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.

http://www.tac.mta.ca/tac/volumes/21/4/21-04.dvi

http://www.tac.mta.ca/tac/volumes/21/4/21-04.ps

http://www.tac.mta.ca/tac/volumes/21/4/21-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/21/4/21-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/21/4/21-04.ps