Identities among relations for higher-dimensional rewriting systems

We generalize the notion of identities among relations, well known for presentations of groups, to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that the polygraph has finite derivation type.

[1]  E. Robertson,et al.  Groups St Andrews 2001 in Oxford , 2003 .

[2]  Friedrich Otto,et al.  For Groups the Property of Having Finite Derivation Type is Equivalent to the Homological Finiteness Condition FP_3 , 1996, J. Symb. Comput..

[3]  M. Kapranov,et al.  Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions. , 1999 .

[4]  Philippe Malbos,et al.  Higher-dimensional categories with finite derivation type , 2008, 0810.1442.

[5]  J. Huebschmann,et al.  Identities among relations , 1982 .

[6]  J. Whitehead,et al.  Combinatorial homotopy. II , 1949 .

[7]  Friedrich Otto,et al.  A Finiteness Condition for Rewriting Systems , 1994, Theor. Comput. Sci..

[8]  H. Baues,et al.  The cohomology of homotopy categories and the general linear group , 1989 .

[9]  Christopher D. Wensley,et al.  Groups St Andrews 2001 in Oxford: Logged rewriting and identities among relators , 2003 .

[10]  R. Peiffer.  Über Identitäten zwischen Relationen , 1949 .

[11]  Elias Gabriel Minian,et al.  Track extensions of categories and cohomology. , 2001 .

[12]  Craig C. Squier,et al.  Word problems and a homological niteness condition for monoids , 1987 .

[13]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[14]  Albert Burroni,et al.  Higher-Dimensional Word Problems with Applications to Equational Logic , 1993, Theor. Comput. Sci..

[15]  J.-L. Loday,et al.  Extensions centrales d'algèbres de Lie , 1982 .

[16]  M. Jibladze,et al.  Classification of Abelian Track Categories , 2002 .

[17]  K. Reidemeister.  Über Identitäten von Relationen , 1949 .

[18]  H. Baues.  Combinatorial homotopy and 4-dimensional complexes , 1990 .

[19]  T. Porter.  Some categorical results in the theory of crossed modules in commutative algebras , 1987 .