# Involutive Functors

Let us call a (co)functor $F:\mathcal{C}\to\mathcal{C}$ on a category $\mathcal{C}$ involutive, if $F^2$ is naturally isomorphic to $1_\mathcal{C}$.

For example, if $\mathcal{C}=\operatorname{Vect}_\mathbb{K}^0$ is the category of finite-dimensional $\mathbb{K}$-vector spaces with $\mathbb{K}$-linear maps, then the cofunctor $F$ of taking duals is idempotent via the natural transformation $\xi_X:X\ni x\mapsto(\alpha\mapsto\alpha(X))\in X^{**}$. One then easily computes that $(\xi_X)^*=(\xi_{X^*})^{-1}$ or equivalently $$F(\xi_X)=(\xi_{F(X)})^{-1}.$$

Is this just by coincidence, or is it true for arbitrary involutive cofunctors? The corresponding statement for involutive functors should be $F(\xi_X)=\xi_{F(X)}$, what can be said about that?

Cheers, Robert

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A generalised version of this was asked on MO just a few days ago. –  Zhen Lin Sep 5 '12 at 1:33
Do you say that my questions is implicitly answered there as well? Actually I don't really understand what they are talking about :) –  Robert Rauch Sep 5 '12 at 7:45
A involutary functor is, in particular, an example of a self-adjoint functor. –  Zhen Lin Sep 5 '12 at 7:50