TY - JOUR
T1 - Doubles for monoidal categories
T2 - Dedicated to Walter Tholen on his 60th birthday
AU - Pastro, Craig
AU - Street, Ross
PY - 2008/6/6
Y1 - 2008/6/6
N2 - 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].
AB - 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].
UR - http://www.scopus.com/inward/record.url?scp=45149087498&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:45149087498
SN - 1201-561X
VL - 21
SP - 61
EP - 75
JO - Theory and Applications of Categories
JF - Theory and Applications of Categories
ER -