The diagonal of the associahedra. (La diagonale de l’associaèdre.) (English. French summary) Zbl 1455.18014
This paper has a threefold purpose.
Since the set-theoretic diagonal of a polytope fails to be cellular in general, the authors need to find a cellular approximation to the diagonal, that is to say, a cellular map from the polytope to its cartesian square homotopic to the diagonal. For a coherent family of polytopes, it is highly challenging to find a family of diagonals compatible with the combinatorics of faces. In the case of the first face-coherent family of polytopes, the geometric simplices, such a diagronal map is given by the classical Alexander-Whitney map [S. Eilenberg and J. A. Zilber, Am. J. Math. 75, 200–204 (1953; Zbl 0050.17301); N. E. Steenrod, Ann. Math. (2) 48, 290–320 (1947; Zbl 0030.41602)]. For the next family given by cubes, a coassociative approximation to the diagonal is straightforward [J.-P. Serre, Ann. Math. (2) 54, 425–505 (1951; Zbl 0045.26003)]. The associahedra form the face-coherent family of polytopes coming next in terms of further truncations of the simplices or of combinatorial complexity. While a face of a simplex or a cube is a simplex or a cube of lower dimension, a face of an associahedra is a product of associahedra of lower dimensions, which makes the problem of the approximation of the diagonal in this turn highly intricate. The two-fold principal result of the paper (Theorem 1) is an explicit operad structure on the Loday realizations of the associahedra together with a compatible approximation to the diagonal.
There is a dichotomy between pointwise and cellular formulas. To investigate their relationship and to make precise the various face-coherent properties, the authors introduce the category of polytopes with subdivision. The definition of the diagonal maps comes from the theory of fiber polytopes so that an induced polytopal subdivision of the associahedra is obtained, for which the authors establishes a magical formula in the verbalism of Jean-Louis Loday. It is made up of the pairs of cells of matching dimensions and comparable order under the Tamari order (Theorem 2).
A synopsis of the paper consisting of four sections goes as follows. §1 recalls the main relevant notions, introducing the category of polytopes in which the authors work. §2 gives a canonical definition of the diagonal map for positively oriented polytopes, addressing their cellular properties. §3 endows the family of Loday realizations of the associahedra with a nonsymmetric operad structure compatible with the diagonal maps. §4 establishes the magical celluar formula for the diagonal map of the associahedra.
- 1.
- to introduce a general machinery to solve the problem of the approximation of the diagonal of face-coherent families of polytopes (§2),
- 2.
- to give a complete proof for the case of the associahedra (Theorem 1), and
- 3.
- to popularize the resulting magical formula (Theorem 2).
- 1.
- There are mathematicians inclined to apply it in their work of computing the homology of fibered spaces in algebraic topology [E. H. Brown jun., Ann. Math. (2) 69, 223–246 (1959; Zbl 0199.58201); A. Prouté, Repr. Theory Appl. Categ. 2011, No. 21, 99 p. (2011; Zbl 1245.55007)], to construct tensor products of string field theories [M. R. Gaberdiel and B. Zwiebach, Nucl. Phys., B 505, No. 3, 569–624 (1997; Zbl 0911.53044); Phys. Lett., B 410, No. 2–4, 151–159 (1997; Zbl 0911.53046)], or to consider the product of Fukaya \(\mathcal{A}_{\infty}\)-categories in symplectic geometry [P. Seidel, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); L. Amorim, Int. J. Math. 28, No. 4, Article ID 1750026, 38 p. (2017; Zbl 1368.53057)].
- 2.
- The analogous result is known, within the kingdom of operad theory and homotopical algebra, in the differential graded context [S. Saneblidze and R. Umble, Homology Homotopy Appl. 6, No. 1, 363–411 (2004; Zbl 1069.55015); L. J. Billera and B. Sturmfels, Ann. Math. (2) 135, No. 3, 527–549 (1992; Zbl 0762.52003)].
- 3.
- This result is appreciated conceptualy as a new development in the theory of fiber polytopes [loc. cit.] by combinatorists and discrete geometers.
Since the set-theoretic diagonal of a polytope fails to be cellular in general, the authors need to find a cellular approximation to the diagonal, that is to say, a cellular map from the polytope to its cartesian square homotopic to the diagonal. For a coherent family of polytopes, it is highly challenging to find a family of diagonals compatible with the combinatorics of faces. In the case of the first face-coherent family of polytopes, the geometric simplices, such a diagronal map is given by the classical Alexander-Whitney map [S. Eilenberg and J. A. Zilber, Am. J. Math. 75, 200–204 (1953; Zbl 0050.17301); N. E. Steenrod, Ann. Math. (2) 48, 290–320 (1947; Zbl 0030.41602)]. For the next family given by cubes, a coassociative approximation to the diagonal is straightforward [J.-P. Serre, Ann. Math. (2) 54, 425–505 (1951; Zbl 0045.26003)]. The associahedra form the face-coherent family of polytopes coming next in terms of further truncations of the simplices or of combinatorial complexity. While a face of a simplex or a cube is a simplex or a cube of lower dimension, a face of an associahedra is a product of associahedra of lower dimensions, which makes the problem of the approximation of the diagonal in this turn highly intricate. The two-fold principal result of the paper (Theorem 1) is an explicit operad structure on the Loday realizations of the associahedra together with a compatible approximation to the diagonal.
There is a dichotomy between pointwise and cellular formulas. To investigate their relationship and to make precise the various face-coherent properties, the authors introduce the category of polytopes with subdivision. The definition of the diagonal maps comes from the theory of fiber polytopes so that an induced polytopal subdivision of the associahedra is obtained, for which the authors establishes a magical formula in the verbalism of Jean-Louis Loday. It is made up of the pairs of cells of matching dimensions and comparable order under the Tamari order (Theorem 2).
A synopsis of the paper consisting of four sections goes as follows. §1 recalls the main relevant notions, introducing the category of polytopes in which the authors work. §2 gives a canonical definition of the diagonal map for positively oriented polytopes, addressing their cellular properties. §3 endows the family of Loday realizations of the associahedra with a nonsymmetric operad structure compatible with the diagonal maps. §4 establishes the magical celluar formula for the diagonal map of the associahedra.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
18M75 | Topological and simplicial operads |
18M70 | Algebraic operads, cooperads, and Koszul duality |
52B11 | \(n\)-dimensional polytopes |
55P35 | Loop spaces |
06A07 | Combinatorics of partially ordered sets |
Keywords:
associahedra; approximation of the diagonal; operads; fiber polytopes; \(\text{A}_\infty\)-algebrasCitations:
Zbl 0199.58201; Zbl 1245.55007; Zbl 0911.53044; Zbl 0911.53046; Zbl 1159.53001; Zbl 1368.53057; Zbl 1069.55015; Zbl 0762.52003; Zbl 0109.24502; Zbl 1389.52013; Zbl 0114.39402; Zbl 0285.55012; Zbl 0244.55009; Zbl 0050.17301; Zbl 0030.41602; Zbl 0045.26003References:
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