4
$\begingroup$

$\def\duevec{2{\rm Vect}}$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 ${\rm Vect}\hookrightarrow 2\rm Vect$?

$\endgroup$
4
  • $\begingroup$ Is there a difference between a lunatic object and a star autonomous structure (usually called a dualizing object in algebraic geometry and algebraic topology)? $\endgroup$
    – Tim Campion
    Jun 25, 2018 at 12:24
  • $\begingroup$ All such ideas are clearly connected; the motivating idea for this investigation was to better understand their interrelation, hopefully avoiding proliferation of standards. $\endgroup$
    – fosco
    Jun 25, 2018 at 12:30
  • $\begingroup$ (also, I do not know how $*$-autonomous structures work, apart from the bare definition, to express myself properly) $\endgroup$
    – fosco
    Jun 25, 2018 at 12:31
  • $\begingroup$ In a sufficiently big $CAT$, the category of sets is lunatic, but $CAT$ isn't $*$-autonomous, or is it? $\endgroup$
    – fosco
    Jun 25, 2018 at 12:43

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.