Post on 21-Jan-2021
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 1/12 1 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 1/12 1 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Φ(X ) =X
X
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 2/12 2 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
Pos0;1 is not l.f.p (no terminal object). It is �nitely accessible andconsistent �nite diagrams have colimits.
Why does such a Φ have a terminal coalgebra?
We claim that it is for the same reasons that every �nitary Φ on an l.f.pcategory has terminal coalgebra!
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 3/12 3 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
Pos0;1 is not l.f.p (no terminal object). It is �nitely accessible andconsistent �nite diagrams have colimits.
Why does such a Φ have a terminal coalgebra?
We claim that it is for the same reasons that every �nitary Φ on an l.f.pcategory has terminal coalgebra!
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 3/12 3 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
Pos0;1 is not l.f.p (no terminal object). It is �nitely accessible andconsistent �nite diagrams have colimits.
Why does such a Φ have a terminal coalgebra?
We claim that it is for the same reasons that every �nitary Φ on an l.f.pcategory has terminal coalgebra!
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 3/12 3 / 12
Freyd's construction of [0; 1] as a terminal coalgebra
Work on Pos0;1 (posets with distinct endpoints), with
Φ : Pos0;1 −→ Pos0;1
Φ(X ) = {(x ; 0) | x ∈ X } ∪ {(1; y) | y ∈ X } := X ∨ X
Pos0;1 is not l.f.p (no terminal object). It is �nitely accessible andconsistent �nite diagrams have colimits.
Why does such a Φ have a terminal coalgebra?
We claim that it is for the same reasons that every �nitary Φ on an l.f.pcategory has terminal coalgebra!
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 3/12 3 / 12
Every �nitely accessible category K is, to within equivalence, Flat(A ; Set)and every �nitary endofunctor Φ : K −→ K is, to within isomorphism,
Φ ∼= M ⊗− ,where M : A op ×A −→ Set is a at module.When K = Pos0;1 then A ∼= (Pos0;1
�n )op and
M (X ;Y ) = {Y −→ Φ(X ) in Pos0;1
�n }.
If A has �nite limits then K is l.f.p and can support interesting examplesof endofunctors, e.g power series
Φ(X ) =⊔n
Pn × X n
Tom's description of the terminal coalgebra can be restated as:T : A −→ Set is the colimit of the diagram(
Complex(M ))op pr
op
0 // A op Y // [A ; Set]
We need to know that T is a at functor, i.e Complex(M ) is co�ltered.
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 4/12 4 / 12
Every �nitely accessible category K is, to within equivalence, Flat(A ; Set)and every �nitary endofunctor Φ : K −→ K is, to within isomorphism,
Φ ∼= M ⊗− ,where M : A op ×A −→ Set is a at module.When K = Pos0;1 then A ∼= (Pos0;1
�n )op and
M (X ;Y ) = {Y −→ Φ(X ) in Pos0;1
�n }.
If A has �nite limits then K is l.f.p and can support interesting examplesof endofunctors, e.g power series
Φ(X ) =⊔n
Pn × X n
Tom's description of the terminal coalgebra can be restated as:T : A −→ Set is the colimit of the diagram(
Complex(M ))op pr
op
0 // A op Y // [A ; Set]
We need to know that T is a at functor, i.e Complex(M ) is co�ltered.
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 4/12 4 / 12
Every �nitely accessible category K is, to within equivalence, Flat(A ; Set)and every �nitary endofunctor Φ : K −→ K is, to within isomorphism,
Φ ∼= M ⊗− ,where M : A op ×A −→ Set is a at module.When K = Pos0;1 then A ∼= (Pos0;1
�n )op and
M (X ;Y ) = {Y −→ Φ(X ) in Pos0;1
�n }.
If A has �nite limits then K is l.f.p and can support interesting examplesof endofunctors, e.g power series
Φ(X ) =⊔n
Pn × X n
Tom's description of the terminal coalgebra can be restated as:T : A −→ Set is the colimit of the diagram(
Complex(M ))op pr
op
0 // A op Y // [A ; Set]
We need to know that T is a at functor, i.e Complex(M ) is co�ltered.
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 4/12 4 / 12
Every �nitely accessible category K is, to within equivalence, Flat(A ; Set)and every �nitary endofunctor Φ : K −→ K is, to within isomorphism,
Φ ∼= M ⊗− ,where M : A op ×A −→ Set is a at module.When K = Pos0;1 then A ∼= (Pos0;1
�n )op and
M (X ;Y ) = {Y −→ Φ(X ) in Pos0;1
�n }.
If A has �nite limits then K is l.f.p and can support interesting examplesof endofunctors, e.g power series
Φ(X ) =⊔n
Pn × X n
Tom's description of the terminal coalgebra can be restated as:T : A −→ Set is the colimit of the diagram(
Complex(M ))op pr
op
0 // A op Y // [A ; Set]
We need to know that T is a at functor, i.e Complex(M ) is co�ltered.
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 4/12 4 / 12
Lemma: Assume that for every �nite non-empty diagram in Complex(M )we have limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
: : : a ′n �m ′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 5/12 5 / 12
Lemma: Assume that for every �nite non-empty diagram in Complex(M )we have limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
: : : a ′n �m ′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 5/12 5 / 12
Lemma: Assume that for every �nite non-empty diagram in Complex(M )we have limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
: : : a ′n �m ′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 5/12 5 / 12
Lemma: Assume that for a �nite non-empty diagram in Complex(M ) wehave limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
a1 × a ′1
66mmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQa0 × a ′0
OO
��
: : : a ′n �m′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 6/12 6 / 12
Lemma: Assume that for a �nite non-empty diagram in Complex(M ) wehave limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
a1 × a ′1
66mmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQ
�
33
�
**
a0 × a ′0
OO
��
: : : a ′n �m′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 7/12 7 / 12
Lemma: Assume that for a �nite non-empty diagram in Complex(M ) wehave limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
a1 × a ′1
66mmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQ
�
33
�
**
� // b0
66
((
a0 × a ′0
OO
��
: : : a ′n �m′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 8/12 8 / 12
Lemma: Assume that for a �nite non-empty diagram in Complex(M ) wehave limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
a1 × a ′1
66mmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQ
�
33
�
**
� // b0
66
((
// a0 × a ′0
OO
��
: : : a ′n �m′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 9/12 9 / 12
Lemma: Assume that for a �nite non-empty diagram in Complex(M ) wehave limits "levelwise", for all n ≥ 0. Then Complex(M ) is co�ltered.
Corollary: (i) Every �nitary endofunctor of an l.f.p category has a terminalcoalgebra.(ii) Freyd's endofunctor has a terminal coalgebra.
Sketch of the Proof:
Let (a•;m•), (a ′•;m′•) in Complex(M ) be a discrete diagram, i.e
: : : an �mn // an−1 � // : : : � // a2 �
m2 // a1 �m1 // a0
: : : � //
55jjjjjjjjjjjjjjjjjjjj
))SSSSSSSSSSSSSSSSSSSS a1 × a ′1
66mmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQ� // a0 × a ′0
OO
��
: : : a ′n �m′n
// a ′n−1 � // : : : � // a ′2
�m ′
2
// a ′1
�m ′
1
// a ′0
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 10/12 10 / 12
One may object that not all �nite colimits exist in Pos0;1
�n∼= A op , so why
should we have "levelwise" limits?
E.g 3u //
d// 2 with u(middle) = 1, d(middle) = 0 can not be coequalized.
But when Yu //
d// X can not be coequalized then there can be no
: : : � // • �m1 //
����
X
����: : : � // • �
m ′1 // Y
in Complex(M )
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 11/12 11 / 12
One may object that not all �nite colimits exist in Pos0;1
�n∼= A op , so why
should we have "levelwise" limits?
E.g 3u //
d// 2 with u(middle) = 1, d(middle) = 0 can not be coequalized.
But when Yu //
d// X can not be coequalized then there can be no
: : : � // • �m1 //
����
X
����: : : � // • �
m ′1 // Y
in Complex(M )
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 11/12 11 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12
More generally terminal coalgebras for �nitary endofunctors on �nitelyaccessible categories exist, provided that
(i) Cones exist "at the head" of �nite diagrams of complexes
(ii) A technical �niteness condition holds, that allows us to infer theexistence of cones at the level of complexes (not just at the head), using atopological version of K�onig's Lemma due to A. StoneThis way
• Tom's modules for topological self-similarity become examples
• Pavlovi�c & Pratt's construction of the continuum and Cantor's space asterminal coalgebras become examples, if we work on the �nitely accessiblecategory Lin with suitable endofunctors.
For details: http://www.math.upatras.gr/∼pkarazer
P.K. , J.V. , A.M. () Cambridge, 4-5 April 2009 12/12 12 / 12