nLab self-dual object

Idea

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.

Properties

Relation to $\dagger$-compact structure

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

$$(-)^\dagger \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (Y \stackrel{h_Y}{\to} Y^\ast \stackrel{f^\ast}{\longrightarrow} X^\ast \stackrel{h_X^{-1}}{\to} X)$$

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.

Graphical language

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

$$(X^\ast \stackrel{h_{X^\ast}}{\to} X^{\ast\ast} \stackrel{h_X^\ast}{\to} X^\ast) \; =\; (X^\ast \stackrel{\theta_{X^\ast}}{\longrightarrow} X^\ast)$$

decomposes a full twist into a pair of half-twists.

This is a special case of half-twists as described by Egger.

Related concepts

• inner product

References

• 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)