R. , E. , and E. R. , where E is the 2-polygraph Com 2 (X) for a finite set X defined in 2.4.2. The 2-cell of the 2-polygraph Com 2 (X) are oriented with respect to a deglex order induced by a total order on X, hence Com 2 (X) is terminating

. Moreover, ? is compatible with R because rewritings with respect to R do not make the dot 2-cell move around a cup or a cap, or create sources of isotopies

, This square extension is made of the ten elements given by the diagrams of the homotopy basis for the 3-polygraph of permutations from

L. Bachmair and N. Dershowitz, Completion for rewriting modulo a congruence, Theoretical Computer Science, vol.67, issue.2, pp.173-201, 1989.

R. Brown, Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, Galois theory, Hopf algebras, and semiabelian categories, vol.43, pp.101-130, 2004.

R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc, vol.36, issue.3, pp.193-212, 1978.

R. Brown, P. J. Higgins, and R. Sivera, Filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics. European Mathematical Society (EMS), vol.15, 2011.

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Différentielle, vol.17, issue.4, pp.343-362, 1976.

J. Brundan, On the definition of heisenberg category, Algebraic Combinatorics, vol.1, issue.4, pp.523-544, 2018.

A. Burroni, Une autre approche des orientaux

A. Burroni, Higher-dimensional word problems with applications to equational logic, 4th Summer Conference on Category Theory and Computer Science, vol.115, pp.43-62, 1991.

J. R. Cockett, J. Koslowski, and R. A. Seely, Introduction to linear bicategories, The Lambek Festschrift: mathematical structures in computer science, vol.10, pp.165-203, 1997.

R. J. Dawson, R. Pare, and D. A. Pronk, Free extensions of double categories, Cah. Topol. Géom. Différ. Catég, vol.45, issue.1, pp.35-80, 2004.

R. Dawson, A forbidden-suborder characterization of binarily-composable diagrams in double categories, Theory Appl. Categ, vol.1, issue.7, pp.146-155, 1995.

R. Dawson and R. Paré, General associativity and general composition for double categories, Cahiers Topologie Géom. Différentielle Catég, vol.34, issue.1, pp.57-79, 1993.

R. Dawson and R. Paré, What is a free double category like?, J. Pure Appl. Algebra, vol.168, issue.1, pp.19-34, 2002.

C. Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup, vol.80, issue.3, pp.349-426, 1963.

S. Gaussent, Y. Guiraud, and P. Malbos, Coherent presentations of Artin monoids, Compos. Math, vol.151, issue.5, pp.957-998, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00682233

Y. Guiraud and P. Malbos, Higher-dimensional categories with finite derivation type, Theory Appl. Categ, vol.22, issue.18, pp.420-478, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00326974

Y. Guiraud and P. Malbos, Coherence in monoidal track categories, Math. Structures Comput. Sci, vol.22, issue.6, pp.931-969, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00470795

Y. Guiraud and P. Malbos, Higher-dimensional normalisation strategies for acyclicity, Adv. Math, vol.231, issue.3-4, pp.2294-2351, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00531242

Y. Guiraud and P. Malbos, Polygraphs of finite derivation type, Math. Structures Comput. Sci, vol.28, issue.2, pp.155-201, 2018.
URL : https://hal.archives-ouvertes.fr/hal-00932845

G. Huet, Confluent reductions: abstract properties and applications to term rewriting systems, J. Assoc. Comput. Mach, vol.27, issue.4, pp.797-821, 1980.

J. and H. Kirchner, Completion of a set of rules modulo a set of equations, Proceedings of the 11th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL '84, pp.83-92, 1984.

J. and J. Li, Church-Rosser properties of normal rewriting, Computer science logic 2012, vol.16, pp.350-365, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00730271

A. Joyal and R. Street, The geometry of tensor calculus. I, Adv. Math, vol.88, issue.1, pp.55-112, 1991.

M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups III, 2008.

M. Khovanov, Heisenberg algebra and a graphical calculus, Fund. Math, vol.225, issue.1, pp.169-210, 2014.

D. Knuth and P. Bendix, Simple word problems in universal algebras, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp.263-297, 1970.

C. Marché, Normalized rewriting: an alternative to rewriting modulo a set of equations, J. Symbolic Comput, vol.21, issue.3, pp.253-288, 1996.

M. Newman, On theories with a combinatorial definition of "equivalence, Ann. of Math, vol.43, issue.2, pp.223-243, 1942.

G. E. Peterson and M. E. Stickel, Complete sets of reductions for some equational theories, J. Assoc. Comput. Mach, vol.28, issue.2, pp.233-264, 1981.

A. J. Power, An n-categorical pasting theorem, Category theory, vol.1488, pp.326-358, 1990.

C. C. Squier, F. Otto, and Y. Kobayashi, A finiteness condition for rewriting systems, Theoret. Comput. Sci, vol.131, issue.2, pp.271-294, 1994.

R. Street, Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra, vol.8, issue.2, pp.149-181, 1976.

P. Viry, BBBB DDDDDD bdupont@math.univ-lyon1, CNRS UMR, vol.5208, 1918.

, F-69622 Villeurbanne cedex

, CNRS UMR, vol.5208, 1918.

F. Villeurbanne-cedex and F. , , 2019.