TY - JOUR

T1 - Dualizations and antipodes

AU - Day, Brian

AU - McCrudden, Paddy

AU - Street, Ross

PY - 2003/6

Y1 - 2003/6

N2 - Because an exact pairing between an object and its dual is extraordinarily natural in the object, ideas of R. Street apply to yield a definition of dualization for a pseudomonoid in any autonomous monoidal bicategory as base; this is an improvement on Day and Street, Adv. in Math. 129 (1997), Definition 11, p. 114. We analyse the dualization notion in depth. An example is the concept of autonomous (which, usually in the presence of a symmetry, also has been called "rigid" or "compact") monoidal category. The antipode of a quasi-Hopf algebra H in the sense of Drinfeld is another example obtained using a different base monoidal bicategory. We define right autonomous monoidal functors and their higher-dimensional analogue. Our explanation of why the category Comodf (H) of finite-dimensional representations of H is autonomous is that the Comodf operation is autonomous and so preserves dualization.

AB - Because an exact pairing between an object and its dual is extraordinarily natural in the object, ideas of R. Street apply to yield a definition of dualization for a pseudomonoid in any autonomous monoidal bicategory as base; this is an improvement on Day and Street, Adv. in Math. 129 (1997), Definition 11, p. 114. We analyse the dualization notion in depth. An example is the concept of autonomous (which, usually in the presence of a symmetry, also has been called "rigid" or "compact") monoidal category. The antipode of a quasi-Hopf algebra H in the sense of Drinfeld is another example obtained using a different base monoidal bicategory. We define right autonomous monoidal functors and their higher-dimensional analogue. Our explanation of why the category Comodf (H) of finite-dimensional representations of H is autonomous is that the Comodf operation is autonomous and so preserves dualization.

UR - http://www.scopus.com/inward/record.url?scp=0038379723&partnerID=8YFLogxK

U2 - 10.1023/A:1024236601870

DO - 10.1023/A:1024236601870

M3 - Article

AN - SCOPUS:0038379723

VL - 11

SP - 229

EP - 260

JO - Applied Categorical Structures

JF - Applied Categorical Structures

SN - 0927-2852

IS - 3

ER -