abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
In a monoidal category a self-duality on a dualizable object $X$ is a choice of equivalence $X \simeq X^\ast$ with its dual object.
If each object $X$ of a compact closed category is equipped with a self-duality structure $h_X : X \simeq X^\ast$, then sending morphisms to their dual morphisms but with these identifications pre- and postcomposed
constitutes a dagger-compact category structure.
See for instance (Selinger, remark 4.5).
Applied for instance to the category of finite-dimensional inner product spaces this dagger-operation sends matrices to their transposed matrix?.
In terms of string diagrams (following Joyal and Street’s conventions for braided monoidal categories), Selinger argues that the isomorphism $h_X : X \simeq X^\ast$ should be depicted as a half-twist. In particular, for a tortile category equipped with a self-duality structure, the coherence condition
decomposes a full twist into a pair of half-twists.
This is a special case of half-twists as described by Egger.
Peter Selinger, Autonomous categories in which $A \simeq A^\ast$, talk at QPL 2012 (pdf)
Jeff Egger, On involutive monoidal categories, Theory and Applications of Categories, Vol. 25, 2011, No. 14, pp 368-393. (TAC)
Last revised on January 27, 2016 at 05:37:01. See the history of this page for a list of all contributions to it.