caa a22 4500 445378611 CHVBK 20180317143010.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10485-009-9210-7 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9210-7 Duals Invert [Elektronische Daten] [Ignacio López Franco, Ross Street, Richard Wood] 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. Springer Science+Business Media B.V., 2009 Monoidal bicategory nationallicence Monoidal object nationallicence Dual nationallicence Frobenius condition nationallicence Cartesian bicategory nationallicence Centre construction nationallicence López Franco Ignacio Laboratorie Prueves, Programmes et Systèmes, Univsersite Paris Diderot—Paris 7, Case 7014, 75205, Paris Cedex 13, France aut Street Ross Department of Mathematics, Macquarie University, 2109, Sydney, New South Wales, Australia aut Wood Richard Department of Mathematics and Statistics, Dalhousie University, B3H 3J5, Halifax, Nova Scotia, Canada aut Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 321-361 0927-2852 19:1<321 2011 19 10485 https://doi.org/10.1007/s10485-009-9210-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9210-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- López Franco Ignacio Laboratorie Prueves, Programmes et Systèmes, Univsersite Paris Diderot—Paris 7, Case 7014, 75205, Paris Cedex 13, France aut NATIONALLICENCE 700 1- Street Ross Department of Mathematics, Macquarie University, 2109, Sydney, New South Wales, Australia aut NATIONALLICENCE 700 1- Wood Richard Department of Mathematics and Statistics, Dalhousie University, B3H 3J5, Halifax, Nova Scotia, Canada aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 321-361 0927-2852 19:1<321 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer