Duals invert
type: Item
instance of: scholarly article
title: Duals invert
zbMATH Open document ID: 1259.18002
author: R. J. Wood
author: Ignacio López Franco
author: Ross H. Street
published in: Applied Categorical Structures
publication date: 2011-03-30T00:00:00Z
review text: The review partly uses the authors' abstract. As the authors notice, monoidal objects (pseudomonoids) in monoidal bicategories share many properties with monoidal categories. The existence of (left or right) duals in monoidal categories permits to define dualization operations; a related definition was suggested for monoidal objects by \textit{B. Day} et al. [Appl. Categ. Struct. 11, No. 3, 229--260 (2003; Zbl 1137.18302)]. The authors define a relative version of this notion called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. They supplement the above-mentioned results of Day et al. and even correct one of them. The existence of biduals in the bicategory is assumed only where necessary. The authors also abstract more recent work of \textit{B. Day} and \textit{C. Pastro} [New York J. Math. 14, 733--742 (2008; Zbl 1178.18006)] on Frobenius monoidal functors to the monoidal bicategory context. As the authors notice, the starting point of this work was the question of invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. A more abstract approach they develop permitted to recover a theorem of \textit{R. F. C. Walters} and the third author [Theory Appl. Categ. 20, 25--47 (2008; Zbl 1153.18006)] asserting that, for objects \(A\) and \(X\) in a Cartesian bicategory, if \(A\) is Frobenius then the corresponding category \(\mathcal Map(X, A)\) of left adjoint arrows is a groupoid. The characterization in [Zbl 1153.18006] of left adjoint arrows between Frobenius objects of a Cartesian bicategory is presented in terms of their present approach. It is shown also that when a monoidal object admits a dualization, its lax centre coincides with the centre defined by the second author [Theory Appl. Categ. 13, 184--190 (2004; Zbl 1058.18004)]. In the end they consider the relationship between lax duals for objects and adjoints for arrows in a mon…
reviewed by: Serguei V. Solov'ev
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18D25
Mathematics Subject Classification ID: 18D35
zbMATH DE Number: 5871642
zbMATH Keywords: monoidal bicategory
zbMATH Keywords: monoidal object
zbMATH Keywords: dual
zbMATH Keywords: Frobenius condition
zbMATH Keywords: Cartesian bicategory
zbMATH Keywords: centre construction
Wikidata QID: Q56687287
MaRDI profile type: Publication
full work available at URL: s10485-009-9210-7
OpenAlex ID: W3189071403
cites work: Introduction to bicategories
cites work: Introduction to linear bicategories
cites work: Cartesian Bicategories II
cites work: Linearly distributive functors
cites work: Note on Frobenius monoidal functors
cites work: Q3514604
cites work: Monoidal bicategories and Hopf algebroids
cites work: Coherence for tricategories
cites work: Braided tensor categories
cites work: The compact closed bicategory of left adjoints
cites work: A coherent approach to pseudomonads
cites work: Formal Hopf algebra theory. I: Hopf modules for pseudomonoids
cites work: Formal Hopf algebra theory. II: Lax centres
cites work: Q4342436
cites work: Q4487536
cites work: Categories tannakiennes
cites work: The monoidal center construction and bimodules
cites work: Q3156504
cites work: Q4534606
cites work: Q4668182
cites work: Frobenius Objects in Cartesian Bicategories
cites work: Quantum groups and representations of monoidal categories
DOI: 10.1007/S10485-009-9210-7
DBLP publication ID: journals/acs/FrancoSW11
Recommended article: Inversions
Recommended article: Chapter 5: Dualities
Recommended article: Inversion, Iteration, and the Art of Dual Wielding
Recommended article: Dualisation of dualities
Recommended article: The dual index and dual core generalized inverse
Recommended article: Q3685650
Recommended article: Q3774076
Recommended article: Inversion Coding
description: scientific article; zbMATH DE number 5871642