An ``almost`` full embedding of the category of graphs into the

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The Functor
Construction of F
Applications
Summary
An “almost” full embedding
of the category of graphs
into the category of groups
Adam J. Przeździecki
Warsaw University of Life Sciences
CAT, 2009
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
The Functor
Choice of categories
F : Graphs → Groups
I
full and faithful:
Hom(X , Y ) ∼
= Hom(FX , FY )
I
“almost” full:
∼
=
HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY )
where Rep(FX , FY ) = Hom(FX , FY )/FY
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
The Functor
Choice of categories
F : Graphs → Groups
I
full and faithful:
(
((
hh
hhh(
((
∼
h
Hom(X
, Y()h
Hom(FX
(
=(
hhh , FY )
(((
h
h
I
“almost” full:
∼
=
HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY )
where Rep(FX , FY ) = Hom(FX , FY )/FY
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
The Functor
Choice of categories
F : Graphs → Groups
Choice of categories: target
I
Groups - is interesting in itself
I
Groups −→ Ho (unpointed homotopy category)
yields, up to constant maps, a full embedding
B
BF : Graphs −→ Ho
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
The Functor
Choice of categories
Choice of categories: source
Graphs is very comprehensive and well researched. Many
“non-homotopy” categories are contained in Graphs as full
subcategories:
I category of groups
I category of fields
I category of R-modules
I category of Hilbert spaces
I category of partially ordered sets
I category of simplicial sets
I category of metrizable spaces and continuous maps
I category of CW-complexes and continuous maps
I category of models of some first order theory
I many more
Tool: Adámek, Rosický Locally presentable and accessible categories, Theorem 2.65.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Bass-Serre theory on groups acting on trees
c
QQ
Q
Q
Qc
M
c
I
I
I
c
G = M ∗N P
N
c
Q
P Q
Q
QQ
c
If A ⊆ G is finite then it stabilizes a vertex of the tree
hence is conjugated to a subgroup of M or P.
Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either
trivial or an inner automorphism.
Let NM (N) = N, N ⊆ M does not extend to P → M,
N ⊆ P extends uniquely to P → P
then M ⊆ G uniquely extends to G → G.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Bass-Serre theory on groups acting on trees
c
QQ
Q
Q
Qc
M
c
I
I
I
c
G = M ∗N P
N
c
Q
P Q
Q
QQ
c
If A ⊆ G is finite then it stabilizes a vertex of the tree
hence is conjugated to a subgroup of M or P.
Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either
trivial or an inner automorphism.
Let NM (N) = N, N ⊆ M does not extend to P → M,
N ⊆ P extends uniquely to P → P
then M ⊆ G uniquely extends to G → G.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Bass-Serre theory on groups acting on trees
c
QQ
Q
Q
Qc
M
c
I
I
I
c
G = M ∗N P
N
c
Q
P Q
Q
QQ
c
If A ⊆ G is finite then it stabilizes a vertex of the tree
hence is conjugated to a subgroup of M or P.
Take A = M finite, s.t. Hom(M, P) = ∗ and M → M is either
trivial or an inner automorphism.
Let NM (N) = N, N ⊆ M does not extend to P → M,
N ⊆ P extends uniquely to P → P
then M ⊆ G uniquely extends to G → G.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
G = (M ∗N P) ∗N P
c
QQ
Q
Q
c
M
Q
c
!!
!
!
c!
P
a
aa
N
aa c
Q
c
c
!!
Q
!
N QQ !!
cP
aa
aa
ac
Q
Rep(G, G) = {∗} ∪ Hom(2, 2)
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Start with a graph Γ
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V
∅
q
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e
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Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
'$
V
c
M
q
?
e
&%
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
'$
P0 c
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e
N
c
M
&%
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Obtain graph of groups GΓ, define F Γ = colim GΓ
P4
N4 chh
N3
hhh
hhhh P3
hc
P0 c
c
Z
Z
Z
Z
N
Z
N0 Z
Z
M
N2
c
((((
(
(
(
P2
Z c((( N
1
P1
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Bass-Serre theory
Construction of the functor F
Reduction to trees
Example
A11 ⊕ A10
chh
hh
hhh
A10
N ⊕ A(S3 ⊕ S7 ) A11
hhh c
N ⊕ A11 c
Z11 o Z5
Z
N
c
M23
Z
Z
Z
Z
N ⊕ A(S4 ⊕ S6 ) Z
Z
A10 (((( c
((
(
Z c(((
A11
A11 ⊕ A10
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Reduction to trees
Bass-Serre theory
Construction of the functor F
Reduction to trees
P4
N4 chh
N3
hhh
P
hhh
hh c 3
P0 c
N
N2
Z
Z
Z
Z
Z
N0 Z
c
c
(((
Z
(((
P2
Z c(((( N
1
M
P1
c
F1 Γ
c
F0 Γ
F2 Γ
F Γ = F 1 Γ ∗ F0 Γ F 2 Γ
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Two definitions of a localization L : C → C
1. L is a left adjoint of an inclusion D ⊆ C of some
subcategory.
2. L is a functor with coaugmentation η : Id → L such that
ηLX = LηX : LX → LLX is an isomorphism
Localizations may be viewed as projections
onto the class of local objects D
along the class of L-equivalences E = {f | Lf is an equivalence}
For every
f : A → B in E, an L-equivalence and
Z in D, an L-local object
∼
we have:
=
Hom(B, Z ) −→ Hom(A, Z )
'
map(B, Z ) −→ map(A, Z )
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Orthogonality classes
If for f : A → B and Z we have
∼
=
Hom(B, Z ) −→ Hom(A, Z )
'
map(B, Z ) −→ map(A, Z )
then we say that f is orthogonal to Z and write f ⊥ Z .
A pair (E, D) is orthogonal if E = D⊥ and D = E ⊥ .
A localization always yields an orthogonal pair.
Whether every orthogonal pair yields a localization depends on
set theory
in Graphs:
NO is consistent with ZFC
weak Vopěnka’s principle is equivalent to YES
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Orthogonality classes
If for f : A → B and Z we have
∼
=
Hom(B, Z ) −→ Hom(A, Z )
'
map(B, Z ) −→ map(A, Z )
then we say that f is orthogonal to Z and write f ⊥ Z .
A pair (E, D) is orthogonal if E = D⊥ and D = E ⊥ .
A localization always yields an orthogonal pair.
Whether every orthogonal pair yields a localization depends on
set theory
in Graphs:
NO is consistent with ZFC
weak Vopěnka’s principle is equivalent to YES
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
More properties of F : Graphs → Groups
∼
=
1. HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY )
2. f ⊥ Z if and only if Ff ⊥ FZ
3. F preserves directed colimits
4. F preserves intersections and countably co-directed limits
5. ∆ ⊆ Γ implies F ∆ ⊆ F Γ
6. for every g ∈ F Γ there exists a finite subgraph ∆ ⊆ Γ s.t.
g ∈ F∆
7. F does not preserve products.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Large localizations of finite groups
Theorem
There exist localizations L : Groups → Groups whose values LM
on a finite group M have arbitrarily large cardinalities.
Proof.
Vopěnka (1965): there exist arbitrarily large graphs Γ s.t.
Hom(Γ, Γ) = {id}.
The inclusion i : ∅ ⊆ Γ is orthogonal to Γ.
Fi ⊥ F Γ
If L = LFi then LM = F Γ.
Proved by Shelah and Göbel (2002) on 28 pages.
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Theorem
The following are equivalent:
1. Every orthogonal pair (E, D) in Groups is associated with a
localization.
2. Every orthogonal pair (E, D) in Graphs is associated with a
localization (weak Vopěnka’s principle).
2 =⇒ 1 was proved by Adámek and Rosický (1994)
1 =⇒ 2 follows from properties of F
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Localizations
More properties of F
Large localizations of finite groups
Orthogonal subcategory problem
Theorem
The following are equivalent:
1. For every orthogonal pair (E, D) in Groups there exists a
homomorphism f such that D = {f }⊥ .
2. For every orthogonal pair (E, D) in Graphs there exists a
map f such that D = {f }⊥ (Vopěnka’s principle).
3. For every orthogonal pair (E, D) in Ho there exists a map f
such that D = {f }⊥ .
2 =⇒ 3 was proved by Casacuberta, Scevenels, Smith (2005)
1 =⇒ 2 and 3 =⇒ 1 follow from properties of F
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Summary
An almost full embedding F : Graphs → Groups
∼
=
HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY )
is a “black box” tool translating some categorical constructions
from many point-set categories to the category of groups
(or to the homotopy category).
I
Question
Is there an embedding F : Graphs → Ab − Groups
such that f ⊥ Z if and only if Ff ⊥ FZ .
Adam J. Przeździecki
An “almost” full embedding ...
The Functor
Construction of F
Applications
Summary
Summary
thank
you
An almost full embedding F : Graphs → Groups
∼
=
HomGraphs (X , Y ) ∪ {∗} −→ Rep(FX , FY )
is a “black box” tool translating some categorical constructions
from many point-set categories to the category of groups
(or to the homotopy category).
I
Question
Is there an embedding F : Graphs → Ab − Groups
such that f ⊥ Z if and only if Ff ⊥ FZ .
Adam J. Przeździecki
An “almost” full embedding ...
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