## Abstract

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 Tamb_{s}(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 D_{s}script A sign and script D_{ls}script A sign of script Dscript A sign such that there are equivalences Tamb_{s}(script A sign) ≃ [script D_{s}script A sign, script V sign] and Tamb _{ls}(script A sign) ≃ [script D_{ls}script 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].

Original language | English |
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Pages (from-to) | 61-75 |

Number of pages | 15 |

Journal | Theory and Applications of Categories |

Volume | 21 |

Publication status | Published - 6 Jun 2008 |