On braiding and dyslexia (Q1346092)

The main theorem of this paper is that, for any commutative algebra \(A\) in a braided monoidal category \(\mathcal V\) (with usual assumptions relating tensor products and colimits), the category \(\text{dys } {\mathcal V}^ A\) of dyslectic (right) \(A\)-modules is also braided monoidal. An \(A\)-module \(M\) is dyslectic when it ``sees'' the braiding at \(M\), \(A\) as symmetric. The inclusion of \(\text{dys } {\mathcal V}^ A\) in the cateogory \({\mathcal V}^ A\) of all right \(A\)-modules is shown to have a right adjoint when \(\mathcal V\) is finitely complete and closed. In fact, \(\text{dys } {\mathcal V}^ A\) is a full sub-braided-monoidal-category of the centre \({\mathcal Z}({\mathcal V}^ A)\) of \({\mathcal V}^ A\) as defined by \textit{A. Joyal} and the reviewer [J. Pure Appl. Algebra 71, No. 1, 43-51 (1991; Zbl 0726.18004)].