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Applied Categorical Structures

, Volume 19, Issue 1, pp 321–361 | Cite as

Duals Invert

  • Ignacio López Franco
  • Ross Street
  • Richard J. Wood
Article

Abstract

Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229–260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229–260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733–742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25–47, 2008) asserting that, for objects A and X in a cartesian bicategory Open image in new window , if A is Frobenius then the category Map Open image in new window (X,A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25–47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184–190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory.

Keywords

Monoidal bicategory Monoidal object Dual Frobenius condition Cartesian bicategory Centre construction 

Mathematics Subject Classification (2000)

Primary 18D05 Secondary 18D10 18D15 18D20 18D25 18D35 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Ignacio López Franco
    • 1
  • Ross Street
    • 2
  • Richard J. Wood
    • 3
  1. 1.Laboratorie Prueves, Programmes et SystèmesUnivsersite Paris Diderot—Paris 7Paris Cedex 13France
  2. 2.Department of MathematicsMacquarie UniversitySydneyAustralia
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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