abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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 2011.
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
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
Last revised on November 7, 2023 at 11:13:35. See the history of this page for a list of all contributions to it.