Braided skew monoidal categories.

*(English)*Zbl 1431.18012The 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.

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.

Reviewer: Hirokazu Nishimura (Tsukuba)

##### 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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\textit{J. Bourke} and \textit{S. Lack}, Theory Appl. Categ. 35, 19--63 (2020; Zbl 1431.18012)

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