# Lunatic objects in the 2-category of 2-vector spaces

$\def\duevec{2{\rm Vect}}$See here for the notation.

The 2-category of 2-vector spaces is extremely rich in structure: I'm interested in studying some of its properties in the following perspective.

Fact 1: there is an involution of $\duevec$ transposing 1-cells and applying transposition entry-wise to 2-cells.

Fact 2: there is a semi-tautological embedding ${\rm Vect}\hookrightarrow \duevec$ which is constant on $\langle 1\rangle$ on 0-cells, and sends $V$ into the endomorphism of $\langle 1\rangle$ determined by the integer $\dim_KV$.

Now.

Let $\cal K$ be a 2-category which is monoidal closed, and endowed with an "involution" (for the purpose of this discussion, an involution is a functor doing the same job that $(-)^{op}$ does on $\duevec$ in Fact 1 above). Let's call an object $\Omega$ of such a monoidal closed 2-category lunatic if it has the following properties:

• the 1-cell $\phi^* = [ \phi^{op},\Omega]$ has a left adjoint for each $\phi : X\to Y$; denote $\phi_! \dashv \phi^*$ this adjunction.
• there is a map $X \to [X^{op},\Omega]$ which is the unit of a monad structure for the 2-functor $T=[\,\_\,^{op},\Omega]$
• the multiplication of this monad is obtained applying $T$ to $\eta_X$, so that $\mu_X = T\eta_X : [[X^{op},\Omega]^{op},\Omega]\to [X^{op},\Omega]$

Question (in case the question here has affirmative answer): does $2{\rm Vect}$ has a lunatic object? Is such an object the ground field under the embedding ${\rm Vect}\hookrightarrow 2\rm Vect$?

• Is there a difference between a lunatic object and a star autonomous structure (usually called a dualizing object in algebraic geometry and algebraic topology)? – Tim Campion Jun 25 '18 at 12:24
• All such ideas are clearly connected; the motivating idea for this investigation was to better understand their interrelation, hopefully avoiding proliferation of standards. – Fosco Jun 25 '18 at 12:30
• (also, I do not know how $*$-autonomous structures work, apart from the bare definition, to express myself properly) – Fosco Jun 25 '18 at 12:31
• In a sufficiently big $CAT$, the category of sets is lunatic, but $CAT$ isn't $*$-autonomous, or is it? – Fosco Jun 25 '18 at 12:43