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;
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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