TITLE: Categories where projective objects are flat 
QUESTION [15 upvotes]: In many categories arising in the theory of algebras or modules, projective objects are flat. For example each projective module over a ring with identity is flat. However, there are categories where this principle is violated: Lewis has shown that the category of Mackey functors for the orthogonal group $O(n)$ has non-flat projectives. 
Q1: Is there a classification of symmetric monoidal abelian categories $\mathcal{A}$  where all projectives are flat ? BTW: Do such categories have a particular name ?  
If a classification is too strong to establish, I would be interested in 
Q2: Are there criteria that ensure that projectives are flat ? 
An example for Q2 is, when $\mathcal{A}$ has direct sums and the unit is a progenerator of $\mathcal{A}$ (this is just the categorification of the module case mentioned above). Note that this criteria doesn't work for instance in the category of $RG$-modules ($R$  commutative ring, $G$ a group) where the tensor product is given by $M \otimes_R N$ with diagonal $G$-operation.

REPLY [4 votes]: This is the sort of question which would interest my advisor, Mark Hovey. In fact, he and Keir Lockridge wrote a paper called Semisimple Ring Spectra which addresses questions like this in triangulated categories (which cannot be abelian but make more sense to work with if you do homotopy theory). In this paper he has two results which answer your question in that context. I'll paraphrase them and highlight the bits that matter, but first some definitions.
A triangulated category come with an endofunctor $\Sigma$ which acts like suspension in topological spaces or shift in chain complexes (there are axioms for this). With this endofunctor you can define homotopy classes of maps and do cohomology theory. Call such a category $T$ a weak stable homotopy theory if every cohomology theory is representable. Suppose you have such a theory and it comes with a compact generator $S$ (i.e. the smallest localizing subcategory containing $S$ is $T$). Call such a theory a stable homotopy theory if there is also an axiom saying that $S$ plays nicely with the smash product. Call it a Brown category if all homology theories and natural transformations between them are representable. Assume below these hypotheses hold on $T$.
Proposition 2.1: The generator $S$ is semisimple iff projective objects and realizable objects coincide.
Remark 2.3: The generator $S$ is von Neumann regular iff flat objects and realizable objects coincide.
For the definitions of semisimple and von Neumann regular, see that paper. These are just the versions for triangulated categories of standard properties of rings. So semisimple implies von Neumann regular and there are classifications for when the converse holds. Furthermore, you can find a classification of semisimple objects in such theories $T$ as Theorem 4.1, which is a purely algebraic classification based on $\pi_*(S)$. So this gives some sense of a classification of theories where projectives and flats are the same. I don't know if anything like this has been done in the abelian setting. I'd be very interested to know!