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Boardman–Vogt tensor products of absolutely free operads

Part of: Discrete geometry Algebraic combinatorics Combinatorics Homological algebra Categories with structure

Published online by Cambridge University Press:  26 January 2019

Murray Bremner  and
Vladimir Dotsenko
Murray Bremner
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada (bremner@math.usask.ca)
Vladimir Dotsenko
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland and Departamento de Matemáticas, CINVESTAV-IPN, Col. San Pedro Zacatenco, México, D.F., CP 07360, Mexico (vdots@maths.tcd.ie)
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Abstract

To the memory of Trevor Evans (1925–1991),

the pioneer of interchange laws in universal algebra

We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as 𝕊-modules. Our results imply that such a tensor product is always a free 𝕊-module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras.

Keywords

absolutely free operadalgebraic operadinterchange lawnonassociative algebraBoardman–Vogt tensor product of symmetric operadsoperad of little d-rectanglesrectangular partitions of the unit d-dimensional cube

MSC classification

Primary: 18D50: Operads
Secondary: 05A15: Exact enumeration problems, generating functions 05E15: Combinatorial aspects of groups and algebras 18D05: Double categories, $2$-categories, bicategories and generalizations 18G10: Resolutions; derived functors 52C22: Tilings in $n$ dimensions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 367 - 385
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Asinowski, A., Barequet, G., Mansour, T. and Pinter, R.. Cut equivalence of d-dimensional guillotine partitions. Discrete Math. 331 (2014), 165–174.CrossRefGoogle Scholar
2Bagherzadeh, F. and Bremner, M.. Commutativity in double interchange semigroups. Appl. Categ. Structures 26 (2018), 1185–1210.CrossRefGoogle Scholar
3Boardman, J. and Vogt, R. M.. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, vol. 347 (Berlin-New York: Springer-Verlag, 1973).CrossRefGoogle Scholar
4Bremner, M. and Madariaga, S.. Permutation of elements in double semigroups. Semigr. Forum 92 (2016), 335–360.CrossRefGoogle Scholar
5Dunn, G.. Tensor product of operads and iterated loop spaces. J. Pure Appl. Algebra 50 (1988), 237–258.CrossRefGoogle Scholar
6Dwyer, W. and Hess, K.. The Boardman-Vogt tensor product of operadic bimodules. In Algebraic topology: applications and new directions (ed. Ulrike Tillmann, Søren Galatius, and Dev Sinha). Contemporary Mathematics, vol. 620, pp. 71–98 (Providence, RI: American Mathematical Society, 2014).Google Scholar
7Eckmann, B. and Hilton, P.. Group-like structures in general categories. I. Multiplications and comultiplications. Mathematische Annalen 145 (1961/1962), 227–255.CrossRefGoogle Scholar
8Evans, T.. Endomorphisms of abstract algebras. Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 66 (1962), 53–64.Google Scholar
9Fiedorowicz, Z. and Vogt, R. M.. An additivity theorem for the interchange of E n-structures. Advances in Math. 273 (2015), 421–484.CrossRefGoogle Scholar
10Fresse, B.. Modules over operads and functors. Lecture Notes in Mathematics, 1967, 314 pp (Berlin: Springer-Verlag, 2009).CrossRefGoogle Scholar
11Godement, R.. Topologie Algébrique et Théorie des Faisceaux. Actualités Scientifiques et Industrielles, 1252, vol. 1, 283 pp (Hermann, Paris: Publications de l'lnstitut de Mathématique de I'Université de Strasbourg XIII, 1958).Google Scholar
12Kock, J.. Note on commutativity in double semigroups and two-fold monoidal categories. J. Homotopy Relat. Str. 2 (2007), 217–228.Google Scholar
13Loday, J.-L. and Vallette, B.. Algebraic operads. Grundlehren der Mathematischen Wissenschaften, vol. 346 (Berlin-Heidelberg: Springer-Verlag, 2012).CrossRefGoogle Scholar
14Maia, M. and Méndez, M.. On the arithmetic product of combinatorial species. Discrete Math. 308 (2008), 5407–5427.CrossRefGoogle Scholar
15Sloane, N. J. A. (ed.) The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.Google Scholar