540 play a prominent role in the definition. |
540 play a prominent role in the definition. |
541 (In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off |
541 (In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off |
542 the tongue as well as ``disk-like''.) |
542 the tongue as well as ``disk-like''.) |
543 |
543 |
544 Another thing we need a name for is the ability to rotate morphisms around in various ways. |
544 Another thing we need a name for is the ability to rotate morphisms around in various ways. |
545 For 2-categories, ``strict pivotal" is a standard term for what we mean. |
545 For 2-categories, ``strict pivotal" is a standard term for what we mean. (See \cite{MR1686423, 0908.3347}, although note there the definition is only for monoidal categories; one can think of a monoidal category as a 2-category with only one $0$-morphism, then relax this requirement, to obtain the sensible notion of pivotal (or strict pivotal) for 2-categories. Compare also \cite{1009.0186} which addresses this issue explicitly.) |
546 A more general term is ``duality", but duality comes in various flavors and degrees. |
546 A more general term is ``duality", but duality comes in various flavors and degrees. |
547 We are mainly interested in a very strong version of duality, where the available ways of |
547 We are mainly interested in a very strong version of duality, where the available ways of |
548 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
548 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
549 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
549 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
550 by ``disk-like". |
550 by ``disk-like". |