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Braided skew monoidal categories. (English) Zbl 1431.18012
The notion of skew monoidal category was introduced in [K. Szlachányi, Adv. Math. 231, No. 3–4, 1694–1730 (2012; Zbl 1283.18006)] in order to deal with bialgebroids. It is, roughly speaking, obtained from the familiar notion of monoidal category by loosening isomorphisms on associativity and unit conditions. This paper answers the question whether there exists a sensible notion of braiding for skew monoidal categories, generalizing the classical theory of braided monoidal categories [A. Joyal and R. Street, Adv. Math. 102, No. 1, 20–78 (1993; Zbl 0817.18007)]. The authors give a notion of braiding on a skew monoidal category which is given by an invertible natural transformation \[ s:(AB)C\rightarrow(AC)B \] pursuant to certain axioms. If \(B\) is a bialgebra, one gets a skew monoidal structure \(\mathbf{Vect}[B]\) on the category \(\mathbf{Vect}\) of vector spaces with product \[ X\bigstar Y=X\otimes B\otimes Y \] and the ground field \(K\) as its unit. Bialgebroids give rise to, and can be characterized by certain skew monoidal categories [K. Szlachányi, Adv. Math. 231, No. 3–4, 1694–1730 (2012; Zbl 1283.18006)].
It is shown in Theorem 4.10 that braidings on the skew monoidal category are in bijection with cobraidings (a.k.a. coquasitriangular structures [C. Kassel, Quantum groups. New York, NY: Springer-Verlag (1995; Zbl 0808.17003); R. Street, Quantum groups. A path to current algebra. Cambridge: Cambridge University Press (2007; Zbl 1117.16031)]) on the bialgebra \(B\). The theorem is presented as a specialized one of the main theorem Theorem 4.7 in §4 claiming that, given a monoidal comonad \(G\) on a monoidal category \(\mathcal{C}\) abiding by a mild hypothesis, there is a bijection between braidings on the monoidal category \(\mathcal{C}^{G}\) of coalgebras and braidings on the cowarped skew monoidal category \(\mathcal{C}[G]\).
The other leading class of examples in the paper arises naturally if one attempts to study \(2\)-categorical structures as strictly as possible, e.g., by considering the \(2\)-category \(\mathbf{FProd}_{s}\) of categories endowed with a choice of finite products and functors which strictly preserve them in place of \(\mathbf{FProd}\) of categories with finite products and functors preserving finite products up to isomorphisms. The skew monoidal structure on \(\mathbf{FProd}_{s}\) is much easier to construct [J. Bourke, J. Homotopy Relat. Struct. 12, No. 1, 31–81 (2017; Zbl 1417.18001)]. Indeed, they are exhibited in §6 after braided skew multicategories are introduced and it is shown how to pass from these to braided skew monoidal categories under the assumption of a representability condition in §5.
MSC:
18M50 Bimonoidal, skew-monoidal, duoidal categories
18M15 Braided monoidal categories and ribbon categories
18N10 2-categories, bicategories, double categories
18N40 Homotopical algebra, Quillen model categories, derivators
16T10 Bialgebras
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References:
[1] John C. Baez and Martin Neuchl. Higher-dimensional algebra. I. Braided monoidal 2-categories.Adv. Math., 121(2):196-244, 1996. · Zbl 0855.18008
[2] John Bourke. Skew structures in 2-category theory and homotopy theory.J. Homotopy Relat. Struct., 12(1):31-81, 2017. · Zbl 1417.18001
[3] John Bourke and Stephen Lack. Skew monoidal categories and skew multicategories. J. Alg.506:237-266, 2018. · Zbl 1401.18019
[4] J. Donin and A. Mudrov. Quantum groupoids and dynamical categories.J. Alg., 296:348-384, 2006. · Zbl 1135.17005
[5] A. D. Elmendorf and M. A. Mandell, Permutative categories, multicategories, and algebraic K-theory.Alg. Geom. Top., 9:2391-2441, 2009. · Zbl 1205.19003
[6] Nick Gurski. Loop spaces, and coherence for monoidal and braided monoidal bicategories.Adv. Math., 226(5):4225-4265, 2011. · Zbl 1260.18008
[7] Martin Hyland and John Power. Pseudo-commutative monads and pseudo-closed 2-categories.Journal of Pure and Applied Algebra 175:141-185, 2002. · Zbl 1009.18003
[8] Andr´e Joyal and Ross Street. An introduction to Tannaka duality and quantum groups. InCategory theory (Como, 1990), volume 1488 ofLecture Notes in Math., pages 413-492. Springer, Berlin, 1991. · Zbl 0745.57001
[9] Andr´e Joyal and Ross Street. Braided tensor categories.Adv. Math., 102(1):20-78, 1993. · Zbl 0817.18007
[10] Christian Kassel. Quantum groups.Graduate Texts in Mathematics155, SpringerVerlag New York 1995.
[11] Stephen Lack and Ross Street. Skew monoidales, skew warpings and quantum categories.Theory Appl. Categ., 26:385-402, 2012. · Zbl 1252.18016
[12] Stephen Lack and Ross Street. Triangulations, orientals, and skew monoidal categories.Adv. Math., 258:351-396, 2014. · Zbl 1350.18012
[13] Ignacio L´opez-Franco. Pseudo-commutativity of KZ 2-monads.Advances in Mathematics 228, 5:2557-2605, 2011. · Zbl 1231.18006
[14] Ross Street. Skew-closed categories.J. Pure Appl. Algebra, 217(6):973-988, 2013. · Zbl 1365.18008
[15] Ross Street. Quantum groups: a path to current algebra.Australian Math. Society Lecture Series 19(Cambridge University Press; 18 January 2007; ISBN-978-0-52169524-4). · Zbl 1117.16031
[16] Korn´el Szlach´anyi.
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