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Duals Invert

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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 , if A is Frobenius then the category Map(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.

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Authors and Affiliations

  1. Laboratorie Prueves, Programmes et Systèmes, Univsersite Paris Diderot—Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Ignacio López Franco

  2. Department of Mathematics, Macquarie University, Sydney, New South Wales, 2109, Australia

    Ross Street

  3. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

    Richard J. Wood

Authors
  1. Ignacio López Franco
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  2. Ross Street
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  3. Richard J. Wood
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Corresponding author

Correspondence to Richard J. Wood.

Additional information

In memory of Max

The first author was supported by an Internal Graduate Studentship at Trinity College, Cambridge, and by the Fondation Sciences Mathématiques de Paris.

The second and third authors gratefully acknowledge financial support from the Australian ARC and the Canadian NSERC. Diagrams typeset using M. Barr’s diagram package, diagxy.tex.

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López Franco, I., Street, R. & Wood, R.J. Duals Invert. Appl Categor Struct 19, 321–361 (2011). https://doi.org/10.1007/s10485-009-9210-7

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  • DOI: https://doi.org/10.1007/s10485-009-9210-7

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