Reflexivity and dualizability in categorified linear algebra

Understand

The "linear dual" of a cocomplete linear category $\mathcal C$ is the category of all cocontinuous linear functors $\mathcal C \to \mathrm{Vect}$.

• We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing).
• Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens.
• Along the way, we prove that the category $\mathrm{Qcoh}(X)$ of quasi-coherent sheaves on a stack $X$ is not dualizable if $X$ is the classifying stack of a semisimple algebraic group in positive characteristic or if $X$ is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if $X$ is the quotient of an affine scheme by a virtually linearly reductive group.