nLab
self-dual object

Contents

Context

Duality

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

In a monoidal category a self-duality on a dualizable object XX is a choice of equivalence XX *X \simeq X^\ast with its dual object.

Properties

Relation to \dagger-compact structure

If each object XX of a compact closed category is equipped with a self-duality structure h X:XX *h_X : X \simeq X^\ast, then sending morphisms to their dual morphisms but with these identifications pre- and postcomposed

() :(XfY)(Yh YY *f *X *h X 1X) (-)^\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:XX *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 *h X *X **h X *X *)=(X *θ X *X *) (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.

References

  • Peter Selinger, Autonomous categories in which AA *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.