identities among relations for polygraphic...
TRANSCRIPT
Identities among relations for polygraphic rewriting
July 6, 2010
References:
• (P. M., Y. Guiraud) Identities among relations for higher-dimensional rewriting systems, arXiv:0910.4538,
to appear;
• (P. M., Y. Guiraud) Coherence in monoidal track categories, arXiv:1004.1055, submitted;
• (P. M., Y. Guiraud) Higher-dimensional categories with finite derivation type, Theory and Applications of
Categories, 2009;
Contents
Two-dimensional homotopy for polygraphs
- Polygraphs and higher track categories
- Critical branchings in 3-polygraphs
Identities among relations for polygraphs
- Abelian track extensions
- Generating identities among relations
Two-dimensional homotopy for polygraphs
Track n-categories and homotopy bases
Definitions.
- A track n-category is an (n−1)-category enriched in groupoids.
- A homotopy basis of an n-category C is a cellular extension Γ such that the track n-category C/Γ
is aspherical
i.e., for every n-spheres ·f
��
g
AA · in C, there exists an (n+1)-cell ·f
��
g
AA ·�� in C(Γ),
i.e., f= g in C/Γ .
Definition. An n-polygraph Σ has finite derivation type (FDT) if
i) Σ is finite,
ii) the free track n-category Σ> = Σ∗n−1(Σn) admits a finite homotopy basis.
Proposition. The property FDT is Tietze invariant for finite polygraphs.
Critical branchings
• A branching is critical when it is a "minimal overlapping" of n-cells, such as:
..
_%9.
..
|t� ||||
||||
||||
||||
||||
|||| .
B�*BB
BBBB
B
BBBB
BBB
BBBB
BBB
. .
• Newman’s diamond lemma (’42) :
Termination + confluence of critical branchings ⇒ Convergence
The homotopy basis of generating confluences
• A generating confluence of an n-polygraph Σ is an (n+1)-cell
uf
~~||||
|| g
!!CC
CCCC
v
f ′ @@
@@@@
⇒ w
g ′~~}}}}
}}
u ′
with (f,g) critical branching.
Theorem. Let Σ be a convergent n-polygraph.
A cellular extension of Σ∗ made of one generating confluence for each critical branching of Σ is a homotopy
basis of Σ>.
Example I : Mac Lane’s coherence theorem
• Monoidal category (C,⊗, I,α,λ,ρ) with natural isomorphisms
αx,y,z : (x⊗y)⊗z∼−→ x⊗ (y⊗z) λx : I⊗x
∼−→ x ρx : x⊗ I∼−→ x
such that the following diagrams commute:
(x⊗(y⊗z))⊗tα
// x⊗((y⊗z)⊗t)
α
""DD
DDDD
DD
((x⊗y)⊗z)⊗t
α<<zzzzzzzz
α// (x⊗y)⊗(z⊗t)
α//
c©
x⊗(y⊗(z⊗t))
x⊗(I⊗y)
λ
��88
8888
8
(x⊗I)⊗y
α??��������
ρ// x⊗y
c©
Theorem. (Mac Lane’s coherence theorem ’63)
In a monoidal category, all the diagrams built from C, ⊗, I, α, λ and ρ are commutative.
Example I : Mac Lane’s coherence theorem
• Let Mon be the 3-polygraph:
- one 0-cell, one 1-cell, two 2-cell : . , .,
- three 3-cells :
..
_%9. .
._%9 .
..
_%9 .
Lemma. Mon terminates and is locally confluent, with the following five generating confluences:
.
._%9
..
B�*BB
BBBB
B
BBBB
BBB
BBBB
BBB
.
. |4H|||||||
|||||||
|||||||
._%9
. ._%9
.��
.
.
.
9�&99
9999
99
9999
9999
9999
9999
.
. �6J�������
�������
�������
._ %9 .
.��
.
.
6�%
.
�9M.
��
.
.
<�'<<
<<<<
<<<
<<<<
<<<<
<
<<<<
<<<<
<
.
. ~5I~~~~~~~
~~~~~~~
~~~~~~~
._%9 .
��
..
<�'<<
<<<<
<<
<<<<
<<<<
<<<<
<<<<
.
. ~5I~~~~~~~
~~~~~~~
~~~~~~~
._%9 .
��
Example I : Mac Lane’s coherence theorem
• Let Γ ={
. , .}
be the cellular extension of Mon∗ with :
.
._%9
. .E�,EEE
EEEE
EEEE
E
.
. y2Fyyyy yyyyyyyy
.S�3SSSSSSSSSSSSSS
SSSSSSSSSSSSSS
SSSSSSSSSSSSSS .
..
k+?kkkkkkkkkkkkkk
kkkkkkkkkkkkkk
kkkkkkkkkkkkkk
.
��
.
.
9�&99
9999
99
9999
9999
9999
9999
.
. �6J�������
�������
�������
.
_%9 .
.
��
• The track 3-category Mon>/Γ is the theory of monoidal categories, i.e., there is an equivalence:
Monoidal categories ↔ 3-functors Mon>/Γ −→ Cat
Theorem. The 3-category Mon>/Γ is aspherical.
Corollary : Mac Lane’s coherence theorem.
Example II : polygraph of permutations
• Perm 3-polygraph of permutations :
- one 0-cell, one 1-cell, one 2-cell :
- two 3-cells :αV and
β
V
Theorem. Perm is finite, convergent and has FDT.
• Perm has an infinite set of critical branchings.
• three regular critical branchings
• a right-indexed critical branching
k
Example II : polygraph of permutations
• regular critical branchings are confluent
α
8�&
α
�8Lαα
α _%9
β 9�&99
9999
99
9999
9999
9999
9999
β_ %9
αβ α
�8L���������
���������
���������
β _%9
α9�&99
9999
999
9999
9999
9
9999
9999
9β
�8L��������
��������
��������
α_%9
βα
• right-indexed have two normal instances
α _%9
α
5�$55
5555
5555
55
5555
5555
5555
5555
5555
5555
β�:N��������
��������
��������
β 6�$66
6666
66
6666
6666
6666
6666
α_%9
α
:N
ββ( )
β _%9 β _%9
β
9�&99
9999
99
9999
9999
9999
9999
β�8L��������
��������
��������
β 9�&99
9999
99
9999
9999
9999
9999
β_%9
β_%9
β
�8L��������
��������
��������
ββ( )
•{αα, αβ, βα, ββ
( ), ββ
( )}is a homotopy base.
Generating confluences and finite derivation type
• Generating confluences of a convergent n-polygraph Σ form a homotopy basis of Σ>.
Corollary. A finite convergent n-polygraph with a finite number of critical branchings has FDT.
Corollary. (Squier ’94) A finite convergent 2-polygraph has FDT.
- Two shapes of branching in a 2-polygraph :
Regular critical branching Inclusion critical branching
//DD//
��//
EY
��
// !!//FF
//
EY
��
Theorem. For n≥ 3, there exist finite convergent n-polygraphs without FDT.
Critical branchings in 3-polygraphs
• Regular critical branching
g
vv
u
sα=
g
v
u
f
h =sβ
u
vf
• Inclusion critical branching
sα = sβ
C
.
• Right-indexed critical branching
k
g
sα
= k
f
h
g= k
sβ
f
;
Critical branchings in 3-polygraphs
• Regular
g
vv
u
sα=
g
v
u
f
h =sβ
u
vfsα
gu v=
f
u h
gvv =
sβu vv
f
g u
vvsα
= h
gu
v f
=fv
sβu
g
u vsα=
u vf
g
h =
v
sβ
fu
• Inclusion
sα = sβ
C
• Indexed
k
g
sα
= k
f
h
g= k
sβ
f
; k
g
sα
= k
g
f
h = k
sβ
f
k0 kn
sα
g
kpk1 = knk0 h1 k1 h2
g
f
hnhp kp =f
knk0
sβ
kpk1
Critical branchings in 3-polygraphs
Theorem. Let Σ be a finite convergent 3-polygraph.
i) If Σ does not have indexed critical branching it has FDT.
ii) If Σ has indexed critical branchings with finitely many normal instances it has FDT.
C
tA fF _%9
Peiff
C
tA g
A ′
<�(<<
<<<<
<<<
<<<<
<<<<
<
<<<<
<<<<
<
C
sA
sBf=
A~4H~~~~~~~
~~~~~~~
~~~~~~~
B @�*@@
@@@@
@
@@@@
@@@
@@@@
@@@
F _%9
C
sBg
sA=
A~4H~~~~~~~
~~~~~~~
~~~~~~~
B @�*@@
@@@@
@
@@@@
@@@
@@@@
@@@
gen.conf.C
h
.
C
tB fF _%9
Peiff
C
tB g
B ′
�6J���������
���������
���������
Example III : pearl necklaces
• Pearl 3-polygraph presenting the 2-category of pearl necklaces:
- one 0-cell, one 1-cell, three 2-cells :
- four 3-cells :αV
β
Vγ
VδV
• Examples of 2-cells
Proposition. Pearl is finite and convergent but does not have FDT.
Example III : pearl necklaces
• Four regular critical branching
γQ�2
δ
m,@γδ
δQ�2
γ
m,@δγ
β _ %9
γ
@�)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
αy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
γ_%9
αγ
α _ %9
δ
@�)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@β y2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
δ_%9
βδ
• One right-indexed critical branching
k k ∈{
, , ,n}
n
Example III : pearl necklaces
β _%9
Peiffγ
! ��
Peiff
δ�5I��������
��������
��������β _%9
δ~5I~~~~~~~~
~~~~~~~~
~~~~~~~~γ _%9
δ
D�+DD
DDDD
DDDD
DDDD
D
DDDD
DDDD
DDDD
DDD
DDDD
DDDD
DDDD
DDD
α
�EY
β ���
γ
h*>hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
αγ
δ
V 4VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
βδ
Peiff
α _%9
γ?�)??
????
??
????
????
????
????
δ _%9
γ
@�)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
Peiff
γ
z3Gzzzzzzzzzzzzzzz
zzzzzzzzzzzzzzz
zzzzzzzzzzzzzzz
α_%9
δ
� DX
Peiff
Example III : pearl necklaces
α _%9
β D�,DDDDDDD
DDDDDDD
DDDDDDDδ
V 4VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVδ _ %9
Peiff
α_%9
βδ
δ_%9
α
�8L���������
���������
���������
β _%9 γ _%9
β
9�&99
9999
999
9999
9999
9
9999
9999
9
αz3Gzzzzzzz
zzzzzzz
zzzzzzz
β_%9
γ
h*>hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
αγ
γ_ %9
Peiff... n
αS�3
β
k+?
... n+1
.
αβ(
n)
Example III : pearl necklaces
γQ�2
δ
m,@γδ
δQ�2
γ
m,@δγ
β _ %9
γ
@�)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
αy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
γ_%9
αγ
α _ %9
δ
@�)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@β y2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
δ_%9
βδ
... n
αS�3
β
k+?
... n+1
.
αβ(
n)
• An infinite homotopy base :{γδ, δγ, αγ, βδ, αβ
(n), n ∈ N
}.
• The 3-polygraph Pearl is finite and convergent but does not have finite derivation type
Identities among relations for polygraphs
The special case of groups
• (Peiffer-Reidemester-Whitehead ’49) the free crossed module
∂ : C(P )−→ F(X)
on a presentation P = (X,R) of a group G.
• Whitehead’s lemma (’49) : the following diagram commutes
ker∂ // C(P )∂
//
( )ab��
F(X) //
[ ]��
G
kerJ // ZG[R]J
// ZG[X] // ZG
• Brown-Huebschmann (’82) : the Abelianisation map induces an isomorphism of G-modules
ker∂' kerJ
Theorem. (Cremanns-Otto ’96) If P is finite, then G has FDT if and only if ker∂ is finitely generated.
Corollary. A finitely presented group is of type FP3 if and only if it has FDT.
Identities among relations
• Let C be a n-category presented by a (n+1)-polygraph Σ.
• Let Σ>ab be the Abelianised free (n+1)-track category on Σ.
Σ>abn+1
s//
t// Σ
∗n
// // C
Theorem A track category is Abelian if and only if it is linear.
• There exists a C-module Π(Σ) of identities among relations of Σ and an isomorphism of
Σ∗n−1-modules
Π̃(Σ)' AutΣ>ab
• Π(Σ) is defined as follows:
- for an n-cell u of C, then Π(Σ)(u) is the quotient of
Z{bfc
∣∣ v fbb in Σ>ab , v= u}
by :
– bf?n gc = bfc+ bgc for every vf"" gbb with v= u
– bf?n gc = bg?n fc for every vf ''
wg
ff with v=w= u
Generating identities among relations
Theorem. Any homotopy basis of the free track category Σ> forms a generating set for the module Π(Σ).
Corollary. If a polygraph Σ has FDT, then the module Π(Σ) is finitely generated.
Corollary. If Σ is convergent, the module Π(Σ) is generated by the generating confluences of Σ.
Example. Let As be the 2-polygraph(∗, ., .
).
- As is a finite convergent presentation of the monoid 〈a | aa= a〉.
- As has one generating confluence:
..
_ %9.
- Hence the following element generates Π(Σ):
⌊.
⌋=
⌊. ?1
(.
)−1⌋=
.
=⌊
.⌋=
⌊.
⌋
Finite Abelian derivation type
Definition. An n-polygraph Σ has finite Abelian derivation type (TDFab) if
i) Σ is finite,
ii) Σ>ab admits a finite homotopy basis.
• TDF implies TDFab,
Proposition. Let C be an n-category presented a polygraph Σ, then Σ has TDFab if and only if the
C-module Π(Σ) is finitely generated.
Corollary. A 1-category has TDFab if and only if it is of homological type FP3.