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Invariance of projective modules in \(\mathsf{Sup}\) under self-duality. (English) Zbl 1457.18006

Given a unital quantale \((Q,\otimes, e)\), there exists the category \(\mathbf{Mod}{}_{r}(Q)\) of right unital \(Q\)-modules, whose objects are pairs \((M,\boxdot)\), where \(M\) is a complete lattice, and \(\boxdot:M\times Q\rightarrow M\) is a map with the following four properties: (1) \((\bigvee S)\boxdot q=\bigvee_{s\in S}(s\boxdot q)\) for every \(S\subseteq M\) and every \(q\in Q\); (2) \(m\boxdot(\bigvee T)=\bigvee_{t\in T}(m\boxdot t)\) for every \(m\in M\) and every \(T\subseteq Q\); (3) \((m\boxdot q_1)\boxdot q_2=m\boxdot(q_1\otimes q_2)\) for every \(m\in M\) and every \(q_1,q_2\in Q\); and (4) \(m\boxdot e=m\) for every \(m\in M\) (see, [D. Kruml and J. Paseka, Handb. Algebra 5, 323–362 (2008; Zbl 1219.06016)] for more detail). In particular, if \(Q\) is the two-element quantale \(\mathsf{2}=\{\bot,\top\}\), then the category \(\mathbf{Mod}{}_{r}(\mathsf{2})\) is isomorphic to the category Sup of complete latices and join-preserving maps. If \(Q\) has additionally an involution \({}^{\prime}\), then there exists a self-duality on the category \(\mathbf{Mod}{}_{r}(Q)\), i.e., a contravariant functor \(S:\mathbf{Mod}{}_{r}(Q)\rightarrow\mathbf{Mod}{}_{r}(Q)\) such that \(S\circ S=1_{\mathbf{Mod}{}_{r}(Q)}\), defined on an object \((M,\boxdot)\) by \(S(M,\boxdot)=(M^{op},\boxdot^{op})\), where \(M^{op}\) is the dual complete lattice of \(M\), and \(m\boxdot^{op}q=\bigvee\{n\in M\,|\, n\boxdot q^{\prime}\leqslant m\,\}\); and on a morphism \(f:(M,\boxdot)\rightarrow(N,\boxdot)\) by \(S(f)=f^{\vdash}\), where \(f^{\vdash:}N^{op}\rightarrow M^{op}\) is the right adjoint map of \(f\) (in the sense of partially ordered sets).
Based in their previous study of quantales and quantale modules in [P. Eklund et al., Semigroups in complete lattices. Quantales, modules and related topics. Cham: Springer (2018; Zbl 1491.06001)] and the paper of I. Stubbe [Theor. Comput. Sci. 373, No. 1–2, 142–160 (2007; Zbl 1111.68073)], who showed that projectivity in the category \(\mathbf{Mod}{}_{r}(Q)\) is equivalent to complete distributivity enriched over \(Q\), the present authors consider preservation of projectivity by the above-mentioned duality \(S\). More precisely, they show that if the underlying quantale \(Q\) is unital and involutive with a designated element (see Definition 4.5 for more detail), then the duality \(S\) preserves projectivity if and only if \(Q\) has a dualizing element (Theorem 4.11). The last section of the paper shows that involutive quantales from the above result are widespread, e.g., they are induced by completely distributive lattices with an order-reversing involution or by arbitrary groups.
The paper is well written, provides most of its required preliminaries, and will be of interest to all the researchers studying categories enriched in quantales.

MSC:

18D20 Enriched categories (over closed or monoidal categories)
06F07 Quantales
03G10 Logical aspects of lattices and related structures
06D10 Complete distributivity
06D75 Other generalizations of distributive lattices
18B25 Topoi
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References:

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