DOUBLES FOR MONOIDAL CATEGORIES

In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal V-category A. When A is autonomous (= rigid = compact), he showed that the V-category (that we call Tamb(A)) of so-equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A, V] of the convolution monoidal V-category [A, V]. Our paper extends these ideas somewhat. For general A, we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence [DA, V] similar or equal to Tamb(A). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V-category Tamb(s)(A) (respectively, Tamb(ls)(A)) which is equivalent to the centre (respectively, lax centre) of [A, V]. We construct localizations D(s)A and D(ls)A of DA such that there are equivalences Tamb(s)(A) similar or equal to [D(s)A, V] and Tamb(ls)(A) similar or equal to [D(ls)A, V]. When A is autonomous, every Tambara module is strong; this implies an equivalence Z[A, V] similar or equal to [DA, V].