In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a *tame* abelian tensor category. An abelian tensor category is called tame if every short exact sequence $0 \to M' \to M \to M'' \to 0$ with $M''$ flat remains exact after tensoring with an arbitrary object. Basically this means that flat objects behave in the usual way. For example it is easy to check that in a tame abelian tensor category every extension of flat objects is flat (Lemma 5.4).

**Question 1:** Do you know any further literature about tame abelian tensor categories?

**Question 2:** Can you give an explicit example of an abelian tensor category which is *not* tame? Jacob Lurie remarks that every abelian tensor category with enough flat objects is tame, because then we can define $Tor_1$ and use symmetry. So in particular, a counterexample does not have enough flat objects.
[Tom Goodwillie has given an example below]

**Question 3:** Can you give an example of a cocontinuous tensor functor between tame abelian tensor categories which is not tame?