# What is a self-dual category?

$\newcommand{\C}{\mathcal{C}}\newcommand{\D}{\mathcal{D}}\newcommand{\op}{\mathrm{op}}$I would like to define the notion of a self-dual category, which should mean a category isomorphic to its opposite in a natural way, and the notion a self-dual functor between such categories. For a category $\C$, I denote by $\C^\op$ its opposite category; for a functor $F \colon \C \to \D$ its opposite functor is $F^\op \colon \C^\op \to \D^\op$; for a natural transformation $\eta \colon F \Rightarrow G$ I denote $\eta^\op \colon G^\op \Rightarrow F^\op$ its opposite natural transformation.

Definition. A self-dual category is a category $\C$, a functor $i_\C \colon \C \to \C^\op$, and a natural isomorphism $$\epsilon_\C \colon i_\C^\op \circ i_\C \Rightarrow \mathrm{id}_\C.$$

I think this is the "correct" definition. (Not quite, see below.) An example would be for $\C$ the category of finite dimensional vector spaces over a field $k$, and $i_\C = \mathrm{Hom}(-,k)$. We would now like to talk about functors being self-dual, which should mean that they commute with taking duals.

Definition. A self-dual functor $\C \to \D$ is a functor $F \colon \C \to \D$ and a natural isomorphism $$\eta \colon i_\D \circ F \Rightarrow F^\op \circ i_\C$$ satisfying the following coherence condition: The diagram of natural isomorphisms

$$\begin{matrix} F \circ i_\C^{\op} & \stackrel{\eta^{\op}}{\Rightarrow} & i_\D^{\op} \circ F^{\op} \\\\ \epsilon_D \Uparrow ~ ~ ~ & & ~ ~ ~ \Downarrow \epsilon_C^{\op} \\\\ i_\D^{\op} \circ i_\D \circ F \circ i_\C^{\op} & \stackrel{\eta}{\Rightarrow} & i_\D^{\op} \circ F^{\op} \circ i_\C \circ i_\C^{\op} \end{matrix}$$

commutes.

This whole definition is quite a mouthful and it feels like someone ought to have defined this carefully somewhere. Googling for self-dual or autodual categories produces several hits where people use this term for categories isomorphic to their opposite, but I haven't seen anyone discuss categories which have such an isomorphism in a coherent way. Does anyone know whether there is such a reference? Maybe this is a special case of a more general construction?

• Side Question: If we are in the situation of the first definition of a self-dual category, is $i_C$ left adjoint to $i_C^{\op}$ with counit $\epsilon_C$ and unit $\epsilon_C^{\op}$? If this was true, we could give a more symmetric definition of a self-dual category, namely as an adjoint equivalence such that the unit and counit morphisms are dual to each other. – Martin Brandenburg Mar 27 '12 at 10:28
• A closely related concept with some basic discussion can be found in section 1 of Paul Balmer's Handbook of K-theory article on Witt groups, publication 14 here: math.ucla.edu/~balmer/research/publications.html – Theo Buehler Mar 27 '12 at 11:12
• In Balmer's definition, cited by Theo, I think that the additional requirement in the definition of a self-category is precisely what I have asked in the Side question above: It is required that $\epsilon_C$ and $\epsilon_C^{op}$ are compatible (i.e. that the triangular identities are satisfied; here one of them suffices). – Martin Brandenburg Mar 27 '12 at 11:32
• Thanks Martin for the diagram and Theo for the reference! I agree with you Martin that one should moreover demand that $\epsilon$ and $\epsilon^\mathrm{op}$ are compatible in this sense. – Dan Petersen Mar 27 '12 at 12:00
• See also in "Quadratic and Hermitian forms over Rings" Knus, p.75, and the QUebbman (1979) article on Bibliography. – Buschi Sergio Mar 27 '12 at 19:46