ct.category theory – Is there a bestiary of “derived 2-vector spaces”?


The appendix “A bestiary of 2–vector spaces” of Bartlett, Douglas, Schommer-Pries, Vicary, “Modular categories as representations of the 3-dimensional bordism 2-category” analyzes a number of symmetric monoidal (weak) 2-categories each deserving the name “the 2-category of 2-vector spaces”. The main result is that in all the analyzed 2-categories, the sub-2-categories of fully dualizable objects all agree. (The subcategories of 1-dualizable objects vary depending on the 2-category.)

What would such a bestiary look like if instead of “vector spaces” we worked with “derived vector spaces”, maybe meaning the $infty$-category of chain complexes? Some options for “the derived 2-category of derived 2-vector spaces” might be various $(infty,2)$-categories of linear $(infty,1)$-categories, for various choices of functors.

Question 0: Has anyone assembled a “bestiary” of such options?

Question 1: Is there a (known or conjectured) bestiary-type result identifying the fully dualizable sub-$(infty,2)$-categories of the different options?

Question 2: How do the derived an underived bestiaries relate?