Cat).Composition is by pullback:

P������

AAAA?�

E������

��???? F~~~~~~

��????

A B C

It is a bicategory enriched in 2-Cat:

2-cells E

{{wwwwwww

##GGGGGGG

��

A B

E ′

ccGGGGGG

;;xxxxxxx

3-cells E

{{wwwwwww

##GGGGGGG

����

____ +3A B

E ′

ccGGGGGG

;;xxxxxxx

The Bicategory of Spans

Monads in Span(K) are internal categories in K.In Span(Cat):

Φ =B2

c

~~~~~~~~~~ d

@@@@@@@@

B B

and Ψ =B2

d

~~~~~~~~~~ c

@@@@@@@@

B B

are monads.

E����� q

��???

1 B

is a pseudo-algebra for composition with Φ on theright if E q−→ B is a cloven fibration.

Ep�����

��===

B 1

is a pseudo-algebra for composition with Φ on theleft if E p−→ B is a cloven opfibration.

The Bicategory of Fibrations

Street (1974)

Pseudo-bimodules for Φ are two-sided fibrations:

Epopfibration

���������� qfibration��

::::::::

A B

Compatibility:For all morphisms a f−→ qe and pe g−→ b,

I the chosen q-cartesian lifting f ∗e → e is p-vertical,I the chosen p-opcartesian lifting e → g!e is q-vertical,I the canonical morphism g!f ∗e → f ∗g!e is an identity.

The Bicategory of Fibrations

Two-sided fibrations form a bicategory Fib enriched in 2-Cat.

2-cells (pseudo-algebra morphisms) are maps of spans preservingcartesian and opcartesian morphisms.3-cells are natural transformations which are vertical over A and B.

E

}}zzzzzzz

!!CCCCCCC

����

____ +3A B

E ′

aaCCCCCCC

=={{{{{{{

The Bicategory of Fibrations

Two-sided fibrations form a bicategory Fib enriched in 2-Cat.

Composition is by pullback and quotient

E������

��???? B2

c~~}}}} d

AAAA E������

��====

A B

����

B C

E��~~~~

BBBB E~~||||

��????

A B C

The identity I is Φ = B2c~~}}} d

AAA

B B; Ψ = B2

d~~}}} c

AAA

B Bis not a

2-sided fibration.

Pseudo-distributive Law

When does Ψ lift to a pseudo-monad on Fib(A,B)?

The spanB

����� ????

1 Bis a pseudo-Φ-algebra.

If there is a lifting,

Ψ

B����� ????

1 B

is a pseudo-Φ-algebra

i.e. B2 c−→ B is a cloven fibration,which is equivalent to B having pullbacks.

Liftings correspond to pseudo-distributive laws ΦΨ→ ΨΦ.

Pseudo-distributive Law

If B has pullbacks, there is a pseudo-distributive law ΦΨ→ ΨΦ:

B2

d

����������c

��88888888 B2

c

����������d

��88888888

ΦΨ

��

.

��555 .

��

B B B.

_

��

B2

c

����������d

��88888888 B2

d

����������c

��88888888

.

��7777

������?�ΨΦ .

��

.

��

B B B.

There is a pseudo-distributive law ΦΨ→ ΨΦ ⇐⇒ B has pullbacks

A two-sided fibration has sums if it is a pseudo-algebra for Ψ.

Adding Sums

A fibration E p−→ B has sums if each reindexing functor f ∗ : EJ → E I

has a left adjoint Σf satisfying a Beck-Chevalley condition.Since

Ψ(F ◦ E) ∼= F ◦Ψ(E),∑��

???�����

B B= ΨI is a pseudo-monad in Fib which freely adds

sums by composition.

∑is the category of spans in B:

I

��

Aoo //

��

J

��

I ′ A′oo // J ′

Opposites

Each two-sided fibration has an opposite fibration

Ep

��������� q

��??????? Eo

po

~~}}}}}}}} qo

AAAAAAAA

A B A B

with the same objects and reversed vertical morphisms.

(−)o is an identity-on-objects pseudo-functor Fib→ Fib such that((−)o)o ∼= 1.

E has products if Eo has sums.

Adding Products

A fibration E p−→ B has products if each reindexing functorf ∗ : EJ → E I has a right adjoint Πf satisfying a Beck-Chevalleycondition.Since

(∑

(Eo))o ∼=∑ o ◦ E ,∏

��???

�����

B B=

∑o is a pseudo-monad in Fib which freely addsproducts by composition.

Objects of∏

are spans. Morphisms:

I

��

Aoo // J

��

EOO

77nnnnnnn

��

_�

I ′ A′oo // J ′.

Pseudo-distributive Law

When is there a pseudo-distributive law∏ ∑

→∑ ∏

?

The spanB

����� ????

1 Bis a pseudo-

∏-algebra.

If there is a lifting of∑

to pseudo-∏-algebras,

∑ B����� ????

1 B

is a pseudo-∏-algebra

i.e. B2 c−→ B has products,which is equivalent to B being locally cartesian closed.

Polynomials

If B is locally cartesian closed, there is a pseudo-distributive law∏ ∑→

∑ ∏: ∑��

��5555555

∏��

��5555555∏ ∑

��

◦

B B B B

∏��

��5555555

∑��

��5555555∑ ∏ ◦

B B B B

. .foo g// . h // .

_

��

. //εzzttttt

��

_� .

Πhg��

.

g $$

fzzttttt

. .h// .

Pseudo-distributive law∏ ∑

→∑ ∏

⇐⇒ B is locally cartesian closed

Polynomials

If B is locally cartesian closed,∑ ∏}}zzzz

!!DDDD

B Bis a pseudo-monad in Fib.

Morphisms:I

��

Boo // A

��

// J

��

EOO

77ooooooo

��

_�

I ′ B′oo // A′ // J ′.

Polynomials

Composition:

(∑ ∏

)(∑ ∏

) ∼=∑

(∏ ∑

)∏ ∑

λ∏

−−−−→∑ ∑ ∏ ∏ µµ−→

∑ ∏

N g∗M M

E

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

Polynomials

Composition:

(∑ ∏

)(∑ ∏

) ∼=∑

(∏ ∑

)∏ ∑

λ∏

−−−−→∑ ∑ ∏ ∏ µµ−→

∑ ∏

N g∗M M

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

Polynomials

Composition:

(∑ ∏

)(∑ ∏

) ∼=∑

(∏ ∑

)∏ ∑

λ∏

−−−−→∑ ∑ ∏ ∏ µµ−→

∑ ∏

N g∗M //

εyyrrrrrr

��

_� M∏g h

����

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

Polynomials

Composition:

(∑ ∏

)(∑ ∏

) ∼=∑

(∏ ∑

)∏ ∑

λ∏

−−−−→∑ ∑ ∏ ∏ µµ−→

∑ ∏

N //

yyrrrrrrrrrrrrrrrrrr _� g∗M //

εyyrrrrrr

��

_� M

��

��111111111111111

Exxrrrrrrrr ?� h

&&LLLLLLL

B f//

sxxrrrrrrrr At &&LLLLLLLL D g

//

uxxrrrrrrrr Cv &&LLLLLLL

I J K .

Polynomials

If B is locally cartesian closed,∑ ∏}}zzzz

!!DDDD

B Bis a pseudo-monad in Fib.

Polynomials with their usual morphisms and composition form a two-sidedfibration, which is a pseudo-monad on fibrations that freely adds productsand then sums.

GeneralisationsFunctors, other monads, internal categories...

Thank you!

Polynomials

If B is locally cartesian closed,∑ ∏}}zzzz

!!DDDD

B Bis a pseudo-monad in Fib.

Polynomials with their usual morphisms and composition form a two-sidedfibration, which is a pseudo-monad on fibrations that freely adds productsand then sums.

GeneralisationsFunctors, other monads, internal categories...

Thank you!