Duals invert
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 monoidal bicategory. A brief outline of the content of the article by section. The Section 1 begins with an outline of the history of creation of this paper and each author`s contribution. Afterwards the notions of a bidual situation and of a monoidal object in a monoidal bicategory are recalled. In Section 2 the authors generalize, to a monoidal bicategory, what is for an objects \(V\) in a monoidal category to be part of a dual situation. Such dual situation is called an exact pairing. In Section 3 the notion of Frobenius monoidal functor due to Day and Pastro is internalized and their theorems (e.g., that Frobenius monoidal arrows preserve exact pairing) are proved. Section 4 provides equivalent formulations of exact pairings. These enable many interesting isomorphisms. Section 5 contains an application of the ``duals invert'' theme. There it is shown that lax braidings for autonomous monoidal objects are necessarily braidings. In Section 6 it is shown that a ``naturally Frobenius map monoidal object'' (in the authors terminology) has itself as a right bidual. In Section 7 the application to Cartesian bicategories is presented by using the \((-)^{\mathrm {coop}}\) dual of previous general results. In [Zbl 1153.18006] there is a characterization of maps between Frobenius objects in a Cartesian bicategory. In Section 8 the authors give a characterization of monoidal maps between Frobenius map monoidal objects in a general monoidal bicategory. From this theorem, taking into account Section 7, the Cartesian theorem of Walters and Wood can be recovered. In Section 9 a brief look at lax duals in a monoidal bicategory is taken. The authors notice that, in a ``slogan-like'' way, one may say ``lax duals adjoin''. They conclude with two theorems providing a justification for their declaration, ``lax transport exists''. A final remark (Corollary 9.6) is that lax duality can be a very weak notion.
scientific article; zbMATH DE number 5871642
scientific article