The word problem for -categories

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					  The word problem for ΣΠ-categories,
i.e. properties of free ΣΠ-categories (II)

               Robin Cockett
       Department of Computer Science
            University of Calgary
                 CANADA

           * Luigi Santocanale *
               LIF, Marseille
                    e
           Universit´ de Provence
                 FRANCE


      Cape Town, Category Theory 2009

                                             1/26
Outline


   Introductory remarks
       ΣΠ-categories and their theory
       Context, some motivations
       The free ΣΠ-category

   “En route” towards a decision procedure
      The CT2006 results

   Softness, cardinals, and (some hints on) a decision procedure
      Understanding softness
      Pushouts-pullbacks, bouncing




                                                                   2/26
Outline


   Introductory remarks
       ΣΠ-categories and their theory
       Context, some motivations
       The free ΣΠ-category

   “En route” towards a decision procedure
      The CT2006 results

   Softness, cardinals, and (some hints on) a decision procedure
      Understanding softness
      Pushouts-pullbacks, bouncing




                                                                   3/26
ΣΠ-categories and their theory

   A ΣΠ-category :
   a category with (chosen) finite products and finite coproducts.


       Binary products:
                                        ,
                   [X,A] × [X,B ] −−  −→ [X,A × B ]
                                       πi
                                       −
                           [ Xi , A ] −→ [ X0 × X1 , A ]

       such that

                πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

       Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
                                             . . . and dual equations.


                                                                         4/26
ΣΠ-categories and their theory

   A ΣΠ-category :
   a category with (chosen) finite products and finite coproducts.


       Binary products:
                                        ,
                   [X,A] × [X,B ] −−  −→ [X,A × B ]
                                       πi
                                       −
                           [ Xi , A ] −→ [ X0 × X1 , A ]

       such that

                πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

       Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
                                             . . . and dual equations.


                                                                         4/26
ΣΠ-categories and their theory

   A ΣΠ-category :
   a category with (chosen) finite products and finite coproducts.


       Binary products:
                                        ,
                   [X,A] × [X,B ] −−  −→ [X,A × B ]
                                       πi
                                       −
                           [ Xi , A ] −→ [ X0 × X1 , A ]

       such that

                πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

       Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
                                             . . . and dual equations.


                                                                         4/26
Units, empty products and coproducts,
                terminal and initial objects


       The units:

                    [ X , 1 ] = { !X }      [ 0, A ] = { ?A }



       Some derived equations:

                       {!X , !Y } = !X +Y         →
                                            :X +Y − 1
                               !0 = ? 1       →
                                            :0− 1




                                                                5/26
Units, empty products and coproducts,
                terminal and initial objects


       The units:

                    [ X , 1 ] = { !X }      [ 0, A ] = { ?A }



       Some derived equations:

                       {!X , !Y } = !X +Y         →
                                            :X +Y − 1
                               !0 = ? 1       →
                                            :0− 1




                                                                5/26
The word problem for ΣΠ-categories?
      Elementary but important mathematical theory,
      Ties to linear logic (additives),

      A challenge for category theorists and logicians:
           Joyal & Hu 96, Hu 98,
           Dosen 99, Dosen & Petric 2009,
           Hughes 2000, Hughes & van Glabbek 03,
           Joyal 95,
           Cockett & Seely 2001,

      From Whitman’s condition to softess:
                mimicking the theory of free lattices,

      Joyal’s theory of communication:
      free ΣΠ-categories are semantics for communication protocols,
                            we tackle equivalence of these protocols.

                                                                        6/26
The word problem for ΣΠ-categories?
      Elementary but important mathematical theory,
      Ties to linear logic (additives),

      A challenge for category theorists and logicians:
           Joyal & Hu 96, Hu 98,
           Dosen 99, Dosen & Petric 2009,
           Hughes 2000, Hughes & van Glabbek 03,
           Joyal 95,
           Cockett & Seely 2001,

      From Whitman’s condition to softess:
                mimicking the theory of free lattices,

      Joyal’s theory of communication:
      free ΣΠ-categories are semantics for communication protocols,
                            we tackle equivalence of these protocols.

                                                                        6/26
The word problem for ΣΠ-categories?
      Elementary but important mathematical theory,
      Ties to linear logic (additives),

      A challenge for category theorists and logicians:
           Joyal & Hu 96, Hu 98,
           Dosen 99, Dosen & Petric 2009,
           Hughes 2000, Hughes & van Glabbek 03,
           Joyal 95,
           Cockett & Seely 2001,

      From Whitman’s condition to softess:
                mimicking the theory of free lattices,

      Joyal’s theory of communication:
      free ΣΠ-categories are semantics for communication protocols,
                            we tackle equivalence of these protocols.

                                                                        6/26
The word problem for ΣΠ-categories?
      Elementary but important mathematical theory,
      Ties to linear logic (additives),

      A challenge for category theorists and logicians:
           Joyal & Hu 96, Hu 98,
           Dosen 99, Dosen & Petric 2009,
           Hughes 2000, Hughes & van Glabbek 03,
           Joyal 95,
           Cockett & Seely 2001,

      From Whitman’s condition to softess:
                mimicking the theory of free lattices,

      Joyal’s theory of communication:
      free ΣΠ-categories are semantics for communication protocols,
                            we tackle equivalence of these protocols.

                                                                        6/26
ΣΠs: a theory of communication
   Objects of ΣΠ(A) are channels,
                          arrows of ΣΠ(A) are mediating protocols.

                                             G
                                                         M        GH



                                            mediating protocol

           1st channel                                                                2nd channel

                                              !"# !"#
                                    #  /// '&%$ '&%$            !"#
                                                                 '&%$        !"#
                                                                            '&%$
                                        /                                                /
                                                        M                              /× //
                               :  G //:        y                      r          :  H //:
            !"#
           '&%$          !"#
                        '&%$                                                                        !"#
                                                                                                   '&%$          !"#
                                                                                                                '&%$
                                                                                  
                  q ˆ                                                                                     p ‡
                                           hooks
   left user                                                                                                     right user

                                                                                                                              7/26
ΣΠ(A), the free ΣΠ category on A

      free ΣΠ-category generated by A:
                         η
             A GG               G ΣΠ(A)
                 GG                    
                   GG
                     GG
                       GGF
                         GG
                           GG            ∃!F o
                             GG              ˜     ΣΠ-functor
                               GG
                                  GG 
                                    5      o
                                      C          ΣΠ-category



      ΣΠ-functor:
      preserves finite products and finite coproducts,



                                                                8/26
ΣΠ(A), the free ΣΠ category on A

      free ΣΠ-category generated by A:
                         η
             A GG               G ΣΠ(A)
                 GG                    
                   GG
                     GG
                       GGF
                         GG
                           GG            ∃!F o
                             GG              ˜     ΣΠ-functor
                               GG
                                  GG 
                                    5      o
                                      C          ΣΠ-category



      ΣΠ-functor:
      preserves finite products and finite coproducts,



                                                                8/26
CS01 : a Lambek-style construction of ΣΠ(A)
                      f
                  →
                x− y
                     η(f )
                 −→
           η(x) − − η(x)
                                                            −
                                                                !
                                                                      R1
                                                       →
                                                     X − 1
                 f                            f                       g
                   →
               Xi − A                          →
                                            X − A                      →
                                                                    X − B
                                     Li ×                                   R×
                      πi (f )                        f ,g
                 −−
         X0 × X1 − − → A                       −−
                                             X −−→ A × B


                 −
                 ?
                                L0
                 →
               0− A
           f                    g                           f
         X − A
            →         Y − A
                         →                            →
                                                   X − Aj
                                      L+                                   Rj +
                     {f ,g }                      σj (f )
          X +Y −−→ A
               −−                             −−
                                            X − − → A0 + A1


                                                                                  9/26
CS01: confluence modulo equations

   Proposition
   The cut-elimination procedure is confluent modulo the equations:

     πi ( f , g ) =    πi (f ), πi (g )   σj ({f , g }) = {σj (f ), σj (g )}

                            πi (σj (f )) = σj (πi (f ))

              { f11 , f12 , f21 , f22 } = {f11 , f21 }, {f12 , f22 }

          πi (!) = !                            σj (?) = ?

         {!, !} = !                              ?, ?     = ?

                                      !0 =?1



                                                                               10/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
                         →
    1. The functor η : A − ΣΠ(A) is full and faithful.
    2. Generators are atomic:
                                                     →
                                        [ η(a), Yj ] − [ η(a),             Yj ]
                                j                                     j

                                                      →
                                        [ Xi , η(b) ] − [       Xi , η(b) ]
                                i                           i

       are isomorphisms.
    3. ΣΠ(A) is soft:

                  i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]


                                                                                    
                 i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

       is a pushout.
                                                                                                       11/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
                         →
    1. The functor η : A − ΣΠ(A) is full and faithful.
    2. Generators are atomic:
                                                     →
                                        [ η(a), Yj ] − [ η(a),             Yj ]
                                j                                     j

                                                      →
                                        [ Xi , η(b) ] − [       Xi , η(b) ]
                                i                           i

       are isomorphisms.
    3. ΣΠ(A) is soft:

                  i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]


                                                                                    
                 i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

       is a pushout.
                                                                                                       11/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
                         →
    1. The functor η : A − ΣΠ(A) is full and faithful.
    2. Generators are atomic:
                                                     →
                                        [ η(a), Yj ] − [ η(a),             Yj ]
                                j                                     j

                                                      →
                                        [ Xi , η(b) ] − [       Xi , η(b) ]
                                i                           i

       are isomorphisms.
    3. ΣΠ(A) is soft:

                  i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]


                                                                                    
                 i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

       is a pushout.
                                                                                                       11/26
Characterization of free ΣΠ-categories (cont.)




   Theorem
   The pair (η, ΣΠ(A)) satisfies 1,2,3.

                                                 →
   If C is a ΣΠ-category “generated” by A, F : A − C,
           and (F , C) satisfies 1,2,3,
                                       ˆ      →
                 then the extension F : ΣΠ(A) − C is an equivalence.




                                                                       12/26
Outline


   Introductory remarks
       ΣΠ-categories and their theory
       Context, some motivations
       The free ΣΠ-category

   “En route” towards a decision procedure
      The CT2006 results

   Softness, cardinals, and (some hints on) a decision procedure
      Understanding softness
      Pushouts-pullbacks, bouncing




                                                                   13/26
A result presented at CT06


   Theorem
   In ΣΠ(A) coproducts are weakly disjoint:
                                 g
                         X                    G Y1


                     f                           σ1


                                σ0           
                     Y0                  G Y0 + Y1




                                                      14/26
A result presented at CT06


   Theorem
   In ΣΠ(A) coproducts are weakly disjoint:
                                                    g
                         X                                  jGR Y1
                                  ∃h                  j jjjj
                                    '           jj?Y1
                                                j
                                            jjjj
                                        0
                     f                                          σ1
                                  
                             ? Y0
                              
                          ×                  σ0             
                     Y0                                   G Y0 + Y1




                                                                      14/26
. . . and some Corollaries


       decide in linear time whether
           an object of ΣΠ(A) is isomorphic to 0 or 1,
           an arrow of ΣΠ(A) factors through 0 and or 1.




       a simple characterization of monic coproduct injections:

                                      →
                              σ0 : A − A + B

       is monic iff either B is not pointed or A is pointed.




                                                                  15/26
. . . and some Corollaries


       decide in linear time whether
           an object of ΣΠ(A) is isomorphic to 0 or 1,
           an arrow of ΣΠ(A) factors through 0 and or 1.




       a simple characterization of monic coproduct injections:

                                      →
                              σ0 : A − A + B

       is monic iff either B is not pointed or A is pointed.




                                                                  15/26
. . . and some Corollaries


       decide in linear time whether
           an object of ΣΠ(A) is isomorphic to 0 or 1,
           an arrow of ΣΠ(A) factors through 0 and or 1.




       a simple characterization of monic coproduct injections:

                                      →
                              σ0 : A − A + B

       is monic iff either B is not pointed or A is pointed.




                                                                  15/26
. . . and some Corollaries


       decide in linear time whether
           an object of ΣΠ(A) is isomorphic to 0 or 1,
           an arrow of ΣΠ(A) factors through 0 and or 1.




       a simple characterization of monic coproduct injections:

                                      →
                              σ0 : A − A + B

       is monic iff either B is not pointed or A is pointed.




                                                                  15/26
Outline


   Introductory remarks
       ΣΠ-categories and their theory
       Context, some motivations
       The free ΣΠ-category

   “En route” towards a decision procedure
      The CT2006 results

   Softness, cardinals, and (some hints on) a decision procedure
      Understanding softness
      Pushouts-pullbacks, bouncing




                                                                   16/26
Softness: the focus of a decision procedure


   A decision procedure focuses on the homset [ X × Y , A + B ].


   For example:
                              →
   l e t equal f g : X − A × B =
       let
          f = f1 , f2
       and
          g = g1 , g2
       in
          e q u a l f1 g1 && e q u a l f2 g2




                                                                   17/26
Softness: the focus of a decision procedure


   A decision procedure focuses on the homset [ X × Y , A + B ].


   For example:
                              →
   l e t equal f g : X − A × B =
       let
          f = f1 , f2
       and
          g = g1 , g2
       in
          e q u a l f1 g1 && e q u a l f2 g2




                                                                   17/26
Understanding softness
   The homset [ X × Y , A + B ] is . . .




                                           18/26
Understanding softness
   The homset [ X × Y , A + B ] is . . .

   the pushout

   [X,A] + [X,B ] + [Y,A] + [Y,B ]         G [X × Y,A] + [X × Y,B ]




                                                      
         [X,A + B ] + [Y,A + B ]               G [X × Y,A + B ]




                                                                  18/26
Understanding softness
   The homset [ X × Y , A + B ] is . . .

   the colimit of the “diagram of cardinals”:
                         π0                     σ0
      [X × Y,A] o                 [X,A]              G [X,A + B ]
                  y                                        y

         π1                                             σ1



        [Y,A]                                          [X,B ]


         σ0                                             π0


                        σ1                     π1
                                                            
      [Y,A + B ] o                [Y,B ]             G [X × Y,B ]

                                                                    18/26
Understanding softness
   The homset [ X × Y , A + B ] is . . .

   the quotient of

         [X,A + B ] + [Y,A + B ] + [X × Y,A] + [X × Y,B ]

   under the equivalence relation generated by elementary pairs (f , g ):

                                h ∈ [X,A]
                                mm         QQQ
                          π0 mmm              QQQσ0
                         mmmm                    QQQ
                                                    QQQ
                     vmmm                              @
         f ∈ [X × Y,A]                             g ∈ [X,A + B ]


            f = π0 (h)                                σ0 (h) = g


                                                                            18/26
(In)definite arrows

   Definition
   An arrow is indefinite if it factors through 0 or through 1.
   Otherwise, it is definite.

   A simple decision procedure for indefinite maps:
   l e t equal f g = (∗f g f a c t o r through 0∗)
       let
          f = f ;?
       and
          g = g ;?
       in
           i f cod f i s p o i n t e d t h e n t r u e
           else
              f =g


                                                                 19/26
(In)definite arrows

   Definition
   An arrow is indefinite if it factors through 0 or through 1.
   Otherwise, it is definite.

   A simple decision procedure for indefinite maps:
   l e t equal f g = (∗f g f a c t o r through 0∗)
       let
          f = f ;?
       and
          g = g ;?
       in
           i f cod f i s p o i n t e d t h e n t r u e
           else
              f =g


                                                                 19/26
A useful Lemma

  If
                           π : f = f0 g1 f2 . . . gn = g
  is a path of elementary pairs crossing a corner,
                       then [f ] = [g ] ∈ [ X × Y , A + B ] is indefinite.

  Consider

  . . . fi−1 = π0 (hi ) σ0 (hi ) = gi   gi = σ1 (hi+1 ) π0 (hi+1 )   = fi+1 . . .

  then hi and hi+1 are copointed and [g ] = [f ] as well.


  Lemma
                                                           ∗
                                                    →
  If [f ] ∈ [ X × Y , A + B ] is definite and π : f − g is a path of
  elementary pairs, then π “bounces” along one side of this diagram.


                                                                                    20/26
A useful Lemma

  If
                           π : f = f0 g1 f2 . . . gn = g
  is a path of elementary pairs crossing a corner,
                       then [f ] = [g ] ∈ [ X × Y , A + B ] is indefinite.

  Consider

  . . . fi−1 = π0 (hi ) σ0 (hi ) = gi   gi = σ1 (hi+1 ) π0 (hi+1 )   = fi+1 . . .

  then hi and hi+1 are copointed and [g ] = [f ] as well.


  Lemma
                                                           ∗
                                                    →
  If [f ] ∈ [ X × Y , A + B ] is definite and π : f − g is a path of
  elementary pairs, then π “bounces” along one side of this diagram.


                                                                                    20/26
A useful Lemma

  If
                        π : f = f0 g1 f2 . . . gn = g
  is a path of elementary pairs crossing a corner,
                       then [f ] = [g ] ∈ [ X × Y , A + B ] is indefinite.

  Consider

  ...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

  then hi and hi+1 are copointed and [g ] = [f ] as well.


  Lemma
                                                        ∗
                                                    →
  If [f ] ∈ [ X × Y , A + B ] is definite and π : f − g is a path of
  elementary pairs, then π “bounces” along one side of this diagram.


                                                                              20/26
A useful Lemma

  If
                        π : f = f0 g1 f2 . . . gn = g
  is a path of elementary pairs crossing a corner,
                       then [f ] = [g ] ∈ [ X × Y , A + B ] is indefinite.

  Consider

  ...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

  then hi and hi+1 are copointed and [g ] = [f ] as well.


  Lemma
                                                        ∗
                                                    →
  If [f ] ∈ [ X × Y , A + B ] is definite and π : f − g is a path of
  elementary pairs, then π “bounces” along one side of this diagram.


                                                                              20/26
A useful Lemma

  If
                        π : f = f0 g1 f2 . . . gn = g
  is a path of elementary pairs crossing a corner,
                       then [f ] = [g ] ∈ [ X × Y , A + B ] is indefinite.

  Consider

  ...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

  then hi and hi+1 are copointed and [g ] = [f ] as well.


  Lemma
                                                        ∗
                                                    →
  If [f ] ∈ [ X × Y , A + B ] is definite and π : f − g is a path of
  elementary pairs, then π “bounces” along one side of this diagram.


                                                                              20/26
Softness for definite maps
   The previous Lemma transforms – for definite maps – the cardinal
   diagram from

                       π0                   σ0
     [X × Y,A] o               [X,A]                G [X,A + B ]
                 y                                        y

        π1                                             σ1



        [Y,A]                                          [X,B ]


        σ0                                             π0


                      σ1                   π1
                                                           
     [Y,A + B ] o              [Y,B ]               G [X × Y,B ]



                                                                     21/26
Softness for definite maps
   The previous Lemma transforms – for definite maps – the cardinal
   diagram to

                        π0                 σ0
       [X × Y,A] o             [X,A]              G [X,A + B ]



       [X × Y,A]                                  [X,A + B ]
                y                                         y
           π1                                        σ1

          [Y,A]                                     [X,B ]
           σ0                                        π0
                                                         
       [Y,A + B ]                                 [X × Y,B ]


                        σ1                 π1
       [Y,A + B ] o            [Y,B ]             G [X × Y,B ]

                                                                     21/26
A new main result
   Theorem
   A pushout diagram

                                   [X,A]
                                   q                MMM
                             π0 qqq                    MMσ0
                             qqq                         MMM
                         xqqq                               MM8
               [X × Y,A]                                   [X,A + B ]

                                       8        x
                                           Po

   is also a pullback.
                         ∗
                         →
   That is: given π : f − g a path “bouncing” on the upper row,
   there exists a unique h such that

                  f = π0 (h) ,                          σ0 (h) = g .

                                                                        22/26
A new main result
   Theorem
   A pushout diagram

                                   [X,A]
                                   q                MMM
                             π0 qqq                    MMσ0
                             qqq                         MMM
                         xqqq                               MM8
               [X × Y,A]                                   [X,A + B ]

                                       8        x
                                           Po

   is also a pullback.
                         ∗
                         →
   That is: given π : f − g a path “bouncing” on the upper row,
   there exists a unique h such that

                  f = π0 (h) ,                          σ0 (h) = g .

                                                                        22/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

   let equivalent f g =
     (∗ f g a r e both d e f i n i t e
                    →           →
         f : X × Y − A , g : X − A + B ∗)
     find h s . t .
       f = σ0 (h) && π0 (h) = g


   Uniqueness of such h “makes it is easy” to find it.


   Theorem
   There exists an algorithm to decide whether two parallel arrow-terms f , g
   of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in




                                                                                23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

   let equivalent f g =
     (∗ f g a r e both d e f i n i t e
                    →           →
         f : X × Y − A , g : X − A + B ∗)
     find h s . t .
       f = σ0 (h) && π0 (h) = g


   Uniqueness of such h “makes it is easy” to find it.


   Theorem
   There exists an algorithm to decide whether two parallel arrow-terms f , g
   of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in




                                                                                23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

   let equivalent f g =
     (∗ f g a r e both d e f i n i t e
                    →           →
         f : X × Y − A , g : X − A + B ∗)
     find h s . t .
       f = σ0 (h) && π0 (h) = g


   Uniqueness of such h “makes it is easy” to find it.


   Theorem
   There exists an algorithm to decide whether two parallel arrow-terms f , g
   of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in


                               size(X ) · size(A) .


                                                                                23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

   let equivalent f g =
     (∗ f g a r e both d e f i n i t e
                    →           →
         f : X × Y − A , g : X − A + B ∗)
     find h s . t .
       f = σ0 (h) && π0 (h) = g


   Uniqueness of such h “makes it is easy” to find it.


   Theorem
   There exists an algorithm to decide whether two parallel arrow-terms f , g
   of ΣΠ(∅) are equal in [ X , A ]. The procedure runs in time linear in


                     (hgt(X ) + hgt(A)) · size(X ) · size(A) .


                                                                                23/26
Background on bouncers
   Lemma
   Consider a pushout in Set

                                              A AA
                                          }            AA g
                                        }}               AA
                                      }}
                                   f
                                     }                     AA
                                  ~}}                        2
                              B                                  C

                                          2        ~
                                              Po

   TFAE:
       the diagram is a weak pullback,
       Ker (f ) and Ker (g ) commute.
   A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that
             (a0 , a1 ) ∈ Ker (f ) ,                      (a1 , a2 ) ∈ Ker (g ) .
                                                                                    24/26
Background on bouncers
   Lemma
   Consider a pushout in Set

                                              A AA
                                          }            AA g
                                        }}               AA
                                      }}
                                   f
                                     }                     AA
                                  ~}}                        2
                              B                                  C

                                          2        ~
                                              Po

   TFAE:
       the diagram is a weak pullback,
       Ker (f ) and Ker (g ) commute.
   A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that
             (a0 , a1 ) ∈ Ker (f ) ,                      (a1 , a2 ) ∈ Ker (g ) .
                                                                                    24/26
Bouncers within ΣΠ(A)



  A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that

                                       f     GA
                              wY X              FF
                        π0 www                    FF σ
                          w                         FF 0
                        ww                            FF
                      ww                                 F4
                   X ×Y                    h             A` + B
                      GG                                 x
                        GG                            xx
                          GG                        xx
                        π0 GGG                     xσ
                               5              # xxx 0
                                  X          GA
                                       g




                                                                     25/26
Bouncers within ΣΠ(A)



  A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that

                                       f     GA
                              x`
                                 X              EE
                        π0 xxx                    EE σ
                          x                         EE 0
                        xx                            EE
                      xx                                 E4
                   X ×0                    h             A+1
                      FF                                 y`
                        FF                            yy
                          FF                        yy
                        π0 FFF                     yσ
                               4              # yyy 0
                                  X          GA
                                       g




                                                                     25/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:




   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                        26/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:


               z    π0 (?)
      f =
             σ0 (!) idx

                             : (0 × 0) + x              G (1 + 1) × x

               z    π1 (?)
      g=
             σ1 (!) idx

   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                        26/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:


               z    π0 (?)
      f =
             σ0 (!) idx
               z    π1 (?)                              G (1 + 1) × x
      h=                     : (0 × 0) + x
             σ0 (!) idx
               z    π1 (?)
      g=
             σ1 (!) idx

   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                        26/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:


               z    π0 (?)
      f =
             σ0 (!) idx
               z    π0 (?)                               G (1 + 1) × x
      h =                    : (0 × 0) + x
             σ1 (!) idx
               z    π1 (?)
      g=
             σ1 (!) idx

   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                         26/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:


               z    π0 (?)
      f =
             σ0 (!) idx
               z    π0 (?)                               G (1 + 1) × x
      h =                    : (0 × 0) + x
             σ1 (!) idx
               z    π1 (?)
      g=
             σ1 (!) idx

   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                         26/26
A minimal nontrivial bouncer
   In ΣΠ(∅) all bouncer are trivial. This is not the case in general:


               z    π0 (?)
      f =
             σ0 (!) idx
               z    π0 (?)                               G (1 + 1) × x
      h =                    : (0 × 0) + x
             σ1 (!) idx
               z    π1 (?)
      g=
             σ1 (!) idx

   Proposition
   In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
   bouncer.

   Proposition
   In ΣΠ(A) there exists at most one bouncer for (f , g ).
                                                                         26/26

				
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