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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

    are monads.

    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.

  • 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

    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.

  • Adding Products

    A fibration E p−→ B has products if each reindexing functor f ∗ : EJ → E I has a right adjoint Πf satisfying a Beck

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