Cantor’s back-and-forth method
in category theory
Wiesław Kubiś
Instytut Matematyki
Akademia Świȩtokrzyska
Kielce, POLAND
http://www.pu.kielce.pl/˜wkubis/
UPEI, Charlottetown, 26 March 2007
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
1 / 23
Outline
1
Cantor’s back-and-forth method
Fraı̈ssé limits
2
Categories
3
Fraı̈ssé sequences
The existence
Cofinality
Homogeneity and uniqueness
The back-and-forth method
4
Example 1: Reversing the arrows
5
Example 2: Countable linear orders
6
Example 3: Retractive pairs
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
2 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Cantor’s back-and-forth method
Theorem (G. Cantor)
Let Q denote the set of rational numbers. Then:
Every countable linearly ordered set embeds into Q.
For every finite sets A, B ⊆ Q, every order preserving injection
f : A → B extends to an order isomorphism F : Q → Q.
Q is a unique (up to order isomorphism) countable linearly
ordered set with the above properties.
Corollary
Q is the unique countable dense linear order with no end-points.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
3 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Proof.
Let Q =
S
S n∈ω
Qn , where each Qn is finite and Qn ⊆ Qn+1 .
Let P = n∈ω Pn be a linearly ordered set, where Pn ⊆ Pn+1 and
each Pn is finite.
Define inductively embeddings fn : Pn → Qkn so that fn+1 Pn = fn .
Now assume P = Q and f : A → B is given, where A, B ⊆ Qk0 .
Extend f to f1 : Qk0 → Qk1 , where k1 > k0 .
Extend f1−1 to a map g1 : Qk1 → Qk2 , where k2 > k1 .
Extend g1−1 to f2 : Qk2 → Qk3 , where k3 > k2 .
And so on ...
...
S
n∈ω fn
is an isomorphism extending f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
4 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (R. Fraı̈ssé 1954)
Let M be a countable class of finitely generated models of a fixed
countable first-order language, satisfying the following conditions:
For every A, B ∈ M there is C ∈ M such that both A and B embed
into C. (Joint Embedding)
For every two embeddings f : E → A and g : E → B, where
E, A, B ∈ M, there exist D ∈ M and embeddings f 0 : A → D,
g 0 : B → D such that f 0 ◦ f = g 0 ◦ g. (Amalgamation)
Then there exists a unique, up to isomorphism, countable model M of
the same language such that:
Every A ∈ M embeds into M.
For every embeddings f : A → M and g : A → B, where A, B ∈ M,
there exists and embedding f : B → M such that f ◦ g = f .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
5 / 23
Theorem (P.S. Urysohn 1927)
There exists a unique complete separable metric space U with the
following properties:
Every separable metric space is isometric to a subset of U.
For every finite sets A, B ⊆ U, every isometry f : A → B extends to
an isometric bijection F : U → U.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
6 / 23
Theorem (P.S. Urysohn 1927)
There exists a unique complete separable metric space U with the
following properties:
Every separable metric space is isometric to a subset of U.
For every finite sets A, B ⊆ U, every isometry f : A → B extends to
an isometric bijection F : U → U.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
6 / 23
Theorem (P.S. Urysohn 1927)
There exists a unique complete separable metric space U with the
following properties:
Every separable metric space is isometric to a subset of U.
For every finite sets A, B ⊆ U, every isometry f : A → B extends to
an isometric bijection F : U → U.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
6 / 23
Categories
Let K be a category.
We say that K has the amalgamation property if for every arrows
f : Z → X and g : Z → Y there are arrows f 0 : X → W and
g 0 : Y → W such that f 0 ◦ f = g 0 ◦ g.
YO
g0
/W
O
g
Z
f0
f
/X
If moreover for every other pair of arrows k : X → V and ` : Y → V
with k ◦ f = ` ◦ g there exists a unique arrow h : W → V such that
f 0 ◦ h = k and g 0 ◦ h = `
then hf 0 , g 0 i is called the pushout of hf , gi.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
7 / 23
Categories
Let K be a category.
We say that K has the amalgamation property if for every arrows
f : Z → X and g : Z → Y there are arrows f 0 : X → W and
g 0 : Y → W such that f 0 ◦ f = g 0 ◦ g.
YO
g0
/W
O
g
Z
f0
f
/X
If moreover for every other pair of arrows k : X → V and ` : Y → V
with k ◦ f = ` ◦ g there exists a unique arrow h : W → V such that
f 0 ◦ h = k and g 0 ◦ h = `
then hf 0 , g 0 i is called the pushout of hf , gi.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
7 / 23
Categories
Let K be a category.
We say that K has the amalgamation property if for every arrows
f : Z → X and g : Z → Y there are arrows f 0 : X → W and
g 0 : Y → W such that f 0 ◦ f = g 0 ◦ g.
YO
g0
/W
O
g
Z
f0
f
/X
If moreover for every other pair of arrows k : X → V and ` : Y → V
with k ◦ f = ` ◦ g there exists a unique arrow h : W → V such that
f 0 ◦ h = k and g 0 ◦ h = `
then hf 0 , g 0 i is called the pushout of hf , gi.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
7 / 23
Pushouts
The pushout of hf , gi
YO
g0
g
Z
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/W
O
f0
f
/X
Cantor’s back-and-forth method
UPEI, 26 March 2007
8 / 23
Pushouts
The pushout of hf , gi
1V
L
`
YO
g0
/W
O
k
g
Z
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
f0
f
/X
Cantor’s back-and-forth method
UPEI, 26 March 2007
8 / 23
Pushouts
The pushout of hf , gi
1
> VL
`
h
YO
g0
/W
O
k
g
Z
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
f0
f
/X
Cantor’s back-and-forth method
UPEI, 26 March 2007
8 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Sequences
By a sequence in a category K we mean a functor ~x from
ω = {0, 1, . . . } into K.
A sequence ~x can be described as {xn }n∈ω together with arrows
inm : xn → xm for n 6 m, such that
1
2
inn = idxn ,
k < ` < m =⇒ ikm = i`m ◦ ik` .
We shall write ~x = hxn , inm , ωi.
Let ~x = hxn , inm , ωi and ~y = hyn , jnm , ωi be sequences in K.
A transformation of ~x into ~y is a pair hϕ, ~f i such that
1
2
3
ϕ : ω → ω is increasing;
~f = {fn }n∈ω , where fn : xn → y
ϕ(n) ;
ϕ(m)
n < m =⇒ fm ◦ inm = jϕ(n) ◦ fn .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
9 / 23
Arrows between sequences
Let ~x , ~y be sequences in K and let hϕ, ~f i, hψ, ~g i be transformations
between them. We say that they are equivalent if all diagrams like
...
...
/ yϕ(n)
/ yψ(n)
cHH
O
HH
HH
gn
fn HHH
/ xn
/ ...
gm
/ ...
/ yϕ(m)
u:
u
u
uu
uu fm
u
u
/ yψ(m)
O
/ xm
/ ...
/ ...
are commutative.
An arrow of sequences ~x → ~y is an equivalence class of this
relation.
We write ~f : ~x → ~y , having in mind the equivalence class of a
transformation ~f = {fn }n∈ω .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
10 / 23
Arrows between sequences
Let ~x , ~y be sequences in K and let hϕ, ~f i, hψ, ~g i be transformations
between them. We say that they are equivalent if all diagrams like
...
...
/ yϕ(n)
/ yψ(n)
cHH
O
HH
HH
gn
fn HHH
/ xn
/ ...
gm
/ ...
/ yϕ(m)
u:
u
u
uu
uu fm
u
u
/ yψ(m)
O
/ xm
/ ...
/ ...
are commutative.
An arrow of sequences ~x → ~y is an equivalence class of this
relation.
We write ~f : ~x → ~y , having in mind the equivalence class of a
transformation ~f = {fn }n∈ω .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
10 / 23
Arrows between sequences
Let ~x , ~y be sequences in K and let hϕ, ~f i, hψ, ~g i be transformations
between them. We say that they are equivalent if all diagrams like
...
...
/ yϕ(n)
/ yψ(n)
cHH
O
HH
HH
gn
fn HHH
/ xn
/ ...
gm
/ ...
/ yϕ(m)
u:
u
u
uu
uu fm
u
u
/ yψ(m)
O
/ xm
/ ...
/ ...
are commutative.
An arrow of sequences ~x → ~y is an equivalence class of this
relation.
We write ~f : ~x → ~y , having in mind the equivalence class of a
transformation ~f = {fn }n∈ω .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
10 / 23
Arrows between sequences
Let ~x , ~y be sequences in K and let hϕ, ~f i, hψ, ~g i be transformations
between them. We say that they are equivalent if all diagrams like
...
...
/ yϕ(n)
/ yψ(n)
cHH
O
HH
HH
gn
fn HHH
/ xn
/ ...
gm
/ ...
/ yϕ(m)
u:
u
u
uu
uu fm
u
u
/ yψ(m)
O
/ xm
/ ...
/ ...
are commutative.
An arrow of sequences ~x → ~y is an equivalence class of this
relation.
We write ~f : ~x → ~y , having in mind the equivalence class of a
transformation ~f = {fn }n∈ω .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
10 / 23
Let K be a fixed category.
A Fraı̈ssé sequence in K is a sequence ~u = hun , inm , ωi satisfying the
following conditions:
(U) For every x ∈ K there exists n ∈ ω such that K(x, un ) 6= ∅.
/ ...
/ un
>
...
∃
x
(A) For every n ∈ ω and for every arrow f ∈ K(un , y), where y ∈ K,
there exist m > n and g ∈ K(y, um ) such that inm = g ◦ f .
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ un
@@
@@
@@
f @@
in?
/ ...
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
11 / 23
Let K be a fixed category.
A Fraı̈ssé sequence in K is a sequence ~u = hun , inm , ωi satisfying the
following conditions:
(U) For every x ∈ K there exists n ∈ ω such that K(x, un ) 6= ∅.
/ ...
/ un
>
...
∃
x
(A) For every n ∈ ω and for every arrow f ∈ K(un , y), where y ∈ K,
there exist m > n and g ∈ K(y, um ) such that inm = g ◦ f .
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ un
@@
@@
@@
f @@
in?
/ ...
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
11 / 23
Let K be a fixed category.
A Fraı̈ssé sequence in K is a sequence ~u = hun , inm , ωi satisfying the
following conditions:
(U) For every x ∈ K there exists n ∈ ω such that K(x, un ) 6= ∅.
/ ...
/ un
>
...
∃
x
(A) For every n ∈ ω and for every arrow f ∈ K(un , y), where y ∈ K,
there exist m > n and g ∈ K(y, um ) such that inm = g ◦ f .
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ un
@@
@@
@@
f @@
in?
/ ...
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
11 / 23
Let K be a fixed category.
A Fraı̈ssé sequence in K is a sequence ~u = hun , inm , ωi satisfying the
following conditions:
(U) For every x ∈ K there exists n ∈ ω such that K(x, un ) 6= ∅.
/ ...
/ un
>
...
∃
x
(A) For every n ∈ ω and for every arrow f ∈ K(un , y), where y ∈ K,
there exist m > n and g ∈ K(y, um ) such that inm = g ◦ f .
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ un
@@
@@
@@
f @@
in?
/ ...
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
11 / 23
Let K be a fixed category.
A Fraı̈ssé sequence in K is a sequence ~u = hun , inm , ωi satisfying the
following conditions:
(U) For every x ∈ K there exists n ∈ ω such that K(x, un ) 6= ∅.
/ un
>
...
/ ...
∃
x
(A) For every n ∈ ω and for every arrow f ∈ K(un , y), where y ∈ K,
there exist m > n and g ∈ K(y, um ) such that inm = g ◦ f .
...
m
in
/ un
@@
@@
@@
f @@
/ um
>
/ ...
∃g
y
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
11 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
f ∈F
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/z
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
Dominating families of arrows
Let F be a set of arrows in K. Let Dom(F) = {dom(f ) : f ∈ F}.
We say that F is dominating in K if the following conditions are
satisfied:
(D1) For every x ∈ K there exists a ∈ Dom(F) such that K(x, a) 6= ∅.
/a
x
(D2) For every arrow g : a → y in K with a ∈ Dom(F) there exist arrows
f , h in K such that f ∈ F and f = h ◦ g.
a>
f ∈F
>>
>>
g >>
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/z
?
h
y
Cantor’s back-and-forth method
UPEI, 26 March 2007
12 / 23
The existence
Theorem
Let K be a category which has the amalgamation property and the joint
embedding property. Assume further that F ⊆ Arr(K) is dominating in
K and |F| 6 ℵ0 .
Then there exists a Fraı̈ssé sequence ~u = hun , inm , ωi in K such that
{un : n ∈ ω} ⊆ Dom(F).
Remark
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in K. Then K has the
joint embedding property and F = {inm : n < m < ω} is dominating in K.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
13 / 23
The existence
Theorem
Let K be a category which has the amalgamation property and the joint
embedding property. Assume further that F ⊆ Arr(K) is dominating in
K and |F| 6 ℵ0 .
Then there exists a Fraı̈ssé sequence ~u = hun , inm , ωi in K such that
{un : n ∈ ω} ⊆ Dom(F).
Remark
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in K. Then K has the
joint embedding property and F = {inm : n < m < ω} is dominating in K.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
13 / 23
The existence
Theorem
Let K be a category which has the amalgamation property and the joint
embedding property. Assume further that F ⊆ Arr(K) is dominating in
K and |F| 6 ℵ0 .
Then there exists a Fraı̈ssé sequence ~u = hun , inm , ωi in K such that
{un : n ∈ ω} ⊆ Dom(F).
Remark
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in K. Then K has the
joint embedding property and F = {inm : n < m < ω} is dominating in K.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
13 / 23
The existence
Theorem
Let K be a category which has the amalgamation property and the joint
embedding property. Assume further that F ⊆ Arr(K) is dominating in
K and |F| 6 ℵ0 .
Then there exists a Fraı̈ssé sequence ~u = hun , inm , ωi in K such that
{un : n ∈ ω} ⊆ Dom(F).
Remark
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in K. Then K has the
joint embedding property and F = {inm : n < m < ω} is dominating in K.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
13 / 23
The existence
Theorem
Let K be a category which has the amalgamation property and the joint
embedding property. Assume further that F ⊆ Arr(K) is dominating in
K and |F| 6 ℵ0 .
Then there exists a Fraı̈ssé sequence ~u = hun , inm , ωi in K such that
{un : n ∈ ω} ⊆ Dom(F).
Remark
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in K. Then K has the
joint embedding property and F = {inm : n < m < ω} is dominating in K.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
13 / 23
Cofinality
Theorem
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in a category with
amalgamation K. Then for every sequence ~x in K there exists an arrow
~f : ~x → ~u .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
14 / 23
Cofinality
Theorem
Assume ~u = hun , inm , ωi is a Fraı̈ssé sequence in a category with
amalgamation K. Then for every sequence ~x in K there exists an arrow
~f : ~x → ~u .
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
14 / 23
Proof.
/ u`0
E
/ x1
x0
/ ...
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ ...
/ ...
Cantor’s back-and-forth method
UPEI, 26 March 2007
15 / 23
Proof.
/ u`0
F CC
CC
CC
CC
!
=w
z
zz
zz
z
zz
/ x1
/ ...
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ ...
/ ...
Cantor’s back-and-forth method
UPEI, 26 March 2007
15 / 23
Proof.
/ u`0
/ u `1
O
F DD
D
DD
D
DD
!
<w
z
zz
zz
z
zz
/ x1
/ ...
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ ...
/ ...
Cantor’s back-and-forth method
UPEI, 26 March 2007
15 / 23
Proof.
/ u `1
/ u`0
E
E
/ x1
/ ...
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ ...
/ ...
Cantor’s back-and-forth method
UPEI, 26 March 2007
15 / 23
Proof.
/ u `0
/ u`1
E
E
/ x1
/ ...
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ u `n
D
/ xn
/ xn+1
Cantor’s back-and-forth method
/ ...
/ ...
UPEI, 26 March 2007
15 / 23
Proof.
/ u`1
/ u `0
F
F
/ ...
/ x1
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ u `n
E FF
FF
FF
FF
"
w
<
x
x
x
x
xx
xx
/ ...
/ xn+1
/ xn
Cantor’s back-and-forth method
/ ...
UPEI, 26 March 2007
15 / 23
Proof.
/ u`0
/ u`1
F
F
/ ...
/ x1
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ ...
/ u`n+1
/ u`n
E HH
O
H
HH
H
HH
H#
:w
vv
v
v
v
vv
vv
/ ...
/ xn
/ xn+1
Cantor’s back-and-forth method
UPEI, 26 March 2007
15 / 23
Proof.
/ u`0
/ u`1
E
E
/ ...
/ x1
x0
...
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
/ u`n+1
/ u`n
D
C
/ ...
/ xn+1
/ xn
Cantor’s back-and-forth method
/ ...
UPEI, 26 March 2007
15 / 23
Homogeneity and Uniqueness
Theorem
n , ωi, ~
n , ωi are Fraı̈ssé sequences in
Assume that ~u = hum , im
v = hvm , jm
a fixed category K.
(a) Let f : uk → v` , where k , ` < ω. Then there exists an isomorphism
F : ~u → ~v such that F ◦ ik = j` ◦ f . In particular ~u ≈ ~v .
(b) Assume K has the amalgamation property. Then for every a, b ∈ K
and for every arrows f : a → b, i : a → ~u , j : b → ~v there exists an
isomorphism F : ~u → ~v such that F ◦ i = j ◦ f .
~uO
F
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
~uO
j`
ik
uk
/~
vO
f
/ v`
F
/~
vO
j
i
a
Cantor’s back-and-forth method
f
/b
UPEI, 26 March 2007
16 / 23
Homogeneity and Uniqueness
Theorem
n , ωi, ~
n , ωi are Fraı̈ssé sequences in
Assume that ~u = hum , im
v = hvm , jm
a fixed category K.
(a) Let f : uk → v` , where k , ` < ω. Then there exists an isomorphism
F : ~u → ~v such that F ◦ ik = j` ◦ f . In particular ~u ≈ ~v .
(b) Assume K has the amalgamation property. Then for every a, b ∈ K
and for every arrows f : a → b, i : a → ~u , j : b → ~v there exists an
isomorphism F : ~u → ~v such that F ◦ i = j ◦ f .
~uO
F
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
~uO
j`
ik
uk
/~
vO
f
/ v`
F
/~
vO
j
i
a
Cantor’s back-and-forth method
f
/b
UPEI, 26 March 2007
16 / 23
Homogeneity and Uniqueness
Theorem
n , ωi, ~
n , ωi are Fraı̈ssé sequences in
Assume that ~u = hum , im
v = hvm , jm
a fixed category K.
(a) Let f : uk → v` , where k , ` < ω. Then there exists an isomorphism
F : ~u → ~v such that F ◦ ik = j` ◦ f . In particular ~u ≈ ~v .
(b) Assume K has the amalgamation property. Then for every a, b ∈ K
and for every arrows f : a → b, i : a → ~u , j : b → ~v there exists an
isomorphism F : ~u → ~v such that F ◦ i = j ◦ f .
~uO
F
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
~uO
j`
ik
uk
/~
vO
f
/ v`
F
/~
vO
j
i
a
Cantor’s back-and-forth method
f
/b
UPEI, 26 March 2007
16 / 23
Homogeneity and Uniqueness
Theorem
n , ωi, ~
n , ωi are Fraı̈ssé sequences in
Assume that ~u = hum , im
v = hvm , jm
a fixed category K.
(a) Let f : uk → v` , where k , ` < ω. Then there exists an isomorphism
F : ~u → ~v such that F ◦ ik = j` ◦ f . In particular ~u ≈ ~v .
(b) Assume K has the amalgamation property. Then for every a, b ∈ K
and for every arrows f : a → b, i : a → ~u , j : b → ~v there exists an
isomorphism F : ~u → ~v such that F ◦ i = j ◦ f .
~uO
F
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
~uO
j`
ik
uk
/~
vO
f
/ v`
F
/~
vO
j
i
a
Cantor’s back-and-forth method
f
/b
UPEI, 26 March 2007
16 / 23
Homogeneity and Uniqueness
Theorem
n , ωi, ~
n , ωi are Fraı̈ssé sequences in
Assume that ~u = hum , im
v = hvm , jm
a fixed category K.
(a) Let f : uk → v` , where k , ` < ω. Then there exists an isomorphism
F : ~u → ~v such that F ◦ ik = j` ◦ f . In particular ~u ≈ ~v .
(b) Assume K has the amalgamation property. Then for every a, b ∈ K
and for every arrows f : a → b, i : a → ~u , j : b → ~v there exists an
isomorphism F : ~u → ~v such that F ◦ i = j ◦ f .
~uO
F
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
~uO
j`
ik
uk
/~
vO
f
/ v`
F
/~
vO
j
i
a
Cantor’s back-and-forth method
f
/b
UPEI, 26 March 2007
16 / 23
The back-and-forth method
/ ...
uO k
f
/ ...
v`
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ ...
uO k B
f
v`
BB g
BB 0
BB
B!
/ v`1
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
/ ...
Cantor’s back-and-forth method
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ uk1
=
AA g
|
f1 ||
AA 0
|
AA
f
||
A
||
/ v`1
v`
uO k A
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
/ ...
/ ...
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ uk1
=
CC
AA g
|
CC g1
f1 ||
AA 0
CC
|
AA
f
|
CC
|
A
!
||
/ v`1
/ v `2
v`
uO k A
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
/ ...
/ ...
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ uk2
/ uk1
=
=
CC
AA g
|
|
C
f2 ||
f1 ||
AA 0
CCg1
|
|
AA
CC
f
||
||
A
C!
||
||
/ v`1
/ v `2
v`
uO k A
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
/ ...
/ ...
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ uk1
/ uk2
=
BB
=
BB
AA g
|
|
BB g2
B
f1 ||
f2 ||
AA 0
BBg1
BB
|
|
AA
B
f
|
|
BB
B
|
|
A
B!
!
||
||
/ v `2
/ v `3
/ v`1
v`
uO k A
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
/ ...
/ ...
UPEI, 26 March 2007
17 / 23
The back-and-forth method
/ uk1
/ uk2
/ .> . .
|
=
BB
=
BB
AA g
|
|
|
|
B
B
f1 ||
f2 ||
AA 0
|
BBg2
BBg1
|
|
|
AA
BB
BB
f
||
||
||
A
B!
B!
||
||
||
/ v `2
/ v `3
/ ...
/ v`1
v`
uO k A
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
17 / 23
Example 1: Reversing the arrows
Let K be the category described as follows:
Objects of K are finite linearly ordered sets.
f ∈ K(P, Q) iff f : Q → P is an order preserving surjection.
Claim
K has the amalgamation property.
Theorem
K has a Fraı̈ssé sequence
P0 ← P1 ← P2 ← . . .
whose limit is the Cantor set with the standard linear ordering.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
18 / 23
Example 1: Reversing the arrows
Let K be the category described as follows:
Objects of K are finite linearly ordered sets.
f ∈ K(P, Q) iff f : Q → P is an order preserving surjection.
Claim
K has the amalgamation property.
Theorem
K has a Fraı̈ssé sequence
P0 ← P1 ← P2 ← . . .
whose limit is the Cantor set with the standard linear ordering.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
18 / 23
Example 1: Reversing the arrows
Let K be the category described as follows:
Objects of K are finite linearly ordered sets.
f ∈ K(P, Q) iff f : Q → P is an order preserving surjection.
Claim
K has the amalgamation property.
Theorem
K has a Fraı̈ssé sequence
P0 ← P1 ← P2 ← . . .
whose limit is the Cantor set with the standard linear ordering.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
18 / 23
Example 1: Reversing the arrows
Let K be the category described as follows:
Objects of K are finite linearly ordered sets.
f ∈ K(P, Q) iff f : Q → P is an order preserving surjection.
Claim
K has the amalgamation property.
Theorem
K has a Fraı̈ssé sequence
P0 ← P1 ← P2 ← . . .
whose limit is the Cantor set with the standard linear ordering.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
18 / 23
Example 1: Reversing the arrows
Let K be the category described as follows:
Objects of K are finite linearly ordered sets.
f ∈ K(P, Q) iff f : Q → P is an order preserving surjection.
Claim
K has the amalgamation property.
Theorem
K has a Fraı̈ssé sequence
P0 ← P1 ← P2 ← . . .
whose limit is the Cantor set with the standard linear ordering.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
18 / 23
Example 2: Countable linear orders
Let K be the category whose objects are countable linear orders hP, 6i
and arrows are left-invertible order preserving maps.
That is: f : hP, 6i → hQ, i is an arrow in K if
f is order preserving, i.e. x 6 y =⇒ f (x) f (y);
there is an order preserving map g : hQ, i → hP, 6i such that
g ◦ f = idP .
Necessarily f is one-to-one.
Lemma
K has the amalgamation property.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
19 / 23
Example 2: Countable linear orders
Let K be the category whose objects are countable linear orders hP, 6i
and arrows are left-invertible order preserving maps.
That is: f : hP, 6i → hQ, i is an arrow in K if
f is order preserving, i.e. x 6 y =⇒ f (x) f (y);
there is an order preserving map g : hQ, i → hP, 6i such that
g ◦ f = idP .
Necessarily f is one-to-one.
Lemma
K has the amalgamation property.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
19 / 23
Example 2: Countable linear orders
Let K be the category whose objects are countable linear orders hP, 6i
and arrows are left-invertible order preserving maps.
That is: f : hP, 6i → hQ, i is an arrow in K if
f is order preserving, i.e. x 6 y =⇒ f (x) f (y);
there is an order preserving map g : hQ, i → hP, 6i such that
g ◦ f = idP .
Necessarily f is one-to-one.
Lemma
K has the amalgamation property.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
19 / 23
Example 2: Countable linear orders
Let K be the category whose objects are countable linear orders hP, 6i
and arrows are left-invertible order preserving maps.
That is: f : hP, 6i → hQ, i is an arrow in K if
f is order preserving, i.e. x 6 y =⇒ f (x) f (y);
there is an order preserving map g : hQ, i → hP, 6i such that
g ◦ f = idP .
Necessarily f is one-to-one.
Lemma
K has the amalgamation property.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
19 / 23
Example 2: Countable linear orders
Let K be the category whose objects are countable linear orders hP, 6i
and arrows are left-invertible order preserving maps.
That is: f : hP, 6i → hQ, i is an arrow in K if
f is order preserving, i.e. x 6 y =⇒ f (x) f (y);
there is an order preserving map g : hQ, i → hP, 6i such that
g ◦ f = idP .
Necessarily f is one-to-one.
Lemma
K has the amalgamation property.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
19 / 23
Lemma
Let π : Q → Q · Q be defined by π(q) = hq, 0i. Then {π} is a
dominating family of arrows in K.
Theorem
K has a Fraı̈ssé sequence ~u = hun , inm , ωi such that each un is
isomorphic to Q and each inm is isomorphic to π.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
20 / 23
Lemma
Let π : Q → Q · Q be defined by π(q) = hq, 0i. Then {π} is a
dominating family of arrows in K.
Theorem
K has a Fraı̈ssé sequence ~u = hun , inm , ωi such that each un is
isomorphic to Q and each inm is isomorphic to π.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
20 / 23
Example 3: Retractive pairs
Fix a category K. Denote by ‡K the following category:
The objects of ‡K are the same as the objects of K.
f ∈ ‡K(a, b) iff f = hr , ei, where r : b → a and e : a → b are arrows
of K such that r ◦ e = ida .
We shall write r (f ) = r , e(f ) = e.
Given compatible arrows f , g in ‡K, their composition is
gf = hr (f ) ◦ r (g), e(g) ◦ e(f )i.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
21 / 23
Example 3: Retractive pairs
Fix a category K. Denote by ‡K the following category:
The objects of ‡K are the same as the objects of K.
f ∈ ‡K(a, b) iff f = hr , ei, where r : b → a and e : a → b are arrows
of K such that r ◦ e = ida .
We shall write r (f ) = r , e(f ) = e.
Given compatible arrows f , g in ‡K, their composition is
gf = hr (f ) ◦ r (g), e(g) ◦ e(f )i.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
21 / 23
Example 3: Retractive pairs
Fix a category K. Denote by ‡K the following category:
The objects of ‡K are the same as the objects of K.
f ∈ ‡K(a, b) iff f = hr , ei, where r : b → a and e : a → b are arrows
of K such that r ◦ e = ida .
We shall write r (f ) = r , e(f ) = e.
Given compatible arrows f , g in ‡K, their composition is
gf = hr (f ) ◦ r (g), e(g) ◦ e(f )i.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
21 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
e(g)
Y
e(g)
Y
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
r (g)
r (f )
Xo
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
e(g)
Y
e(g)
Y
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
r (g)
r (f )
Xo
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Y o
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
e(g)
Y
e(g)
r (g)
Z
W
r (f )
Xo
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Y o
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
e(g)
Y
e(g)
r (g)
Z
k
W
h
r (f )
Xo
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
idX
Y o
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
e(g)
Y
e(g)
r (g)
k
W
h
r (f )
Xo
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
idX
`
e(g)
Y o
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Y
k
W
h
r (f )
Xo
e(g)
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
idX
w
Y o
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
idY
`
e(g)
Y
k
W
h
r (f )
Xo
e(g)
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
Xo
e(f )
Z
idX
w
Y o
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
idY
`
e(g)
w
k
Xo
e(g)
Y
j
W
h
r (f )
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
e(f )
Z
idX
e(g)
idY
`
nY
nnn
n
n
nnnn
wnnnn j
e(g)
w
Y o
Xo
k
W
h
r (f )
Xo
r (g)
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Claim
If K has pullbacks then ‡K has the amalgamation property.
Proof.
r (f )
Z o
e(g)
Y o
r (g)
Z o
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
r (g 0 )
r (f )
Xo
e(f )
Z
e(g)
e(f 0 ) nn Y
nnn
n
n
nnn 0
wnnnn e(g )
W
r (g)
r (f 0 )
Xo
e(f )
Cantor’s back-and-forth method
Z
UPEI, 26 March 2007
22 / 23
Selected bibliography
F RA ÏSS É , R., Sur quelques classifications des systèmes de
relations, Publ. Sci. Univ. Alger. Sér. A. 1 (1954) 35–182.
J ÓNSSON , B., Homogeneous universal relational systems, Math.
Scand. 8 (1960) 137–142.
U RYSOHN , P.S., Sur un espace metrique universel, Bull. Sci.
Math. 51 (1927) 1–38.
W.Kubiś (http://www.pu.kielce.pl/∼wkubis/)
Cantor’s back-and-forth method
UPEI, 26 March 2007
23 / 23