Doubles for monoidal categories

Craig Pastro and Ross Street

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 of [DA, V] with 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_sA and D_{ls}A of DA such that there are equivalences of Tamb_s(A) with [D_sA, V] and of Tamb_{ls}(A) with [D_{ls}A, V]. When A is autonomous, every Tambara module is strong; this implies an equivalence of Z[A, V] with [DA,V].

Keywords: monoidal centre, Drinfeld double, monoidal category, Day convolution

2000 MSC: 18D10

Theory and Applications of Categories, Vol. 21, 2008, No. 4, pp 61-75.

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