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 2011.

Related concepts

• inner product

References

• Peter Selinger, Autonomous categories in which $A \simeq A^\ast$, 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 $n = 1$), self-dual objects in symmetric monoidal $\infty$-categories correspond equivalently to un-oriented 1-dimensional TQFTs:

• Jacob Lurie: Ex. 2.4.28 in: On the Classification of Topological Field Theories, Current Developments in Mathematics 2008 (2009) 129-280 [arXiv:0905.0465, doi:10.4310/CDM.2008.v2008.n1.a3, euclid:cdm/1254748657]

• Mikhail Khovanov (notes by You Qi), Thm. 4 on p. 7 in §2 of: Introduction to categorification, lecture notes, Columbia University (2010, 2020) [web, web, full:pdf]

• Constantin Teleman, Rem. 1.7 in: Five lectures on topological field theory, in Geometry and Quantization of Moduli Spaces, CRM Advanced Courses in Mathematics, Birkhäuser (2016) [doi:10.1007/978-3-319-33578-0_3, pdf, pdf]

same non-topological functorial field theory

• Daniel Grady, Dmitri Pavlov, Thm 6.1.5 in: The geometric cobordism hypothesis [arXiv:2111.01095]