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$\newcommand{\C}{\mathcal{C}}\newcommand{\D}{\mathcal{D}}\newcommand{\op}{\mathrm{op}}$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?

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    $\begingroup$ 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. $\endgroup$
    – Martin Brandenburg
    Commented Mar 27, 2012 at 10:28
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    $\begingroup$ 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 $\endgroup$
    – Theo Buehler
    Commented Mar 27, 2012 at 11:12
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    $\begingroup$ 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). $\endgroup$
    – Martin Brandenburg
    Commented Mar 27, 2012 at 11:32
  • $\begingroup$ 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. $\endgroup$
    – Dan Petersen
    Commented Mar 27, 2012 at 12:00
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    $\begingroup$ See also in "Quadratic and Hermitian forms over Rings" Knus, p.75, and the QUebbman (1979) article on Bibliography. $\endgroup$
    – Buschi Sergio
    Commented Mar 27, 2012 at 19:46

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The question has essentially been answered in the comments, I am recording this here so that the question does not get bumped back to the top. Theo Buehler and Buschi Sergio both gave nice references, and it seems that this notion is well known in K-theory under the name "category with duality". Martin's remarks were also helpful.

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    $\begingroup$ You should summarise the answer arising via comments, and then accept this answer. $\endgroup$
    – David Roberts ♦
    Commented Mar 28, 2012 at 10:00
  • $\begingroup$ Dan, I think you should give this definition in this post, because it would save people the trouble of searching. $\endgroup$
    – Sergei Akbarov
    Commented Nov 2, 2022 at 4:49
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Grothendieck defined the notion of "autodualité" in SGA 1 VI 11 d) "Exemples divers" in his formalism of fibred categories.

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