nLab self-dual object




Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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.


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 2011.


  • Peter Selinger, Autonomous categories in which AA *A \simeq A^\ast, talk at QPL 2012 (pdf)

  • Matthew B. Young, §3.1 in Self-Dual Hall modules, PhD thesis, Stony Brook (2013) [pdf, pdf]

    (in a context relating to orientifold-BPS-algebras)

  • Chenjing Bu, Def. 3.2 in: Enumerative invariants in self-dual categories. I. Motivic invariants [arXiv:2302.00038]

On half-twists as above

  • Jeff Egger, On involutive monoidal categories, Theory and Applications of Categories 25 14 (2011) 368-393 [tac:25-14]

Under the cobordism hypothesis (which is a theorem certainly for the relevant case n=1n = 1), self-dual objects in symmetric monoidal \infty -categories correspond equivalently to un-oriented 1-dimensional TQFTs:

same non-topological functorial field theory

Last revised on November 7, 2023 at 11:13:35. See the history of this page for a list of all contributions to it.