Fibrations and Polynomial Functors - University of...
Transcript of Fibrations and Polynomial Functors - University of...
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!