# Examples and non-examples of Tannakian $\infty$-categories

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.