Shuffle polygraphic resolutions for operads. (English) Zbl 1519.18015
Algebraic rewriting theory provides methods to compute cofibrant replacements of algebraic structures from presentations that take into account computational properties of these structures. This rewriting approach gives algebraic algorithmic methods to solve decidability and computational problems, sush as the ideal membership problem, and the computation of linear bases and (co)homological properties. The machinery consists in presenting an algebraic structure by a system of generators and rewriting rules, and producing a cofibrant replacement that involves the overlappings occurring in the applications of the rewriting rules. The rewriting systems are formulated in terms of Gröbner bases, being defined with respect to a given minimal order. Rewriting approaches have also been used in the categorical context to present higher categories by higher dimensional rewriting systems, called polygraphs or computads [A. Burroni, Theor. Comput. Sci. 115, No. 1, 43–62 (1993; Zbl 0791.08004); R. Street, J. Pure Appl. Algebra 8, 149–181 (1976; Zbl 0335.18005)], where the cofibrant replacements of a higher category are generated by polygraphic resolutions [Y. Guiraud and P. Malbos, Adv. Math. 231, No. 3–4, 2294–2351 (2012; Zbl 1266.18008); Y. Lafont et al., Adv. Math. 224, No. 3, 1183–1231 (2010; Zbl 1236.18017); F. Métayer, Theory Appl. Categ. 11, 148–184 (2003; Zbl 1020.18001)].
This paper combines the polygraphic and the Gröbner bases approaches in order to compute higher dimensional presentations of shuffle operads using the polygraphic machinerty. The main construction of this paper extends a confluent and terminating shuffle polygraph presenting a shuffle operad into a shuffle polygraphic resolution generated by the overlapping branchings of the original polygraph. To address the question of minimal resolutions, the authors make explicit these overlappings in all dimensions of the polygraphic resolution, giving an inductive method to compute a bimodule resolution that allows of stating a minimality result for shuffle operads, as well as a condition for Koszulness.
This paper combines the polygraphic and the Gröbner bases approaches in order to compute higher dimensional presentations of shuffle operads using the polygraphic machinerty. The main construction of this paper extends a confluent and terminating shuffle polygraph presenting a shuffle operad into a shuffle polygraphic resolution generated by the overlapping branchings of the original polygraph. To address the question of minimal resolutions, the authors make explicit these overlappings in all dimensions of the polygraphic resolution, giving an inductive method to compute a bimodule resolution that allows of stating a minimality result for shuffle operads, as well as a condition for Koszulness.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18M70 | Algebraic operads, cooperads, and Koszul duality |
| 18N30 | Strict omega-categories, computads, polygraphs |
| 68Q42 | Grammars and rewriting systems |
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