Doubles for monoidal categories

Abstract

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] \simeq 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) \simeq [D_s A,V] and Tamb_{ls}(A) \simeq [D_{ls} A,V]. When A is autonomous, every Tambara module is strong; this implies an equivalence Z[A,V] \simeq [DA,V].