In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal script V sign-category script A sign. When script A sign is autonomous (= rigid = compact), he showed that the script V sign-category (that we call Tamb(script A sign)) of so-equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre script Z sign[script A sign, script V sign] of the convolution monoidal script V sign-category [script A sign, script V sign]. Our paper extends these ideas somewhat. For general script A sign, we construct a promonoidal script V sign-category script Dscript A sign (which we suggest should be called the double of script A sign) with an equivalence [script Dscript A sign, script V sign] ≃ Tamb(script A sign). When script A sign is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a script V sign-category Tambs(script A sign) (respectively, Tamb ls(script A sign)) which is equivalent to the centre (respectively, lax centre) of [script A sign, script V sign]. We construct localizations script Dsscript A sign and script Dlsscript A sign of script Dscript A sign such that there are equivalences Tambs(script A sign) ≃ [script Dsscript A sign, script V sign] and Tamb ls(script A sign) ≃ [script Dlsscript A sign, script V sign]. When script A sign is autonomous, every Tambara module is strong; this implies an equivalence script Z sign[script A sign, script V sign] ≃ [script Dscript A sign, script V sign].
|Number of pages||15|
|Journal||Theory and Applications of Categories|
|Publication status||Published - Jun 6 2008|
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