My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge).
(p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-ring, $C$ a rigid $R$-linear symmetric monoidal $\infty$-category and $\omega\colon C\to\mathrm{Mod}^{\mathrm{rig}}_R$ be an $R$-linear symmetric monoidal functor ($\mathrm{Mod}^{\mathrm{rig}}_R$ is the $\infty$-category of rigid $R$-modules). Then $\omega$ is said to be
a finite fiber functor if $\mathrm{Ind}(\omega)$ is conservative and preserves small limits,
if $R$ is connective, then $\omega$ is a flat fiber functor if $\mathrm{Ind}(\omega)$ is concervative, creates a $t$-structure on $C$, is exact and whose right adjoint is $t$-exact.
if $R$ is connective and bounded, then $\omega$ is a positive fiber functor if $\mathrm{Ind}(\omega)$ is conservative, creates a $t$-structure on $C$ and is exact.
(p.$59$, definition $9.1$) Let $R$ be an $E_{\infty}$-ring. A rigid $R$-tensor $\infty$-category $T$ is
finite $R$-Tannakian if there exists a finite fiber functor,
flat $R$-Tannakian if there exists a flat fiber functor,
positive $R$-Tannakian if there exists a positive fiber functor.
I would like to know which rigid symmetric monoidal $\infty$-categories are examples and which are counterexamples to this definition. Also, I would be interested in figuring out how strong these conditions are on an arbitrary rigid symmetric monoidal $\infty$-category. For starters, take the following rigid $\infty$-categories categories:
The derived $\infty$-category $\mathrm{Perf}(X)$ of perfect complex complexes on a quasicompact quasiseparated scheme.
The stable homotopy category of finite pointed CW-complexes.
The category $\mathrm{stab(k[G])}$ of finitely-generated $k[G]$-modules (modulo the projectives) where $G$ is a finite group (or a finite group scheme) and $k$ is a field of positive characteristic.
Let $G$ be a compact Lie group. I wonder if the subcategory spanned by the compact objects of the equivariant stable homotopy category of $G$-spectra is Tannakian a-la Wallbridge.
The category $\mathrm{DM_{gm}}(S)$ of geometric $S$-motives for a base scheme $S$.
The subcategory $\mathrm{SH}^{\mathbb{A}^1}_{\mathrm{gm}}(S)$ generated by smooth $S$-schemes in the stable motivic homotopy category $\mathrm{SH}^{\mathbb{A}^1}(S)$.
I would be interesting to know whether these are (finite, flat, positive) Tannakian.
