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• 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 // s

zzttt ttt

tt A

t \$\$J

JJJ JJJ

J

I J .

F represents the polynomial functor

PF = B/B ∏

f // B/A ∑ t

\$\$JJ JJJ

JJ

B/I

s∗ ::uuuuuuu B/J .

In the internal language of B,

(Xi )i∈I 7→ ( ∑ a∈Aj

∏ b∈Ba

Xs(b))j∈J .

• Polynomial Functors Examples:

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

A

ww ww

ww

ww ww

ww A

HH HH

HH

HH HH

HH

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

A

{{xxx xxx

x A ##G

GG GG

G

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

{(i , n) | i ≤ n ∈ N} π2 //

xxppp ppp

pp N

""F FF

FF FF

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

��

E OO

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 //

sxxrrr rrr

rr A t &&LL

LLL LLL D g

//

uxxrrr rrr

rr C v &&LL

LLL LL

I J K . F G

PG◦F ∼= PGPF

• Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrr rrr

rrr rrr

rrr rrr _

� g∗M //

εyyrrr rrr

��

_� M∏

g h

��

E xxrrr

rrr rr ? h

&&LL LLL

LL

B f //

sxxrrr rrr

rr A t &&LL

LLL LLL D g

//

uxxrrr rrr

rr C v &&LL

LLL LL

I J K . F G

PG◦F ∼= PGPF

• Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrr rrr

rrr rrr

rrr rrr _

� g∗M //

εyyrrr rrr

��

_� M

��

�� 11

11 11

11 11

11 11

1

E xxrrr

rrr rr ? h

&&LL LLL

LL

B f //

sxxrrr rrr

rr A t &&LL

LLL LLL D g

//

uxxrrr rrr

rr C v &&LL

LLL LL

I J K . G◦F

PG◦F ∼= PGPF

• Polynomials

Polynomials are the horizontal arrows of a double category Poly.

Composition:

N //

yyrrr rrr

rrr rrr

rrr rrr _

� g∗M //

εyyrrr rrr

��

_� M

��

�� 11

11 11

11 11

11 11

1

E xxrrr

rrr rr ? h

&&LL LLL

LL

B f //

sxxrrr rrr

rr A t &&LL

LLL LLL D g

//

uxxrrr rrr

rr C v &&LL

LLL LL

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

AA AA?

E

��

?? ??

F ~~~~

~~ ��

?? ??

A B C

It is a bicategory enriched in 2-Cat:

2-cells E

{{ww ww

ww w

##G GG

GG GG

��

A B

E ′

ccGGGGGG

;;xxxxxxx

3-cells E

{{ww ww

ww w

##G GG

GG GG

����

____ +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

E ����

� q ��

?? ?

1 B

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

Ep ��

� ��

== =

B 1

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

• The Bicategory of Fibrations

Street (1974)

Pseudo-bimodules for Φ are two-sided fibrations:

Ep opfibration

���� ��

�� ��

q fibration ��

:: ::

:: ::

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 preserving cartesian and opcartesian morphisms. 3-cells are natural transformations which are vertical over A and B.

E

}}zz zz

zz z

!!C CC

CC CC

����

____ +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

��

?? ?? B

2 c ~~}}

}} d

AA AA

E ����

�� ��

== ==

A B

����

B C

E ~~

~~ B

BB B E

~~|| ||

�� ??

??

A B C

The identity I is Φ = B 2

c ~~}}

} d

AA A

B B ; Ψ = B

2 d ~~}}

} c

AA A

B B is not a

2-sided fibration.

• Pseudo-distributive Law

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

The span B

����� ? ???

1 B is 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

�� 88

88 88

88 B2

c

���� ��

�� �� d

�� 88

88 88

88

ΦΨ

��

.

�� 55

5 .

��

B B B .

_

��

B2 c

���� ��

�� �� d

�� 88

88 88

88 B2

d

���� ��

�� �� c

�� 88

88 88

88 .

�� 77

77

���� ��?

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.

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

E p

q

�� ??

?? ??

? Eo po

~~}} }}

}} }} qo

AA

AA AA

AA

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.