We formulate an elementary condition on an involutive quantaloid $Q$ under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of $Q$-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric $Q$-enriched category is again symmetric. Examples include Lawvere's quantale of non-negative real numbers and Walters' small quantaloids of closed cribles.
Keywords: Quantaloid, enriched category, symmetry, Cauchy completion
2000 MSC: 06F07, 18C15, 18D05, 18D20
Theory and Applications of Categories, Vol. 25, 2011, No. 11, pp 276-294.