Extending Tannakian "dictionary" to gerbes

Question

The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".

Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the corresponding functor $\operatorname{Rep}_k(G') \to \operatorname{Rep}_k(G)$.

a) $f$ is faithfully flat if and only if $\omega^f$ is fully faithful and every subobject of $\omega^f(X')$ for $X' \in \operatorname{ob}(\operatorname{Rep}_k(G'))$, is isomorphic to the image of a subobject of $X'$.

b) $f$ is a closed immersion if and only if every object of $\operatorname{Rep}_k(G)$ is isomorphic to a subquotient of an object of the form of $\omega^f(X'), X' \in \operatorname{ob}(\operatorname{Rep}_k(G'))$.

The proof of the above proposition relies on the fact that $G$ and $G'$ are affine and therefore the spectra of Hopf algebras.

Are there analogous results for non-neutral Tannakian categories? The groups $G$ and $G'$ would have to be replaced by affine gerbes, the theory of which is much less developed than that of affine algebraic groups. Homomorphisms of gerbes are homomorphisms of the underlying stacks, and I unfortunately know very little about the algebraic geometry of stacks. Maybe these results are known to experts already. I would appreciate any help.

Answer 1

Yes, the statement extends to general tannakian categories --- you just have to replace the affine group schemes in the statement with the bands (liens) of the associated gerbes.

The following is a translation of Saavedra 1972, III, 3.3.3, p.205.

Let $\omega\colon\mathsf{C}^{\prime}\rightarrow\mathsf{C}$ be a morphism of tannakian categories over a field $k$, bound by a morphism of bands $u\colon L\rightarrow L^{\prime}$.

(a) $u$ is faithfully flat (i.e., an epimorphism of bands) if and only if $\omega$ is fully faithful and its essential image is stable under taking subobjects.

(b) $u$ is injective (i.e., locally represented by a monomorphism of groups) if and only if every object of $\mathsf{C}{}$ is a subobject of a quotient of $\omega(X^{\prime})$ for some $X^{\prime}\in\mathrm{ob}\mathsf{C}^{\prime}$.